Search Results for theory

Biographies

1. Bell John biography
   
   • John Bell's great achievement was that during the 1960s he was able to breathe new and exciting life into the foundations of quantum theory, a topic seemingly exhausted by the outcome of the Bohr-Einstein debate thirty years earlier, and ignored by virtually all those who used quantum theory in the intervening period.
     
   • Indeed, almost wholly due to Bell's pioneering efforts, the subject of quantum foundations, experimental as well as theoretical and conceptual, has became a focus of major interest for scientists from many countries, and has taught us much of fundamental importance, not just about quantum theory, but about the nature of the physical universe.
     
   • In addition, and this could scarcely have been predicted even as recently as the mid-1990s, several years after Bell's death, many of the concepts studied by Bell and those who developed his work have formed the basis of the new subject area of quantum information theory, which includes such topics as quantum computing and quantum cryptography.
     
   • Attention to quantum information theory has increased enormously over the last few years, and the subject seems certain to be one of the most important growth areas of science in the twenty-first century.
     
   • Bell was already thinking deeply about quantum theory, not just how to use it, but its conceptual meaning.
     
   • He would also have liked to study the conceptual basis of quantum theory more thoroughly.
     
   • Economic considerations, though, meant that he had to forget about quantum theory, at least for the moment, and get a job, and in 1949 he joined the UK Atomic Research Establishment at Harwell, though he soon moved to the accelerator design group at Malvern.
     
   • Through his career he gained much from discussions with Mary, and when, in 1987, his papers on quantum theory were collected [Speakable and Unspeakable in Quantum Mechanics (Cambridge, 1987).
     
   • During his time in Birmingham, Bell did work of great importance, producing his version of the celebrated CPT theorem of quantum field theory.
     
   • Bell published around 80 papers in the area of high-energy physics and quantum field theory.
     
   • The most important work was that of 1969 leading to the Adler-Bell-Jackiw (ABJ) anomaly in quantum field theory.
     
   • This work solved an outstanding problem in particle physics; theory appeared to predict that the neutral pion could not decay into two photons, but experimentally the decay took place, as explained by ABJ.
     
   • Reinhold Bertlmann, who himself did important work with Bell, has written a book titled Anomalies in Quantum Field Theory [Anomalies in Quantum Field Theory (Oxford, 2000).',10)">10], and the two surviving members of ABJ, Adler and Jackiw shared the 1988 Dirac Medal of the International Centre for Theoretical Physics in Trieste for their work.
     
   • While particle physics and quantum field theory was the work Bell was paid to do, and he made excellent contributions, his great love was for quantum theory, and it is for his work here that he will be remembered.
     
   • As we have seen, he was concerned about the fundamental meaning of the theory from the time he as an undergraduate, and many of his important arguments had their basis at that time.
     
   • A complete theory of hidden variables must actually be more complicated than this -- we must remember that we wish to predict the results of measuring not just sz, but also sx and sy, and any other component of s.
     
   • They were therefore pleased when John von Neumann proved a theorem claiming to show rigorously that it is impossible to add hidden variables to the structure of quantum theory.
     
   • This was his Copenhagen interpretation of quantum theory.
     
   • (3) Quantum theory is not exactly true in these rather special experiments.
     
   • He therefore concluded that if quantum theory was correct, so one ruled out possibility (3), then (2) must be true.
     
   • In Einstein's terms, quantum theory was not complete but needed to be supplemented by hidden variables.
     
   • He was delighted by the creation in 1952 by David Bohm of a version of quantum theory which included hidden variables, seemingly in defiance of von Neumann's result.
     
   • In 1964, Bell made his own great contributions to quantum theory.
     
   • Von Neumann had illegitimately extended to his putative hidden variables a result from the variables of quantum theory that the expectation value of A + B is equal to the sum of the expectation values of A and of B.
     
   • (The expectation value of a variable is the mean of the possible experimental results weighted by their probability of occurrence.) Once this mistake was realised, it was clear that hidden variables theories of quantum theory were possible.
     
   • Bell had showed rigorously that one could not have local realistic theories of quantum theory.
     
   • So it has been assumed that quantum theory is exactly true, but of course this can never be known.
     
   • In EPR-type experiments, this inequality is obeyed by local hidden variables, but may be violated by other theories, including quantum theory.
     
   • While at least one loophole still remains to be closed [in August 2002], it seems virtually certain that local realism is violated, and that quantum theory can predict the results of all the experiments.
     
   • For the rest of his life, Bell continued to criticise the usual theories of measurement in quantum theory.
     
   • Gradually it became at least a little more acceptable to question Bohr and von Neumann, and study of the meaning of quantum theory has become a respectable activity.
     
   • Since that date, the amount of interest in his work, and in its application to quantum information theory has been steadily increasing.
     

2. Burnside biography
   
   • His work quickly turned to the study of groups and from 1894 onwards he was to be occupied almost entirely with the study of group theory.
     
   • Burnside was elected a Fellow of the Royal Society in 1893 for his work on hydrodynamics and complex function theory.
     
   • This paper was the first of a series which Burnside described himself as follows (see for example [Pioneers of representation theory : Frobenius, Burnside, Schur, and Brauer (Providence, RI, 1999).',3)">3]):- .
     
   • His work on group theory quickly progressed and in 1897 he published The Theory of Groups of Finite Order, the first treatise on group theory in English.
     
   • The present treatise is intended to introduce to the reader the main outlines of the theory of groups of finite order apart from any applications.
     
   • This book was to have a major influence in the development of group theory.
     
   • It has been suggested to me that I should take advantage of the present occasion to give an account of the recent progress of the theory of groups of finite order.
     
   • any attempt on my part to give, on the present occasion, an account of the recent advance in the theory ..
     
   • It is undoubtedly the fact that the theory of groups of finite order has failed, so far, to arouse the interest of any but a very small number of English mathematicians; and this want of interest in England, as compared with the amount of attention devoted to the subject both on the Continent and in America, appears to me very remarkable.
     
   • Let us now examine some more of Burnside's contributions to group theory.
     
   • Frobenius started his development of the representation theory of groups and character theory in 1896.
     
   • Burnside quickly recognised the importance of Frobenius's methods and he began to use character theory.
     
   • Much of group theory today still moves in directions set by Burnside.
     
   • His famous 'Burnside Problem' on the finiteness of groups when the elements have fixed finite orders is still a major area of group theory research today.
     
   • If the first edition of The Theory of Groups of Finite Order was important, the second edition published in 1911 which contains a systematic development of the subject including Frobenius's character theory and Burnside's work using these methods, was a classic which is still widely read today.
     
   • Very considerable advances in the theory of groups of finite order have been made since the appearance of the first edition of this book.
     
   • In particular the theory of groups of linear substitutions has been the subject of numerous and important investigations by several writers; and the reason given in the original preface for omitting any account of it no longer holds good.
     
   • in fact it is now true to say that for further advances in the abstract theory one must look largely to the representation of a group as a group of linear substitutions.
     
   • During his life Burnside was to publish around 150 papers of which about 50 were on group theory.
     
   • In fact in the latter years of his life he turned to probability theory and his first paper on the subject appeared in 1918.
     
   • He left a complete manuscript of a book on probability which was published as The Theory of Probability in the year after his death.
     
   • Before he died he replied to Philip Hall who wrote to him asking for suggestions for the most profitable group theory problems to study.
     
   • Hall was to prove a very worthy successor to Burnside as the promoter of group theory in England.
     
   • W Burnside - Theory of Groups of Finite Order .
     
   • History Topics: The development of group theory .
     

3. Ito biography
   
   • It was during his student years that he became attracted to probability theory.
     
   • In [My Sixty Years in Studies of Probability Theory : acceptance speech of the Kyoto Prize in Basic Sciences (1998).',3)">3] he explains how this came about:- .
     
   • Although I knew that probability theory was a means of describing such phenomena, I was not satisfied with contemporary papers or works on probability theory, since they did not clearly define the random variable, the basic element of probability theory.
     
   • At that time, few mathematicians regarded probability theory as an authentic mathematical field, in the same strict sense that they regarded differential and integral calculus.
     
   • Accordingly, I was able to continue studying probability theory, by reading Kolmogorov's Basic Concept of Probability Theory and Levy's Theory of Sum of Independent Random Variables.
     
   • At that time, it was commonly believed that Levy's works were extremely difficult, since Levy, a pioneer in the new mathematical field, explained probability theory based on his intuition.
     
   • My first paper was thus developed; today, it is common practice for mathematicians to use my method to describe Levy's theory.
     
   • Ito began to reconstruct from scratch the concept of stochastic integrals, and its associated theory of analysis.
     
   • He created the theory of stochastic differential equations, which describe motion due to random events.
     
   • Ito, who still did not have a doctorate at this time, would have to wait several years before the importance of his ideas would be fully appreciated and mathematicians would begin to contribute to developing the theory.
     
   • Volume 20 of the Proceedings of the Imperial Academy of Tokyo contains six papers by Ito: (1) On the ergodicity of a certain stationary process; (2) A kinematic theory of turbulence; (3) On the normal stationary process with no hysteresis; (4) A screw line in Hilbert space and its application to the probability theory; (5) Stochastic integral; and (6) On Student's test.
     
   • In the following year he published his famous text Probability theory.
     
   • In this book, Ito develops the theory on a probability space using terms and tools from measure theory.
     
   • This book contained five chapters, the first providing an introduction, then the remaining ones studying processes with independent increments, stationary processes, Markov processes, and the theory of diffusion processes.
     
   • Ito gives a wonderful description mathematical beauty in [My Sixty Years in Studies of Probability Theory : acceptance speech of the Kyoto Prize in Basic Sciences (1998).',3)">3] which he then relates to the way in which he and other mathematicians have developed his fundamental ideas:- .
     
   • Music by Mozart, for instance, impresses greatly even those who do not know musical theory; the cathedral in Cologne overwhelms spectators even if they know nothing about Christianity.
     
   • Ito's theory is used in various fields, in addition to mathematics, for analysing phenomena due to random events.
     
   • Calculation using the "Ito calculus" is common not only to scientists in physics, population genetics, stochastic control theory, and other natural sciences, but also to mathematical finance in economics.
     
   • A recent monograph entitled Ito's Stochastic Calculus and Probability Theory (1996), dedicated to Ito on the occasion of his eightieth birthday, contains papers which deal with recent developments of Ito's ideas:- .
     
   • Professor Kiyosi Ito is well known as the creator of the modern theory of stochastic analysis.
     
   • Although Ito first proposed his theory, now known as Ito's stochastic analysis or Ito's stochastic calculus, about fifty years ago, its value in both pure and applied mathematics is becoming greater and greater.
     

4. Bauer biography
   
   • His main interests at this time were in measure and integration, and in the work submitted for his habilitation he studied an abstract Riemann integral, introduced by L H Loomis, from the point of view of the theory of Radon measure.
     
   • It was at this time that he became interested in potential theory and convexity theory, two areas to which he was to make major contributions over the rest of his career.
     
   • The first part is a standard development of measure theory, containing three chapters dealing with measure theory, integration theory, and product measure spaces in that order.
     
   • The second part of the book is devoted to probability theory.
     
   • Generally speaking, only probability theory as it pertains to product measure spaces is discussed.
     
   • An English version Probability Theory and Elements of Measure Theory was published in 1972.
     
   • Because of the great popularity the book enjoyed, an extensive reworking and expansion of the sections on probability appeared in English translation as Probability theory in 1996, with the same treatment was given to the sections of measure theory, published in English translation as Measure and integration theory in 2001.
     
   • This was Mass- und Integrationstheorie (1990) which provided an introduction to measure theory and the theory of integration.
     
   • the theory of multidimensional Lebesgue integration as a tool for handling integrals involved in problems of analysis and mathematical statistics (the gamma function, the Gauss distribution function, potential theory, the volume of the n-dimensional sphere, etc.).
     
   • His contribution ti the Proceedings starts as follows: "In 1964 Pierre Jacquinot opened a colloquium on potential theory in Orsay, France, by comparing potential theory with a road intersection in mathematics.
     
   • Meanwhile traffic has increased, and crossroads had to be converted into interchanges of highways - also in potential theory." The first part of the contribution addresses three aspects of classical potential theory: superharmonic functions, Newtonian kernel and potentials, Brownian motion.
     
   • The second part reflects two main sspects of potential theory of the early seventies: harmonic spaces and Markov processes.
     
   • The last part is devoted to Fuglede's theory of finely harmonic functions, including an application to asympototic paths for subharmonic functions.
     
   • Professor Bauer is involved in research in integration theory, functional analysis (convexity and approximation theory), potential theory, and Markov processes.
     

5. Bass biography
   
   • Hyman Bass writes [Algebra, K-theory, groups, and education, New York, 1997 (Contemp.
     
   • He [Algebra, K-theory, groups, and education, New York, 1997 (Contemp.
     
   • It was to completely change the direction of Bass's interests [Algebra, K-theory, groups, and education, New York, 1997 (Contemp.
     
   • Although he found Chicago very different from Princeton, nevertheless it was "again an environment that lived and breathed mathematics." [Algebra, K-theory, groups, and education, New York, 1997 (Contemp.
     
   • Bass writes in [Algebra, K-theory, groups, and education, New York, 1997 (Contemp.
     
   • Appointed as Ritt Instructor at Columbia [Algebra, K-theory, groups, and education, New York, 1997 (Contemp.
     
   • So I tried to learn what others were doing, attending many of the graduate courses: number theory and algebraic geometry from Lang, Lie groups and class field theory from Harish-Chandra, differential algebra from Kolchin, category theory from Eilenberg, and fiber bundles from Albrecht Dold.
     
   • The paper The homotopy theory of projective modules (1962) was written jointly with S Schanuel.
     
   • With this point of departure we have attempted to adapt some of the results and methods of homotopy theory to certain purely arithmetic and even noncommutative settings.
     
   • We have mentioned some areas of Bass's work above, but let us note that he himself gives his research interests as algebraic K-theory; number theory; group theory (geometric methods); and algebraic geometry.
     
   • To many people, Bass is best known for his classic text Algebraic K-theory published in 1968.
     
   • Algebraic K-theory flows from two sources.
     
   • The latter notion was applied by Atiyah and Hirzebruch in order to construct a new cohomology theory which has been enormously fruitful in topology.
     
   • The observation that these two ideas could be unified in a beautiful and powerful theory with widespread applications in algebra is due to the author, who is also responsible for a major portion of those applications.
     
   • In [ K-Theory 30 (3) (2003), 203-209.
     
   • These informal reminiscences, presented at the ICTP 2002 Conference on algebraic K-theory, recount the trajectory in the author's early research, from work on the Serre conjecture (on projective modules over polynomial algebras), via ideas from algebraic geometry and topology, to the ideas and constructions that eventually contributed to the founding of algebraic K-theory.
     
   • He has received many other honours and prizes, in addition to that for Algebraic K-theory, such as the Cole Prize in Algebra from the American Mathematical Society in 1975.
     
   • There he gave the course Topics in Algebraic K-theory which was published in Lecture notes, Tata Institute of Fundamental Research (Bombay, 1966).
     
   • His description of the relation between mathematicians and educators given in [Algebra, K-theory, groups, and education, New York, 1997 (Contemp.
     
   • Examples of such articles are Interweaving content and pedagogy in teaching and learning to teach: Knowing and using mathematics (2000), Making believe: The collective construction of public mathematical knowledge in the elementary classroom (2000), Making mathematics reasonable in school (2003), A practice-based theory of mathematical knowledge for teaching: The case of mathematical reasoning (2003), and Knowing mathematics for teaching (2003).
     

6. Weyl biography
   
   • At the end of my first year I went home with the "Zahlbericht" under my arm, and during the summer vacation I worked my way through it - without any previous knowledge of elementary number theory or Galois theory.
     
   • His habilitation thesis Uber gewohnliche Differentialgleicklungen mit Singularitaten und die zugehorigen Entwicklungen willkurlicher Funktionen investigated the spectral theory of singular Sturm-Liouville problems.
     
   • It united analysis, geometry and topology, making rigorous the geometric function theory developed by Riemann.
     
   • has undoubtedly had a greater influence on the development of geometric function theory than any other publication since Riemann's dissertation.
     
   • In his first academic year in this new post he was a colleague of Einstein who was at this time working out the details of the theory of general relativity.
     
   • It was an event which had a large influence on Weyl who quickly became fascinated by the mathematical principles lying behind the theory.
     
   • These later ideas included a gauge metric (the Weyl metric) which led to a gauge field theory.
     
   • Also over this period Weyl also made contributions on the uniform distribution of numbers modulo 1 which are fundamental in analytic number theory.
     
   • In particular his theory of representations of semisimple groups, developed during 1924-26, was very deep and considered by Weyl himself to be his greatest achievement.
     
   • The ideas behind this theory had already been introduced by Hurwitz and Schur, but it was Weyl with his general character formula which took them forward.
     
   • He was not the only mathematician developing this theory, however, for Cartan also produced work on this topic of outstanding importance.
     
   • He produced the first unified field theory for which the Maxwell electromagnetic field and the gravitational field appear as geometrical properties of space-time.
     
   • With his application of group theory to quantum mechanics he set up the modern subject.
     
   • It was his lecture course on group theory and quantum mechanics in Zurich in session 1927-28 which led to his third major text Gruppentheorie und Quantenmechanik published in 1928.
     
   • Wheeler's theory, like Weyl's, lacks the connection with quantum phenomena that is so important for interactions other than gravitation.
     
   • These include Elementary Theory of Invariants (1935), The classical groups (1939), Algebraic Theory of Numbers (1940), Philosophy of Mathematics and Natural Science (1949), Symmetry (1952), and The Concept of a Riemannian Surface (1955).
     
   • Is the occurrence in nature of one of the two enantiomorphous forms of an optically active substance characteristic of living matter? At what stage in the development of an embryo is the plane of symmetry determined? The second lecture contains a neat exposition of the theory of groups of transformations, with special emphasis on the group of similarities and its subgroups: the groups of congruent transformations, of motions, of translations, of rotations, and finally the symmetry group of any given figure.
     
   • [In the fourth lecture he] shows how the special theory of relativity is essentially the study of the inherent symmetry of the four-dimensional space-time continuum, where the symmetry operations are the Lorentz transformations; and how the symmetry operations of an atom, according to quantum mechanics, include the permutations of its peripheral electrons.
     
   • Turning from physics to mathematics, he gives an extraordinarily concise epitome of Galois theory, leading up to the statement of his guiding principle: "Whenever you have to do with a structure-endowed entity, try to determine its group of automorphisms".
     
   • Preface to H Weyl's Theory of groups and quantum mechanics - First Edition .
     
   • Preface to H Weyl's Theory of groups and quantum mechanics - Second Edition .
     
   • Introduction to H Weyl's theory of groups and quantum mechanics .
     

7. Chebyshev biography
   
   • Brashman was particularly interested in mechanics but his interests were wide ranging and, in addition to courses on mechanical engineering and hydraulics, he taught his students the theory of integration of algebraic functions and the calculus of probability.
     
   • Theory 96 (1) (1999), 111-138.',12)">12] the authors suggest that Chebyshev may have visited Paris in 1842 accompanying the Russian geographer Chikhachev who certainly met Catalan (who assisted Liouville in producing his journal) in December of that year.
     
   • The thesis was on the theory of probability, and in it he developed the main results of the theory in a rigorous but elementary way.
     
   • Between arriving in St Petersburg and this 1853 publication Chebyshev published some of his most famous results on number theory.
     
   • He wrote an important book Teoria sravneny on the theory of congruences which he submitted for his doctorate, defending it on 27 May 1849.
     
   • He collaborated with Bunyakovsky in producing a complete edition of Euler's 99 number theory papers which they published in two volumes in 1849.
     
   • [I] found an occasion each day to talk with this geometer concerning [applications of calculus to number theory] as well as other questions on pure and applied analysis.
     
   • In fact Chebyshev's interest both in the theory of mechanisms and in the theory of approximation stem from his 1852 trip.
     
   • In [Topics in polynomials of one and several variables and their applications (River Edge, NJ, 1993), 495-512.',31)">31] Tikhomirov studied Chebyshev's work on approximation theory and writes:- .
     
   • set the foundations of the Russian school of approximation theory: we show the relation of Chebyshev's ideas in approximation theory to applied problems (theory of mechanisms and computational mathematics).
     
   • It was in this work that his famous Chebyshev polynomials appeared for the first time but he later went on to develop a general theory of orthogonal polynomials.
     
   • Laplace had found and studied the Hermite polynomials in the course of his discoveries in probability theory during the early nineteenth century.
     
   • It was Chebyshev who saw the possibility of a general theory and its applications.
     
   • His work arose out of the theory of least squares approximation and probability; he applied his results to interpolation, approximate quadrature and other areas.
     
   • Theory 96 (1) (1999), 111-138.',12)">12]).
     
   • We have mentioned some contributions that Chebyshev made to the theory of probability.
     
   • Twenty years later Chebyshev published On two theorems concerning probability which gives the basis for applying the theory of probability to statistical data, generalising the central limit theorem of de Moivre and Laplace.
     
   • In the theory of integrals he generalised the beta function and examined integrals of the form .
     
   • The closer mutual approximation of the points of view of theory and practice brings most beneficial results, and it is not exclusively the practical side that gains; under its influence the sciences are developing in that this approximation delivers new objects of study or new aspects in subjects long familiar.
     
   • And if theory gains much when new applications or developments of old methods occur, the gain is still greater when new methods are discovered; and here science finds a reliable guide in practice.
     

8. Von Neumann biography
   
   • Von Neumann received his doctorate in mathematics from the University of Budapest, also in 1926, with a thesis on set theory.
     
   • Veblen invited von Neumann to Princeton to lecture on quantum theory in 1929.
     
   • In his youthful work, he was concerned not only with mathematical logic and the axiomatics of set theory, but, simultaneously, with the substance of set theory itself, obtaining interesting results in measure theory and the theory of real variables.
     
   • It was in this period also that he began his classical work on quantum theory, the mathematical foundation of the theory of measurement in quantum theory and the new statistical mechanics.
     
   • As a result, the mathematical framework of the theory was developed and the formal aspects of its entirely novel rules of interpretation were analysed by one single man in two years (1927-1929).
     
   • His interest in ergodic theory, group representations and quantum mechanics contributed significantly to von Neumann's realisation that a theory of operator algebras was the next important stage in the development of this area of mathematics.
     
   • 64 (1958), 1-49.',35)">35] how he was led towards game theory:- .
     
   • ideas which culminated later in one of his most original creations, the theory of games.
     
   • In game theory von Neumann proved the minimax theorem.
     
   • He gradually expanded his work in game theory, and with co-author Oskar Morgenstern, he wrote the classic text Theory of Games and Economic Behaviour (1944).
     
   • His work, from the beginnings of the Second World War, concerns a study of the equations of hydrodynamics and the theory of shocks.
     
   • Von Neumann spent a considerable part of the last few years of his life working in [automata theory].
     
   • It represented for him a synthesis of his early interest in logic and proof theory and his later work, during World War II and after, on large scale electronic computers.
     
   • Involving a mixture of pure and applied mathematics as well as other sciences, automata theory was an ideal field for von Neumann's wide-ranging intellect.
     
   • He advanced the theory of cellular automata, advocated the adoption of the bit as a measurement of computer memory, and solved problems in obtaining reliable answers from unreliable computer components.
     
   • History Topics: The beginnings of set theory .
     
   • Paul Walker (A history of Game Theory) .
     

9. Linnik biography
   
   • He worked in Leningrad for the rest of his life organising the chair of probability theory there and founding the world famous Leningrad school of probability and mathematical statistics.
     
   • His main research topics were number theory, probability theory and mathematical statistics.
     
   • He introduced ergodic methods into number theory in his first work on the analytic theory of quadratic forms.
     
   • In a 1941 paper he introduced the large sieve method in number theory.
     
   • After his early concentration on number theory, from 1947 onwards Linnik embarked on a deep study of probability.
     
   • From that time on he undertook research in three areas, namely probability, mathematical statistics and the analytic theory of numbers.
     
   • In 1950 he introduced the concepts of probability into number theory and introduced the dispersion method in number theory.
     
   • He devised the dispersion method to attack additive problems in number theory of binary type.
     
   • for treating certain types of problems in additive number theory which have previously resisted all attacks.
     
   • In 1948-49 Linnik obtained results which contained, in principle, a complete solution to two central problems in the theory of the summation of variables forming a Markov chain.
     
   • One of these, raised by Markov, the creator of the theory of chains, was: to find the conditions for the application of the integral limit theorem to the case of a singular chain.
     
   • An important feature of the method used in this paper, which was largely responsible for its success, is the use of arguments from the study of trigonometric sums in the theory of numbers.
     
   • The use of ergodic methods in metrical number theory is well known; part of the latter theory is essentially a special case of general ergodic theorems.
     
   • These theorems allow one to obtain asymptotic expressions for the distribution of lattice points on these varieties and to arrive at results which are not accessible by the usual methods of analytical number theory.
     
   • These lectures concentrated on applications of the theory of functions of one and several complex variables to the theory of similar tests and unbiased estimation.
     
   • Many problems of statistical theory have the basic property that some special distributions have important properties that allow a reduction of the initial problem to a simpler one.
     
   • This would lead step-by-step to the creation of a theory and of general methods.
     
   • Two volumes of his works in number theory have been published (1979-1980), the first subtitled The ergodic method and L-functions and the second L-functions and the dispersion method.
     
   • A volume has also been published of his work on Probability theory (1981) and on Mathematical statistics (1982).
     

10. Doob biography
    
    • His first student was Paul Halmos who completed his doctorate with the thesis Invariants of certain stochastic transformation: The mathematical theory of gambling systems in 1938.
      
    • Doob's work was in probability and measure theory, in particular he studied the relations between probability and potential theory.
      
    • Monthly 105 (1) (1998), 28-35.',1)">1] looks at many of the areas of probability to which Doob made major contributions such as separability, stochastic processes, martingales, optimal stopping, potential theory, and classical potential theory and its probabilistic counterpart.
      
    • Doob built on work by Paul Levy and, during the 1940's and 1950's, he developed basic martingale theory and many of its applications.
      
    • In 1953 he published a book which gives a comprehensive treatment of stochastic processes, including much of his own development of martingale theory.
      
    • probability theory is simply a branch of measure theory, with its own special emphasis and field of application ..
      
    • Another classic text by Doob is Classical potential theory and its probabilistic counterpart first published in 1984 and reprinted in 2001.
      
    • His interest in potential theory went back to 1955 when he was invited to speak at the Berkeley Symposium on Probability and Statistics.
      
    • He decided to speak on Axiomatic Potential Theory and from then on he undertook research on the subject.
      
    • He corresponded with Brelot, a leading expert on the topic, and many years later Brelot said that he wanted to write a text on modern potential theory.
      
    • He asked Doob to cooperate with him in writing the sections on probability theory but in the end Doob wrote the whole book.
      
    • This is the long-awaited book by the author, developing in parallel potential theory and part of the theory of stochastic processes.
      
    • The first half concerns the potential theory of the Laplace operator (i.e.
      
    • classical potential theory) and of the heat operator and its adjoint (i.e.
      
    • parabolic potential theory), while the second half treats the probabilistic counterparts (interpreted liberally) to the objects in the first half.
      
    • Doob is also the author of a well known book on measure theory published in 1994 when he was 84 years old.
      
    • what measure theory every would-be analyst should learn.
      
    • this text, written by one of the most illustrious probabilists alive, is an interesting addition to the textbook literature in measure theory; every serious mathematical library should acquire it and teachers of measure theory - especially those who are analysts by profession - should not fail to consult it for their future courses.
      

11. Frobenius biography
    
    • The positive side of his appointment was undoubtedly his remarkable contributions to the representation theory of groups, in particular his development of character theory, and his position as one of the leading mathematicians of his day.
      
    • In a letter to Hurwitz, who was a product of the Gottingen system, he wrote on 3 February 1896 (see [Pioneers of representation theory : Frobenius, Burnside, Schur, and Brauer (Providence, Rhode Island, 1999).',4)">4]):- .
      
    • Fairly extensive quotes from this document, and another similar document from Fuchs and Helmholtz, are given in [Pioneers of representation theory : Frobenius, Burnside, Schur, and Brauer (Providence, Rhode Island, 1999).',4)">4] but we quote a short extract to show the power, variety and high quality of Frobenius's work in his Zurich years.
      
    • The theory of linear differential equations.
      
    • The theory of elliptic and Jacobi functions..
      
    • The theory of biquadratic forms.
      
    • On the theory of surfaces with a differential parameter.
      
    • In his work in group theory, Frobenius combined results from the theory of algebraic equations, geometry, and number theory, which led him to the study of abstract groups.
      
    • In that year he published five papers on group theory and one of them Uber die Gruppencharactere on group characters is of fundamental importance.
      
    • I shall develop the concept [of character for arbitrary finite groups] here in the belief that through its introduction, group theory will be substantially enriched.
      
    • Hence 1897 is the year in which the representation theory of groups was born.
      
    • It was a burst of activity which set up the foundations of the whole of the machinery of representation theory.
      
    • He continued his applications of character theory in papers of 1900 and 1901 which studied the structure of Frobenius groups.
      
    • Frobenius's character theory was used with great effect by Burnside and was beautifully written up in Burnside's 1911 edition of his Theory of Groups of Finite Order.
      
    • Frobenius collaborated with Schur in representation theory of groups and character theory of groups.
      
    • Frobenius's representation theory for finite groups was later to find important applications in quantum mechanics and theoretical physics which may not have entirely pleased the man who had such "pure" views about mathematics.
      
    • He introduced the concept of irreducibility for matrices and the papers which he wrote containing this theory around 1910 remain today the fundamental results in the discipline.
      
    • History Topics: The development of group theory .
      

12. Tutte biography
    
    • Finding a copy of Rouse Ball's book Mathematical Recreations and Essays in the library while he was at this school he began to acquire a fascination with the graph theory problems he read about in this book but it was not enough to change his mind about studying chemistry at university.
      
    • In the following year he gave the lecture "FISH and I", which is reproduced in [Coding theory and cryptography, Annapolis, MD, 1998 (Berlin, 2000), 9-17.',8)">8], giving a fascinating account.
      
    • He worked on algebra and graph theory, combining the two to produce his first outstanding contribution to matroid theory.
      
    • Tutte was soon publishing papers on many different aspects of graph theory.
      
    • In 1946 he published On Hamiltonian circuits, and in the following year the two papers A family of cubical graphs and A ring in graph theory.
      
    • In the same year he published a paper on perhaps the most famous of all graph theory problems On the four-colour conjecture.
      
    • Among his books are: Connectivity in graphs published in 1966; Introduction to the theory of matroids (1971), based on a series of lectures given by Tutte at the Rand Corporation in 1965; Graph Theory (1984); and Graph Theory as I Have Known It (1998) which gives a fascinating account of how he discovered his many fundamental results.
      
    • The pace at which graph theory developed was quite remarkable so that when Tutte wrote Connectivity in graphs in 1966 he stated in the preface:- .
      
    • Graph theory is now too extensive a subject for adequate presentation in a book of this size.
      
    • Faced with the alternatives of writing a shallow survey of the greater part of graph theory or of giving a reasonably deep account of a small part, I have chosen the latter.
      
    • In fact he chose to cover results from the general theory of undirected graphs such as Euler paths, the symmetry of graphs, the girth, and results on non-separability and triple connection.
      
    • In 1984 Tutte published Graph Theory which contains a foreword written by C St J A Nash-Williams:- .
      
    • It is both fitting and fortunate that the volume on graph theory in the Encyclopedia of Mathematics and its Applications has an author whose contributions to graph theory are - in the opinion of many - unequalled.
      
    • [T]his is by no means 'just another book on graph theory', since the treatment of [many of the central themes of graph theory] is unified into a coherent whole by Professor Tutte's highly individual approach.
      
    • Moreover, the more customary topics are leavened with some 'pleasant surprises', such as the author's attractive theory of decomposition of graphs into 3-connected '3-blocks', an interesting and remarkable approach to electrical networks, and - perhaps particularly - the classification theorem for closed surfaces.
      
    • In graph theory he established fundamental results for matching, connectivity, symmetry in graphs, reconstruction, colouring, Hamiltonian circuits, graphs on higher surfaces, graph enumeration and graph polynomials.
      
    • In matroid theory, he is the single most important pioneer.
      
    • In his quiet way he enjoyed the recognition that accompanied the growth in popularity and status of graph theory, the subject he had built.
      

13. Clifford Alfred biography
    
    • In fact it turned out to be a remarkably fruitful start for the structure theory of semigroups.
      
    • In this work he considered the arithmetic and ideal theory of abstract multiplication.
      
    • He later extended the work of his thesis and published the two papers Arithmetic and ideal theory of abstract multiplication (1934) and Arithmetic and ideal theory of commutative semigroups (1938).
      
    • Rhodes writes [Semigroup theory and its applications, New Orleans, LA, 1994 (Cambridge Univ.
      
    • as an assistant professor, a paper famous in the semigroup community about union of groups semigroups, that Clifford learned about Rees' Theorem determining the structure of completely 0-simple semigroups, generalizing the Wedderburn theory of rings.
      
    • Rees' theory generalized much earlier results of Suschkiewitsch unknown to Clifford until 1941.
      
    • Rhodes writes [Semigroup theory and its applications, New Orleans, LA, 1994 (Cambridge Univ.
      
    • Preston writes [Semigroup theory and its applications, New Orleans, LA, 1994 (Cambridge Univ.
      
    • The first volume of Clifford and Preston, The algebraic theory of semigroups, was published in 1961.
      
    • Before, there has been no systematic treatment on semigroups at all, with the exception of the book of Suchkewitsch, 'The theory of generalized groups' (1937) containing naturally a very limited number of results.
      
    • Volume II of Clifford and Preston, The algebraic theory of semigroups was published in 1967.
      
    • Volume II has been eagerly awaited by those who are working in semigroup theory and related subjects (e.g., automata theory).
      
    • deals with additional branches of the theory to which there was at most passing reference in Volume I.
      
    • These two volumes have had an enormous impact on the development of semigroup theory.
      
    • Clear in exposition, broad and deep in its coverage of the field, the book has had, and continues to have, a profound influence on the development of the theory of semigroups.
      
    • Rhodes describes Clifford's research following the publication of the second volume of 'Clifford and Preston' in [Semigroup theory and its applications, New Orleans, LA, 1994 (Cambridge Univ.
      
    • He has produced significant results in group theory and semigroup theory.
      
    • Many of his results are of fundamental importance and, especially in semigroup theory, he initiated many techniques and approaches that are peculiar to, and are now part and parcel of, the theory.
      

14. Kurosh biography
    
    • I attended his lectures on the theory of sets, the theory of functions, and topology.
      
    • However, although Kurosh's first results were in topology, solving problems posed by Aleksandrov, he was already interested in the theory of groups.
      
    • He had read O Yu Schmidt's group theory papers while still in Smolensk so when he found himself able to attend Schmidt's seminar at Moscow State University, his interest in groups increased further.
      
    • However, after attending Schmidt's group theory course in 1930, he found himself taking over some of Schmidt's duties when he left the university in the autumn of that year.
      
    • Quickly he moved to research in group theory and his first paper on this topic appeared in 1932 on direct decompositions of groups.
      
    • Kurosh is best known for his book The Theory of Groups which was written in two volumes.
      
    • has been widely acclaimed as the first modern text on the general theory of groups, with major emphasis on infinite groups.
      
    • However, Kurosh did not spend all his research efforts on group theory.
      
    • Gradually, along with papers on group theory, Kurosh began to publish papers on ring theory, linear algebra and lattices; later, also papers on category theory and the theory of multi-operator groups, rings and linear algebras.
      
    • As with The Theory of Groups this text was also translated into English by Hirsch.
      
    • During the 1950s he concentrated on Universal algebra and category theory, organising a major seminar on category theory in 1958.
      
    • Many mathematicians participated in this seminar and it led to the birth of the Moscow School of Category Theory.
      
    • between the theory of universal algebras and the classical branches of general algebra there exists a big uncultivated space.
      
    • In that year he gave his first lecture to the Society on Fundamental trends in finite group theory.
      
    • Kurosh's book The theory of groups 2nd edition .
      
    • Kurosh's book The theory of groups 1st edition .
      

15. Cantor biography
    
    • He spent the summer term of 1866 at the University of Gottingen, returning to Berlin to complete his dissertation on number theory De aequationibus secundi gradus indeterminatis in 1867.
      
    • During this time he worked on his habilitation and, immediately after being appointed to Halle in 1869, he presented his thesis, again on number theory, and received his habilitation.
      
    • At Halle the direction of Cantor's research turned away from number theory and towards analysis.
      
    • Between 1879 and 1884 Cantor published a series of six papers in Mathematische Annalen designed to provide a basic introduction to set theory.
      
    • Firstly Cantor realised that his theory of sets was not finding the acceptance that he had hoped and the Grundlagen was designed to reply to the criticisms.
      
    • The correspondence between Mittag-Leffler and Cantor all but stopped shortly after this event and the flood of new ideas which had led to Cantor's rapid development of set theory over about 12 years seems to have almost stopped.
      
    • He turned from the mathematical development of set theory towards two new directions, firstly discussing the philosophical aspects of his theory with many philosophers (he published these letters in 1888) and secondly taking over after Clebsch's death his idea of founding the Deutsche Mathematiker-Vereinigung which he achieved in 1890.
      
    • His last major papers on set theory appeared in 1895 and 1897, again in Mathematische Annalen under Klein's editorship, and are fine surveys of transfinite arithmetic.
      
    • However, it was not to be, but the second paper describes his theory of well-ordered sets and ordinal numbers.
      
    • In their lectures at the Congress [A history of set theory (Boston, Mass., 1972).',4)">4]:- .
      
    • Hurwitz openly expressed his great admiration of Cantor and proclaimed him as one by whom the theory of functions has been enriched.
      
    • Jacques Hadamard expressed his opinion that the notions of the theory of sets were known and indispensable instruments.
      
    • By the time of the Congress, however, Cantor had discovered the first of the paradoxes in the theory of sets.
      
    • For example in his illness of 1884 he had requested that he be allowed to lecture on philosophy instead of mathematics and he had begun his intense study of Elizabethan literature in attempting to prove his Bacon-Shakespeare theory.
      
    • He did continue to work and publish on his Bacon-Shakespeare theory and certainly did not give up mathematics completely.
      
    • He lectured on the paradoxes of set theory to a meeting of the Deutsche Mathematiker-Vereinigung in September 1903 and he attended the International Congress of Mathematicians at Heidelberg in August 1904.
      
    • He also corresponded with Jourdain on the history of set theory and his religious tract.
      
    • Extract from Cantor's Uber einen die trigonometrischen Reihen betreffenden Lehrsatz which is one of his first publications on the theory of functions.
      
    • History Topics: The beginnings of set theory .
      

16. Siegel biography
    
    • Initially his intention had been to study astronomy, but Frobenius's influence took him towards number theory which would became the main research topic of his career.
      
    • Number Theory 20 (3) (1985), 373-404.
      
    • Geometry of numbers and its applications to algebraic number theory.
      
    • Quadratic forms: analytic theory and modular forms.
      
    • Siegel is especially famed for his work on the theory of numbers where he held an eminent role.
      
    • Schneider, who was a student of Siegel's, gave three lectures on Siegel's contributions to number theory to the German Mathematical Union in 1982.
      
    • 85 (4) (1983), 147-157.',13)">13] and describe Siegel's most important results in number theory.
      
    • In the 1929 paper Siegel made a substantial contribution to transcendence theory, especially a new method for the algebraic independence of values of certain E-functions.
      
    • Siegel's research on the analytic theory of quadratic forms in 1935/37 was of fundamental importance and he broke new ground in considering quadratic forms in which the coefficients were from an algebraic number field.
      
    • In this general area Siegel considered the theory of discontinuous groups and their fundamental domains, algebraic relations between modular functions and between modular forms, and Fourier series of modular forms.
      
    • Siegel's work in celestial mechanics, which came next to number theory in his list of favourite topics, is discussed by Russmann in [Jahresber.
      
    • Siegel gave a much improved version of lunar theory as developed by Hill.
      
    • He examined Birkhoff's work on perturbation theory solutions for analytical Hamiltonian differential equations near an equilibrium point using formal power series.
      
    • contributions to stability theory.
      
    • The whole style of the author contradicts the sense for simplicity and honesty which we admire in the works of the masters in number theory - Lagrange, Gauss, or on a smaller scale, Hardy, Landau.
      
    • Unfortunately there are many "fellow-travellers" who have already disgraced a large part of algebra and function theory; however, until now, number theory had not been touched.
      
    • I am afraid that mathematics will perish before the end of this century if the present trend for senseless abstraction - as I call it: theory of the empty set - cannot be blocked up.
      
    • Siegel enjoyed teaching, however, even elementary courses, and he published textbooks on the theory of numbers, celestial mechanics, and the theory of functions of several complex variables.
      

17. Karlin biography
    
    • From about 1950 his research interests changed and he began to work on game theory.
      
    • This interest had arisen through his years at Princeton when he had been influenced by John von Neumann who published his classic text Theory of Games and Economic Behaviour, written with Oskar Morgenstern, at about the time Karlin arrived at Princeton.
      
    • His interest in game theory continued and he worked for the RAND Corporation, in addition to lecturing at Stanford.
      
    • applied game theory to the analysis of games of pursuit and evasion like a dogfight between warplanes.
      
    • In 1958 he published the book Studies in the mathematical theory of inventory and production which was written jointly with Kenneth Arrow and Herbert Scarf.
      
    • In 1959 he published the book Mathematical methods and theory in games, programming and economics.
      
    • This was a major two volume work, the first volume being Matrix games, programming, and mathematical economics while the second volume was The theory of infinite games.
      
    • These ideas play a basic role in problems involving convexity, moment spaces, orthogonal polynomials, Chebyshev systems, the oscillation properties of linear differential equations, and the theory of approximation.
      
    • They figure in an essential and elegant way in the theory of stochastic processes, especially in linear diffusion processes.
      
    • Finally, they are important in various applications - in statistics where they are fundamental to the understanding of statistical decision procedures, and also in such topics as inventory control and reliability theory.
      
    • Karlin received many honours throughout his career including a lifetime achievement award from the National Academy of Sciences in 1973, the John von Neumann Theory Prize in 1987, and the National Medal of Science in 1989.
      
    • The citation for the John von Neumann Theory Prize reads:- .
      
    • Samuel Karlin has been awarded the ORSA/TIMS John von Neumann Theory Prize for 1987 for his contributions to the theory of games, inventory theory, decision theory, birth-death and diffusion processes, total positivity and the theory of approximations.
      
    • Over the years, his vitality, scholarship and industry have generated more than fifty research students with spin-off in such diverse areas as reliability theory and queuing theory.
      
    • As a tribute to this breadth and vitality, we award Samuel Karlin the John von Neumann Theory Prize.
      

18. Ore biography
    
    • He then worked on non-commutative ring theory and proved his celebrated embedding theorem for a non-commutative integral domain into a division ring.
      
    • He then turned his attention to lattice theory and, together with Garrett Birkhoff, led the increasing activity in lattice theory throughout the 1930s.
      
    • Ore's work on lattices led him to the study of equivalence relations, closure relations and Galois connections, and then to the study of graph theory which occupied him to the end of his life.
      
    • These include Les Corps Algebraique et la Theorie des Ideaux (1934), L'Algebre Abstraite (1936), Number Theory and its History (1948), Theory of Graphs (1962), Graphs and Their Uses (1963), The Four-Color Problem (1967), and Invitation to Number Theory (1969).
      
    • In Number Theory and its History Ore states that his aim is to present:- .
      
    • the results of the theory integrated more fully in the historical and cultural framework [than is usual].
      
    • In its own sphere, the book gives a very readable account of the history of (classical) number theory with much serious mathematical thought.
      
    • The Theory of Graphs was published by the American Mathematical Society.
      
    • The present book has grown out of courses on graph theory given from time to time at Yale University.
      
    • The present century has witnessed a steady development of graph theory which in the last ten to twenty years has blossomed out into a new period of intense activity.
      
    • Clearly discernible in this process are the effects of the demands from new fields of application: game theory and programming, communications theory, electrical networks and switching circuits as well as problems from biology and psychology.
      
    • The second volume will be devoted to more special topics: planar graphs, the four-color conjecture, the theory of flow, games, electrical networks, as well as applications to a number of other fields in which graph theory is a principal tool.
      
    • an interesting experiment - a book on graph theory for high school students.
      
    • It would not be hard to present the history of graph theory as an account of the struggle to prove the four colour conjecture, or at least to find out why the problem is difficult.
      
    • Combinatorial Theory 8 (1970), i-iii.',2)">2] Ore's interests outside mathematics are described:- .
      

19. Artin biography
    
    • His thesis concerned applying the methods of the theory of quadratic number fields to quadratic extensions of a field of rational functions of one variable taken over a finite prime field of constants.
      
    • He made a major contribution to field theory, the theory of braids and, around 1928, he worked on rings with the minimum condition on right ideals, now called Artinian rings.
      
    • Field theory had been created by Steinitz in 1910.
      
    • By this stage he had proved, using very clever arguments with Galois theory and Cauchy's theorem on subgroups of prime order, that O had to be an extension of K of degree 2 and that the subfield K had to have the property that -1 could not be expressed as a sum of squares.
      
    • It is also worth noting that the theory of real-closed fields directly influenced Abraham Robinson in his contributions to model theory, particularly for the concepts of model completeness and model completion, see for example [Arch.
      
    • In 1920 Takagi published his fundamental paper on class field theory in which he built the theory around a remarkable fact which he had discovered, namely that the set of class fields, as defined by Heinrich Weber, over a fixed ground field k is identical to the set of abelian extension fields over k.
      
    • The theorems of Artin's 1927 paper have became central results in abelian class field theory.
      
    • The situation is similar to that with Galois Theory which, today, is formulated in the framework of abstract algebra, and in this form opens new applications and generalizations.
      
    • Similarly, Artin's Reciprocity Law opens the way to new applications and progress.The most striking application was given by Furtwangler's proof of the principal ideal theorem of class field theory, given one year after the publication of Artin's Reciprocity Law.
      
    • Another important piece of work done by Artin during his first period in Hamburg was the theory of braids which he presented in 1925.
      
    • He again showed his originality by introducing this new area of research which today is being studied by an increasing number of mathematicians working in group theory, semigroup theory, and topology.
      
    • Again, Artin's conjecture triggered a lot of interesting activities in number theory.
      
    • Among Artin's main books are Galois theory (1942), Rings with minimum condition (1948) written jointly was C J Nesbitt and R M Thrall, Geometric algebra (1957) and Class field theory (1961) written with J T Tate.
      
    • Artin was honoured by the award of the American Mathematical Society's Cole Prize in number theory.
      
    • History Topics: The development of Ring Theory .
      

20. Thomson biography
    
    • At the end of session 1839-40 Thomson read Fourier's The Analytical Theory of Heat a work on the application of abstract mathematics to heat flow.
      
    • Not only did his professors put him in touch with much modern experimental and mathematical research, but they also articulated the ideal of mathematising physical theory, even though none of them was himself a master of that craft.
      
    • A more important paper On the uniform motion of heat and its connection with the mathematical theory of electricity was published in 1842 while Thomson was studying for the mathematical tripos examinations at Cambridge.
      
    • It was at Liouville's request that Thomson began to try to bring together the ideas of Faraday, Coulomb and Poisson on electrical theory.
      
    • Thomson was led to study the whole methodology of a physical science, distinguishing 'physical' parts of a theory from 'mathematical' parts.
      
    • In 1847-49 he collaborated with Stokes on hydrodynamical studies, which Thomson applied to electrical and atomic theory.
      
    • Many of these letters discuss the mathematical similarities in the theory of heat and the theory of fluids.
      
    • The absolute scale that he proposed was based on his studies of the theory of heat, in particular the theory proposed by Sadi Carnot and later developed by Clapeyron.
      
    • Thomson published between 1849 and 1852 three influential papers on the theory of heat.
      
    • Thomson came to believe in a dynamical theory of heat and, in 1872, he wrote about how his views were led towards that approach (see for example [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]):- .
      
    • At the 1847 meeting of the British Association in Oxford, I learned from Joule the dynamical theory of heat, and was forced to abandon at once many, and gradually from year to year all other, statical preconceptions regarding the ultimate causes of apparently statical phenomena.
      
    • The dynamical theory of heat led Thomson to also think of a dynamical theory for electricity and magnetism.
      
    • This work by Thomson in 1856 on electricity and magnetism is important for it was these ideas which led Maxwell to develop his remarkable new theory of electromagnetism.
      
    • One might think that Thomson would have eagerly supported Maxwell's theory which his own work had helped to create, but this was not so.
      
    • Thomson had ideas of his own which he hoped would lead to a unifying theory, and his ideas took him further and further from accepting those of Maxwell.
      
    • However, Thomson's initial hope that his theory could explain electromagnetism, light, gravity, and chemical processes slowly faded.
      
    • We wish to stress Thomson's often under-estimated merits in the theory of the electromagnetic field.
      

21. Schrodinger biography
    
    • In theoretical physics he studied analytical mechanics, applications of partial differential equations to dynamics, eigenvalue problems, Maxwell's equations and electromagnetic theory, optics, thermodynamics, and statistical mechanics.
      
    • In mathematics he was taught calculus and algebra by Franz Mertens, function theory, differential equations and mathematical statistics by Wilhelm Wirtinger (whom he found uninspiring as a lecturer).
      
    • He was able to continue research and it was at this time that he published his first results on quantum theory.
      
    • From 1918 to 1920 he made substantial contributions to the theory of colour vision.
      
    • He had also made important contributions to the kinetic theory of solids, studying the dynamics of crystal lattices.
      
    • During these years of changing from one place to another, Schrodinger studied physiological optics, in particular he continued his work on the theory of colour vision.
      
    • I have been intensely concerned these days with Louis de Broglie's ingenious theory.
      
    • Schrodinger published his revolutionary work relating to wave mechanics and the general theory of relativity in a series of six papers in 1926.
      
    • Wave mechanics, as proposed by Schrodinger in these papers, was the second formulation of quantum theory, the first being matrix mechanics due to Heisenberg.
      
    • I am simply fascinated by your [wave equation] theory and the wonderful new viewpoint it brings.
      
    • There he studied electromagnetic theory and relativity and began to publish on a unified field theory.
      
    • Schrodinger was so entranced by his new theory that he threw caution to the winds, abandoned any pretence of critical analysis, and even though his new theory was scarcely hatched, he presented it to the Academy and to the Irish press as an epoch-making advance.
      
    • Now the Einstein Theory becomes simply a special case..
      
    • Einstein, however, realised immediately that there was nothing of merit in Schrodinger's 'new theory' [Schrodinger : Life and Thought (New York, 1989).',8)">8]:- .
      
    • Einstein wrote immediately saying that he was breaking off their correspondence on unified field theory.
      
    • Unified field theory was, however, not the only topic to interest him during his time at the Institute for Advanced Study in Dublin.
      
    • During his last few years Schrodinger remained interested in mathematical physics and continued to work on general relativity, unified field theory and meson physics.
      

22. Hopf Eberhard biography
    
    • Eberhard Hopf, an Austrian mathematician who made significant contributions in topology and ergodic theory, was born in Salzburg.
      
    • While in the Harvard College Observatory he worked on many mathematical and astronomical subjects including topology and ergodic theory.
      
    • In particular he studied the theory of measure and invariant integrals in ergodic theory and his paper On time average theorem in dynamics which appeared in the Proceedings of the National Academy of Sciences is considered by many as the first readable paper in modern ergodic theory.
      
    • While at MIT, Hopf did much of his work on ergodic theory which he published in papers such as Complete Transitivity and the Ergodic Principle (1932), Proof of Gibbs Hypothesis on Statistical Equilibrium (1932) and On Causality, Statistics and Probability (1934).
      
    • Using these concepts Hopf was able to give a unified presentation of many results in ergodic theory that he and others had found since 1931.
      
    • In that book containing only 81 pages, Hopf made a precise and elegant summary of ergodic theory.
      
    • Coming out of this lecture was a paper Ergodic theory and the geodesic flow on surfaces of constant negative curvature which he published in the Bulletin of the American Mathematical Society.
      
    • Famous investigations on the theory of surfaces of constant negative curvature have been carried out around the turn of the century by F Klein and H Poincare in connection with complex function theory.
      
    • The theory of the geodesics in the large on such surfaces was developed later in the famous memoirs by P Koebe.
      
    • This theory is purely topological.
      
    • The measure-theoretical point of view became dominant in the later thirties after the advent of ergodic theory, and the papers of G A Hedlund and E Hopf on the ergodic character of the geodesic flow came into being.
      
    • As a result most of his work to ergodic theory and topology was neglected or even attributed to others in the years following the end of World War II.
      
    • His interests and principal achievements were in the fields of partial and ordinarydifferential equations, calculus of variations, ergodic theory, topological dynamics, integral equations, differential geometry, complex function theory and functional analysis.
      
    • Hopf's work is also of the greatest importance to the hydrodynamics, theory of turbulence and radiative transfer theory.
      

23. Rademacher biography
    
    • His initial mathematical interests were in the theory of real functions which he was taught by Caratheodory who also taught him the calculus of variations.
      
    • At Gottingen he also studied number theory with Landau but the outbreak of World War I in 1914 meant that Rademacher had to undertake research while serving in the German army, which he did from 1914 to 1916.
      
    • Continuing his interest in the theory of real functions he completed his doctorate in 1916, written under Caratheodory's supervision, and submitted a dissertation on single-valued mappings and mensurability.
      
    • It examined some delicate problems in the theory of differentiation and integration of real-valued functions of real variables and it was in the same spirit as work being carried out by Denjoy and William Young.
      
    • Rademacher changed his area of mathematical interest from the theory of real functions to number theory in 1922 when he accepted the position of extraordinary professor at the University of Hamburg.
      
    • He was led towards number theory by Hecke who had been appointed to Hamburg three years before Rademacher.
      
    • About 1928 he began research on the topics for which he is best known among mathematicians today, namely his work in connection with questions concerning modular forms and analytic number theory.
      
    • Three years later Rademacher was again invited to address the American Mathematical Society and this time he chose the topic Trends in research: the analytic number theory.
      
    • methods and results in analytic number theory after the work of Landau, Hardy and Littlewood.
      
    • The first is that analytic number theory is not restricted to asymptotic formulas and estimates but that it has another side which is concerned with the derivation of identities, the use of group theoretical arguments, etc.
      
    • The second point is that analytic number theory is not merely a device for proving number theoretical results with the aid of analysis, but that it is really a thorough fusion of analysis and arithmetic in which the main interest is often as much on the analytical part as on the arithmetical part.
      
    • In addition to the significant contributions to real analysis and measure theory which we have briefly mentioned above, he contributed to complex analysis, geometry, and numerical analysis.
      
    • serve not only as a fitting memorial to a great mathematician and human being, but also provide excellent examples of how mathematics should be presented, and serve as leisurely but authentic introductions to some fascinating parts of analysis and number theory.
      
    • He lectured on Additive algebraic number theory.
      
    • He also wrote a number of textbooks such as Lectures on analytic number theory (1955), Lectures on elementary number theory (1964), Dedekind sums (1972), Topics in analytic number theory (1973), and Higher mathematics from an elementary point of view which was only published in 1983 but was based on a series of lectures he delivered at Stanford University in 1947.
      
    • The book Topics in analytic number theory was also published after Rademacher's death.
      

24. Carmeli biography
    
    • The significance of the "tail" and the relation to other equations of motion (1965), Motion of a charge in a gravitational field (1965), The equations of motion of slowly moving particles in the general theory of relativity (1965), and Equations of motion without infinite self-action terms in general relativity (1965).
      
    • Carmeli's field included gravitation and gauge field theory, the theory of spinors as applied to physics, Einstein special and general relativity, and astrophysics.
      
    • He developed his own cosmological relativity theory, both special and general, in which the age of Universe is postulated as constant, just as the speed of light is in Einstein's theory, and the velocity of receding galaxies is considered as a new independent variable.
      
    • In this five-dimensional brane world theory the results of Einstein's theory are obtained and the tests are fulfilled.
      
    • The theory is used by scientists in their research in different fields of theoretical physics, such as astrophysics and hydrodynamics.
      
    • written as a textbook for undergraduate students of mathematics and natural sciences who are studying group theory; .
      
    • Group theory and general relativity : Representations of the Lorentz group and their applications to the gravitational field (1977):- .
      
    • the first book to found the theory of general relativity on the principle of gauge invariance; .
      
    • Classical fields : general relativity and gauge theory (1982):- .
      
    • It contains an exposition of the theory of classical gravitational and gauge fields; .
      
    • Statistical theory and random matrices (1983):- .
      
    • students) Gravitation: SL(2, C) gauge theory and conservation laws (1990):- .
      
    • devoted to the formulation of Einstein's theory of general relativity as a gauge theory with the SL(2, C) group as the gauge group; .
      
    • and (with Shimon Malin) Theory of spinors : An introduction (2000):- .
      
    • which gives an introduction to the theory of spinors for the general physicist, not only for workers in general relativity.
      

25. Gruenberg biography
    
    • He was awarded a doctorate in 1954 for his thesis A Contribution to the Theory of Commutators in Groups and Associative Rings.
      
    • However the direction of his research moved towards cohomology theory, particularly its applications to group theory.
      
    • Typical of this is his famous Some cohomological topics in group theory which appeared in the Queen Mary College Mathematics Notes series in 1967.
      
    • The subject of these notes - which are based on the lectures the author gave at Queen Mary College, London, in 1965 - 6 and at Cornell University in 1966 - 7 - is "group theory with a cohomological flavour".
      
    • These notes are of great interest to workers in group theory who have some background in homological algebra.
      
    • The chapters are: (1) Fixed point free action; (2) The cohomology and homology groups; (3) Presentations and resolutions; (4) Free groups; (5) Classical extension theory; (6) More cohomological machinery; (7) Finite p-groups.
      
    • In 1970 these notes were republished as Cohomological topics in group theory by Springer-Verlag with four additional chapters: (8) Cohomological dimension; (9) Extension categories: general theory; (10) More module theory; (11) Extension categories: finite groups.
      
    • On the one hand, I wanted to show group theorists how the presentation theory of finite groups can nowadays be successfully approached with the help of integral representation theory.
      
    • On the other hand, I hoped to persuade ring theorists that here was an area of group theory well suited to applications of integral representation theory.
      
    • As a result, the course had to be constructed so that only a modicum of either group theory or module theory would be presupposed of the audience.
      
    • The author gives an excellent account of the theory of the relation modules of finite groups.
      
    • Practically all of the theory has previously appeared in papers of the author and his collaborators.
      
    • I gave a talk in the 'Group Theory' splinter group session on work which I was doing for my doctorate.
      

26. Vinogradov biography
    
    • Two of his teachers there, A A Markov and Ya V Uspenskii, both had interests in probability and number theory and Vinogradov's interest in number theory stems from this period.
      
    • While he was successfully preparing for the Master's examination with its very broad syllabus, Vinogradov was working on very difficult problems in the theory of numbers ..
      
    • He gave a course on number theory at the university which was to be the basis for his famous text on the subject Foundations of number theory.
      
    • He was promoted to professor at the university in 1925, becoming head of the probability and number theory section.
      
    • These were: fundamental questions of analysis and mathematical physics; special areas of function theory of real variables; number theory and Galois theory; probability theory; theoretical mechanics; applied methods of analysis.
      
    • The importance of trigonometric sums in the theory of numbers was first shown by Weyl in 1916.
      
    • In the 1920s the work of Hardy and Littlewood developed Weyl's methods to attack other problems in analytic number theory.
      
    • His methods reached their height in Some theorems concerning the theory of prime numbers written in 1937 which provides a partial solution to the Goldbach conjecture.
      
    • For example, in what is probably his most celebrated piece of work [Some theorems concerning the theory of prime numbers (1937)], he was able to combine the bilinear form technique with the Hardy-Littlewood method so as to reduce the Goldbach ternary problem to that of checking a finite number of cases.
      
    • Vinogradov made many other contributions, for example to the theory of distribution of power residues, non-residues, indices and primitive roots.
      
    • Two of his monographsThe method of trigonometric sums in the theory of numbers, and Special variants of the method of trigonometric sums are also in the book.
      
    • Even in Edmund Landau's three volume work on number theory, published in 1927, prominence is given to Vinogradov's methods.
      
    • One hundred years after his birth, on 14 September, a conference on analytic number theory was organised, followed by 'Vinogradov lectures'.
      
    • Nauk SSSR (9) (1991), 91-103.',26)">26] gives summaries of the 10 one hour 'Vinogradov lectures' devoted to number theory and related problems in algebraic geometry.
      

27. Taussky-Todd biography
    
    • Furtwangler became her thesis supervisor and after enjoying his course on number theory in her first year and his algebraic number theory seminar in her second year she asked him if she could write her thesis on number theory.
      
    • He said that she would work on class field theory which, she wrote [Mathematical People: Profiles and Interviews (Boston, 1985), 309-336.',1)">1]:- .
      
    • She wrote her thesis on algebraic number fields just as class field theory was being developed.
      
    • Encouraged to attend two meetings of the German Mathematical Society, she lectured there on her results and began a collaboration on group theory with A Scholzy, a student of Schur.
      
    • Courant had been looking for someone to work with Wilhelm Magnus and Helmut Ulm editing the first volume of Hilbert's complete works on number theory and Taussky fitted the bill perfectly.
      
    • While in Gottingen Taussky also edited Artin's lectures in class field theory (1932), assisted Emmy Noether in her class field theory and Courant with his differential equations course.
      
    • At this time Taussky-Todd wrote some papers on group theory, for example studying groups in which, in today's terminology, every subnormal subgroup is normal.
      
    • thesis on combinatorial group theory during 1942 and 1943.
      
    • Secondly I learned a lot of matrix theory.
      
    • In the following year she published On some boundary value problems in the theory of the non-uniform supersonic motion of an aerofoil in which she gives rigorous proofs of methods to find the velocity potential due to a two-dimensional airfoil in a supersonic stream whose shape and motion are given.
      
    • She continued to write papers on matrix theory, group theory, algebraic number theory but she also wrote on numerical analysis.
      
    • In 1955 Taussky-Todd and her husband spent a year's leave at the Courant Institute in New York where she taught a matrix theory course and her husband taught a numerical analysis course.
      
    • Olga's best-known and most influential work was in the field of matrix theory, though she also made important contributions to number theory.
      

28. Klein biography
    
    • Felix Klein is best known for his work in non-euclidean geometry, for his work on the connections between geometry and group theory, and for results in function theory.
      
    • In his dissertation Klein classified second degree line complexes using Weierstrass's theory of elementary divisors.
      
    • Leipzig seemed to be a superb outpost for building the kind of school he now had in mind: one that would draw heavily on the abundant riches offered by Riemann's geometric approach to function theory.
      
    • At Gottingen he taught a wide variety of courses, mainly on the interface between mathematics and physics, such as mechanics and potential theory.
      
    • The journal specialised in complex analysis, algebraic geometry and invariant theory.
      
    • It also provided an important outlet for real analysis and the new area of group theory.
      
    • However Klein himself saw his work on function theory as his major contribution to mathematics.
      
    • Klein considered his work in function theory to be the summit of his work in mathematics.
      
    • He owed some of his greatest successes to his development of Riemann's ideas and to the intimate alliance he forged between the later and the conception of invariant theory, of number theory and algebra, of group theory, and of multidimensional geometry and the theory of differential equations, especially in his own fields, elliptic modular functions and automorphic functions.
      
    • He wrote Riemanns Theorie der algebraischen Funktionen und ihre Integrals in 1882 which treats function theory in a geometric way connecting potential theory and conformal mappings.
      
    • He developed a theory of automorphic functions, connecting algebraic and geometric results in his important 1884 book on the icosahedron.
      
    • However Poincare began publishing an outline of his theory of automorphic functions in 1881 and, as explained in [Osiris (2) 5 (1989), 186-213.',13)">13], this led to a competition between the two:- .
      
    • Klein initiated a correspondence with Poincare, and soon a friendly rivalry ensued as both sought to formulate and prove a grand uniformization theorem that would serve as a capstone to this theory.
      
    • History Topics: The development of group theory .
      

29. Dedekind biography
    
    • There he learnt number theory which was the most advanced material he studied.
      
    • Dedekind did his doctoral work in four semesters under Gauss's supervision and submitted a thesis on the theory of Eulerian integrals.
      
    • He attended courses by Dirichlet on the theory of numbers, on potential theory, on definite integrals, and on partial differential equations.
      
    • Around this time Dedekind studied the work of Galois and he was the first to lecture on Galois theory when he taught a course on the topic at Gottingen during this period.
      
    • As well as his analysis of the nature of number, his work on mathematical induction, including the definition of finite and infinite sets, and his work in number theory, particularly in algebraic number fields, is of major importance.
      
    • On one such holiday in 1874 he met Cantor while staying in the beautiful city of Interlaken and the two discussed set theory.
      
    • Dedekind was sympathetic to Cantor's set theory as is illustrated by this quote from Was sind und was sollen die Zahlen (1888) regarding determining whether a given element belongs to a given set :- .
      
    • Dedekind edited Dirichlet's lectures on number theory and published these as Vorlesungen uber Zahlentheorie in 1863.
      
    • It was in the third and fourth editions of Vorlesungen uber Zahlentheorie, published in 1879 and 1894, that Dedekind wrote supplements in which he introduced the notion of an ideal which is fundamental to ring theory.
      
    • Dedekind formulated his theory in the ring of integers of an algebraic number field.
      
    • Dedekind, in a joint paper with Heinrich Weber published in 1882, applies his theory of ideals to the theory of Riemann surfaces.
      
    • presented a logical theory of number and of complete induction, presented his principal conception of the essence of arithmetic, and dealt with the role of the complete system of real numbers in geometry in the problem of the continuity of space.
      
    • Among other things, he provides a definition independent of the concept of number for the infiniteness or finiteness of a set by using the concept of mapping and treating the recursive definition, which is so important to the theory of ordinal numbers.
      
    • History Topics: The beginnings of set theory .
      
    • History Topics: A history of group theory .
      
    • History Topics: The development of Ring Theory .
      

30. Euler biography
    
    • Nowhere else could he have been surrounded by such a group of eminent scientists, including the analyst, geometer Jakob Hermann, a relative; Daniel Bernoulli, with whom Euler was connected not only by personal friendship but also by common interests in the field of applied mathematics; the versatile scholar Christian Goldbach, with whom Euler discussed numerous problems of analysis and the theory of numbers; F Maier, working in trigonometry; and the astronomer and geographer J-N Delisle.
      
    • The core of his research program was now set in place: number theory; infinitary analysis including its emerging branches, differential equations and the calculus of variations; and rational mechanics.
      
    • Studies of number theory were vital to the foundations of calculus, and special functions and differential equations were essential to rational mechanics, which supplied concrete problems.
      
    • He made decisive and formative contributions to geometry, calculus and number theory.
      
    • He studied continuum mechanics, lunar theory with Clairaut, the three body problem, elasticity, acoustics, the wave theory of light, hydraulics, and music.
      
    • He laid the foundation of analytical mechanics, especially in his Theory of the Motions of Rigid Bodies (1765).
      
    • Firstly his work in number theory seems to have been stimulated by Goldbach but probably originally came from the interest that the Bernoullis had in that topic.
      
    • Some of Euler's number theory results have been mentioned above.
      
    • Further important results in number theory by Euler included his proof of Fermat's Last Theorem for the case of n = 3.
      
    • This work bases the calculus on the theory of elementary functions rather than on geometric curves, as had been done previously.
      
    • He published his full theory of logarithms of complex numbers in 1751.
      
    • Euler made substantial contributions to differential geometry, investigating the theory of surfaces and curvature of surfaces.
      
    • determination of the orbits of comets and planets by a few observations, methods of calculation of the parallax of the sun, the theory of refraction, consideration of the physical nature of comets, ..
      
    • In fact Euler's lunar theory was used by Tobias Mayer in constructing his tables of the moon.
      
    • Euler also published on the theory of music, in particular he published Tentamen novae theoriae musicae in 1739 in which he tried to make music:- .
      
    • History Topics: The development of group theory .
      
    • History Topics: The development of Ring Theory .
      

31. Iwasawa biography
    
    • At this time Tokyo University had become a centre for the study of algebraic number theory as a result of Teiji Takagi's remarkable contributions.
      
    • Although the great tradition in number theory at Tokyo inspired him to an interest in that topic, some of his early research contributions were to group theory.
      
    • Artin was at the Institute during Iwasawa's two years there and he was one of the main factors in changing the direction of Iwasawa's research to algebraic number theory.
      
    • In 1952 Iwasawa published Theory of algebraic functions in Japanese.
      
    • The book begins with an historical survey of the theory of algebraic functions of one variable, from analytical, algebraic geometrical, and algebro-arithmetical view points.
      
    • A proof of the Riemann-Roch theorem is given, and the theory of Riemann surfaces and their topology is studied.
      
    • a general method in arithmetical algebraic geometry, known today as Iwasawa theory, whose central goal is to seek analogues for algebraic varieties defined over number field of the techniques which have been so successfully applied to varieties defined over finite fields by H Hasse, A Weil, B Dwork, A Grothendieck, P Deligne, and others.
      
    • The dominant theme of his work in number theory is his revolutionary idea that deep and previously inaccessible information about the arithmetic of a finite extension F of Q can be obtained by studying coarser questions about the arithmetic of certain infinite Galois towers of number fields lying above F.
      
    • The ideas were taken up immediately by Serre who saw their great potential and gave lectures to the Seminaire Bourbaki in Paris on Iwasawa theory.
      
    • The theory had become richer, and at the same time, more mysterious.
      
    • Even though only a few mathematicians had studied the theory thoroughly at that time, there was a general feeling that the theory was very promising.
      
    • This conjecture became known as "the main conjecture on cyclotomic fields" and it remained one of the most outstanding conjectures in algebraic number theory until it was solved by Mazur and Wiles in 1984 using modular curves.
      
    • He published Local class field theory in the year that he retired:- .
      
    • This carefully written monograph presents a self-contained and concise account of the modern formal group-theoretic approach to local class field theory.
      
    • AMS Cole Prize in Number Theory1962 .
      
    • AMS Cole Prize for number theory1962 .
      

32. Takagi biography
    
    • This is manifested, for instance, in the theory of groups, which is one of the most important fields of present day mathematics, and is related to various other branches.
      
    • It started as the theory of permutation groups, but now the general theory of groups does not suppose that elements of groups should be permutations.
      
    • He then read Hilbert's Zahlbericht, a report on algebraic number theory which had been published in 1897.
      
    • At the time when I studied in Germany, Gottingen was perhaps the only place in the world where research in algebraic number theory was going on.
      
    • Thus, when I told Hilbert that I wanted to study this theory, he did not seem to believe me immediately ..
      
    • If Takagi expected Hilbert to be actively engaged in algebraic number theory then he would have been disappointed.
      
    • I did not have a heavy workload so you might imagine that I did research on class field theory in those carefree days, but it was not so.
      
    • Concerning class field theory, I confess that I was misled by Hilbert.
      
    • from the standpoint of the theory of algebraic functions which are defined by Riemann surfaces, it is natural to limit consideration to the unramified case ..
      
    • finally I made my theory confirming this conjecture, but I could not rid myself of the doubt that it might contain an error which would invalidate the whole theory.
      
    • Takagi spoke of his work on class field theory, building on Heinrich Weber's work, at the International Congress of Mathematicians in Strasbourg in 1920.
      
    • He wrote his most important paper in 1920 which introduced the Takagi class-field theory generalising Hilbert's class field.
      
    • It became the framework of algebraic number theory.
      
    • Hasse included Takagi's theory in his treatise on class field theory a few years later.
      
    • His two most important books from this time are Introduction to analysis (1938), Algebraic number theory (1948) and an important work on the history of mathematics in the 19th century.
      

33. Chevalley biography
    
    • His papers of 1936 and 1941 where he introduced the concepts of adele and idele led to major advances in class field theory and also in algebraic geometry.
      
    • He did pioneering work in the theory of local rings in 1943, developing ideas due to Krull.
      
    • spinors were a well-established tool in theoretical physics, and Elie Cartan had already published his account of the theory.
      
    • Chevalley's exposition of the algebraic theory of spinors contains a number of interesting innovations.
      
    • He wrote Theory of Lie Groups in three volumes which appeared in 1946, 1951 and 1955.
      
    • Chevalley's most important contribution to mathematics is certainly his work on group theory ..
      
    • [The Theory of Lie Groups] was the first systematic exposition of the foundations of Lie group theory consistently adopting the global viewpoint, based on the notion of analytic manifold.
      
    • G D Mostow, reviewing Volume 2 of the Theory of Lie Groups writes:- .
      
    • The book is essentially self-contained and puts the theory on a clear-cut foundation.
      
    • Chevalley also published Theory of Distributions (1951), Introduction to the theory of algebraic functions of one variable (1951), The algebraic theory of spinors (1954), Class field theory (1954), The construction and study of certain important algebras (1955), Fundamental concepts of algebra (1956) and Foundations of algebraic geometry (1958).
      
    • Zariski reviewing Introduction to the theory of algebraic functions of one variable writes:- .
      
    • In this book the author develops systematically the theory of fields R of algebraic functions of one variable over arbitrary fields K of constants ..
      
    • The approach is therefore very general, and the treatment incorporates most of the new ideas and methods which have been introduced into the purely algebraic theory of function fields since the appearance of the classical treatise of Hensel-Landsberg.
      
    • AMS Cole Prize in Number Theory1941 .
      

34. Robertson biography
    
    • They began to study the effectiveness of explosives and the mathematical theory of explosion damage.
      
    • However he did make outstanding contributions to differential geometry, quantum theory, the theory of general relativity, and cosmology [H P Robertson : January 27, 1903-August 26, 1961.
      
    • he was interested in the foundations of physical theories, differential geometry, the theory of continuous groups, and group representations.
      
    • His contributions to differential geometry came in papers such as: The absolute differential calculus of a non-Pythagorean non-Riemannian space (1924); Transformation of Einstein space (1925); Dynamical space-times which contain a conformal Euclidean 3-space (1927); Note on projective coordinates (1928); (with H Weyl) On a problem in the theory of groups arising in the foundations of differential geometry (1929); Hypertensors (1930); and Groups of motion in space admitting absolute parallelism (1932).
      
    • In fact his association with Weyl went much further and in 1931 he published an English translation of the second edition of Weyl's classic text The theory of groups and quantum mechanics.
      
    • In 1928 and 1929 Robertson developed fully the "postulate of uniformity" so as to obtain the complete family of line-elements from the theory of continuous groups in Riemannian space.
      
    • In three papers entitled Kinematic and world structure published in 1935 and 1936 Robertson looked at Milne's theory and Einstein's theory of gravitation and their respective application to cosmology, arguing strongly that Milne's objections to Einstein's theory are not valid.
      
    • The author's purpose is to give "a study of the physical geometry of the gravitational and of the thermal fields, and of the reasons for the success of the former as opposed to the latter, as a basis for a tenable physical theory." The discussion of gravitation is given in terms of a four-dimensional scalar theory but the main ideas of Einstein's general theory are stressed.
      
    • The results of this theory regarding the motion of test particles and the behaviour of light rays are stated and compared to that of Einstein's theory and differ from that theory.
      
    • However the author's purpose is not to give a physical theory of gravitation, but to discuss the geometry of a theory of the gravitational field.
      

35. Krull biography
    
    • He attended Klein's seminar in the session 1920-21 and he then returned to Freiburg and presented his doctoral thesis on the theory of elementary divisors in 1922.
      
    • Ring theory results from this thesis have recently been found important in the area of coding theory.
      
    • his earlier studies, but also dealt with other fields of mathematics: group theory, calculus of variations, differential equations, Hilbert spaces.
      
    • He then studied Galois theory and extended the classical results on Galois theory of finite field extensions to infinite field extensions.
      
    • In 1928 he defined the Krull dimension of a commutative Noetherian ring and brought ring theory into in new setting in which he was able to show that the principal ideal theorem held.
      
    • was quickly recognised as a decisive advance in Noether's programme of emancipating abstract ring theory from the theory of polynomial rings.
      
    • Krull carried his work forward, defining further concepts which are today central to modern research in ring theory.
      
    • He then wrote the remarkable treatise Ideal Theory which remains a beautiful introduction to ring theory but is simply a theory built from the results that Krull had himself proved.
      
    • One could say that Krull had achieved the goal he had in some sense set himself in his Erlangen address and arranged his theory to be self-evident.
      
    • Another major topic in ring theory is the study of local rings, that is rings having a unique maximal ideal, and they are used in the study of local properties of algebraic varieties.
      
    • The concept was introduced by Krull in 1938 and his fundamental results were developed into a major theory by mathematicians such as Chevalley and Zariski.
      
    • Indeed much of modern ring theory is still following the path which Krull took, building on the foundations which Emmy Noether had laid.
      
    • History Topics: The development of Ring Theory .
      

36. Russell biography
    
    • Bertrand Russell published a large number of books on logic, the theory of knowledge, and many other topics.
      
    • His contributions relating to mathematics include his discovery of Russell's paradox, his defence of logicism (the view that mathematics is, in some significant sense, reducible to formal logic), his introduction of the theory of types, and his refining and popularizing of the first-order predicate calculus.
      
    • A large amount of work throughout the early part of this century in logic, set theory, and the philosophy and foundations of mathematics was thus prompted.
      
    • Russell's paradox arises as a result of naive set theory's so-called unrestricted comprehension (or abstraction) axiom.
      
    • Russell's own response to the paradox came with the introduction of his theory of types.
      
    • Using the vicious circle principle also adopted by Henri Poincare, together with his so-called "no class" theory of classes, Russell was then able to explain why the unrestricted comprehension axiom fails: propositional functions, such as the function "x is a set", should not be applied to themselves since self-application would involve a vicious circle.
      
    • Although first introduced by Russell in 1903 in the Principles, his theory of types finds its mature expression in his 1908 article Mathematical Logic as Based on the Theory of Types and in the monumental work he co-authored with Alfred North Whitehead, Principia Mathematica (1910, 1912, 1913).
      
    • Thus, in its details, the theory admits of two versions, the "simple theory" and the "ramified theory".
      
    • Both versions of the theory later came under attack.
      
    • Russell's response to the second of these objections was to introduce, within the ramified theory, the axiom of reducibility.
      
    • Of equal significance during this same period was Russell's defence of logicism, the theory that mathematics was in some important sense reducible to logic.
      
    • In Principia Mathematica, Whitehead and Russell were able to provide detailed derivations of many major theorems in set theory, finite and transfinite arithmetic, and elementary measure theory.
      
    • History Topics: The beginnings of set theory .
      

37. Lyndon biography
    
    • His supervisor was Saunders Mac Lane and his thesis was entitled The Cohomology Theory of Group Extensions.
      
    • Mac Lane explained that the thesis addressed [Contributions to group theory, Contemporary Mathematics 33 (Prividence, 1984), 10-23.',4)">4]:- .
      
    • Lyndon's second paper The Cohomology Theory of Group Extensions was based on his doctoral thesis and appeared in print in 1948.
      
    • After attending a course by Alfred Tarski, Lyndon and Tarski became good friends and Lyndon was later to work on model theory as a result of attending these lectures.
      
    • Returning to the United States in 1948 he decided that to make further progress in cohomology theory he needed to learn more about current work in topology and clearly Princeton was the leading institution for research in that area.
      
    • Accepting a position as an Instructor in Mathematics at Princeton, he attended a course on knot theory by R Fox and from this his interest was aroused in combinatorial group theory.
      
    • Kenneth Appel, writes in [Contributions to group theory, Contemporary Mathematics 33 (Prividence, 1984), 1-10.',2)">2]:- .
      
    • Lyndon made numerous major contributions to combinatorial group theory.
      
    • These include the development of 'small cancellation theory', work on Fuchsian groups and the Riemann-Hurwitz formula, his introduction of 'aspherical' presentations of groups and his work on length functions in free products of groups.
      
    • Lyndon was the coauthor of one of the most important works on combinatorial group theory.
      
    • Together with Paul Schupp, he wrote Combinatorial group theory (1976).
      
    • Combinatorial group theory was not Lyndon's first book, however, for he had published Notes on logic ten years earlier in 1966.
      
    • This book is a very readable introduction to group theory, geometry, and the connections between them.
      
    • They include: Groups rings and dimension subgroups; Two investigations on the borderline of logic and algebra; Decision problems of finite automata design and related arithmetic; On Dehn's algorithm and the conjugacy problem; Projectivities of free products; Continuous model theory and set theory; Real length functions in groups; Automorphisms of the fundamental group of an orientable 2-manifold; Some algorithmic problems for semigroups; and Groups acting on trees.
      
    • student, writes [Contributions to group theory, Contemporary Mathematics 33 (Prividence, 1984), 1-10.',2)">2]:- .
      

38. Khinchin biography
    
    • He was an outstanding student being particularly interested in the metric theory of functions and before he graduated in 1916 he had already written his first paper on a generalisation of the Denjoy integral.
      
    • Around 1922 Khinchin took up new mathematical interests when he began to study the theory of numbers and probability theory.
      
    • In 1927 Khinchin was appointed as a professor at Moscow University and, in the same year, he published Basic laws of probability theory.
      
    • Between 1932 and 1934 he laid the foundations for the theory of stationary random processes culminating in a major paper in Mathematische Annalen in 1934.
      
    • Khinchin left Moscow in 1935 to spend two years at Saratov University but returned to Moscow University in 1937 to continue his role of building the school of probability theory there in partnership with Kolmogorov and others, including in particular their student Gnedenko.
      
    • From the 1940s his work changed direction again and this time he became interested in the theory of statistical mechanics.
      
    • In the last few years of his life his interests turned to developing Shannon's ideas on information theory.
      
    • The book consists of three chapters, the first two of which present the classical theory of continued fractions.
      
    • The third chapter, the longest and most important, contains an account of Khinchin's own contributions to the topic of the metrical theory of Diophantine approximations.
      
    • Another contribution by Khinchin to number theory is the short book Three pearls of number theory which appeared in an English translation in 1952.
      
    • Khinchin's book Mathematical Foundations of Information Theory, translated into English from the original Russian in 1957, is important.
      
    • It consists of English translations of two articles: The entropy concept in probability theory and On the basic theorems of information theory which were both published earlier in Russian.
      
    • II (Berkeley, 1961), 1-15.',6)">6] Gnedenko, who was a student of Khinchin, lists 151 publications by Khinchin on the mathematical theory of probability (the list is given again in [Ann.
      
    • A I Khinchin on Information Theory .
      

39. Preston biography
    
    • The work there was all aimed at cracking German codes and Preston worked there with Alan Turing [Monash Conference on Semigroup Theory, Melbourne, 1990 (World Sci.
      
    • This was my first experience of research - it was a mixture of algebra and statistics, or probability theory, and I greatly enjoyed it.
      
    • In 1946 Preston returned to Oxford to complete his undergraduate studies and he graduated with first class honours in mathematics in 1948 [Monash Conference on Semigroup Theory, Melbourne, 1990 (World Sci.
      
    • It was through the connections that he had made in Bletchley Park that Preston first became interested in semigroups [Monash Conference on Semigroup Theory, Melbourne, 1990 (World Sci.
      
    • on a part-time basis and was awarded the degree in 1954 for his thesis Some Problems in the Theory of Ideals.
      
    • He writes in [Monash Conference on Semigroup Theory, Melbourne, 1990 (World Sci.
      
    • [In my thesis] I tried to extend properties of groups and rings to a more general context, for example some of the ideal theory of Noetherian rings.
      
    • when he wrote his three hugely influential papers laying the foundations of inverse semigroup theory it is not at all surprising that he was completely unaware of the closely similar work of Vagner in Russia.
      
    • He writes in [Semigroup theory and its applications, New Orleans, LA, 1994 (Cambridge Univ.
      
    • The first volume of Clifford and Preston's The algebraic theory of semigroups was published in 1961.
      
    • Howie writes [Monash Conference on Semigroup Theory, Melbourne, 1990 (World Sci.
      
    • We mentioned above that the first volume of Clifford and Preston, The algebraic theory of semigroups, was published in 1961.
      
    • Before, there has been no systematic treatment on semigroups at all, with the exception of the book of Suchkewitsch, 'The theory of generalized groups' (1937) containing naturally a very limited number of results.
      
    • Volume II of Clifford and Preston, The algebraic theory of semigroups was published in 1967.
      
    • Volume II has been eagerly awaited by those who are working in semigroup theory and related subjects (e.g., automata theory).
      
    • deals with additional branches of the theory to which there was at most passing reference in Volume I.
      

40. Petryshyn biography
    
    • from Columbia University for his thesis Linear Transformations Between Hilbert Spaces and the Application of the Theory to Linear Partial Differential Equations.
      
    • His major results include the development of the theory of iterative and projective methods for the constructive solution of linear and nonlinear abstract and differential equations.
      
    • The theory of A-proper maps was developed by Petryshyn and this work is described in [Encyclopedia of Ukraine (Toronto-Buffalo-London, 1993).',1)">1]:- .
      
    • Petryshyn is a founder and principal developer of the theory of approximation-proper (A-proper) maps, a new class of maps which attracted considerable attention in the mathematical community.
      
    • He has shown that the theory of A-proper type maps not only extends and unifies the classical theory of compact maps with some recent theories of condensing and monotone-accretive maps, but also provides a new approach to the constructive solution of nonlinear abstract and differential equations.
      
    • The theory has been applied to ordinary and partial differential equations.
      
    • This outstanding reference/text develops an essentially constructive theory of solvability on linear and nonlinear abstract and differential equations involving A-proper operator equations in separable Banach spaces, treats the problem of existence of a solution for equations involving pseudo-A-proper and weakly-A-proper mappings, and illustrates their applications.
      
    • Facilitating the understanding of the solvability of equations in infinite dimensional Banach space through finite dimensional approximations, Approximation - solvability of Nonlinear Functional and Differential Equations: offers an important elementary introduction to the general theory of A-proper and pseudo-A-proper maps; develops the linear theory of A-proper maps; furnishes the best possible results for linear equations; establishes the existence of fixed points and eigenvalues for P-gamma-compact maps, including classical results; provides surjectivity theorems for pseudo-A-proper and weakly-A-proper mappings that unify and extend earlier results on monotone and accretive mappings; shows how Friedrichs' linear extension theory can be generalized to the extensions of densely defined nonlinear operators in a Hilbert space; presents the generalized topological degree theory for A-proper mappings; and applies abstract results to boundary value problems and to bifurcation and asymptotic bifurcation problems.
      
    • In this monograph we develop the generalised degree theory for densely defined A-proper mappings, and then use it to study the solubility (sometimes constructive) and the structure of the solution set of [an] important class of semilinear abstract and differential equations ..
      
    • The theory subsumes classical theory involving compact vector fields as well as more recent theories of condensing vector fields and strongly monotone and strongly accretive maps.
      
    • The book begins with an outline of Brouwer degree theory and a description of some basic constructive results.
      
    • These abstract results are then applied to boundary value problems of ODEs and PDEs with general nonlinearities, problems that are intractable under any other existing theory.
      

41. Ptolemy biography
    
    • One of the most influential Greek astronomers and geographers of his time, Ptolemy propounded the geocentric theory in a form that prevailed for 1400 years.
      
    • The Almagest is the earliest of Ptolemy's works and gives in detail the mathematical theory of the motions of the Sun, Moon, and planets.
      
    • The Almagest was not superseded until a century after Copernicus presented his heliocentric theory in the De revolutionibus of 1543.
      
    • For the sake of completeness in our treatment we shall set out everything useful for the theory of the heavens in the proper order, but to avoid undue length we shall merely recount what has been adequately established by the ancients.
      
    • [After introducing the mathematical concepts] we have to go through the motions of the sun and of the moon, and the phenomena accompanying these motions; for it would be impossible to examine the theory of the stars thoroughly without first having a grasp of these matters.
      
    • Our final task in this way of approach is the theory of the stars.
      
    • In examining the theory of the sun, Ptolemy compares his own observations of equinoxes with those of Hipparchus and the earlier observations Meton in 432 BC.
      
    • This theory of the sun forms the subject of Book 3 of the Almagest.
      
    • In Books 4 and 5 Ptolemy gives his theory of the moon.
      
    • An interesting discussion of Ptolemy's theory of the moon is given in [Nuncius Ann.
      
    • Having given a theory for the motion of the sun and of the moon, Ptolemy was in a position to apply these to obtain a theory of eclipses which he does in Book 6.
      
    • The final five books of the Almagest discuss planetary theory.
      
    • The planetary theory which Ptolemy developed here is a masterpiece.
      
    • a man working [on map-construction] without the support of a developed theory but within a mathematical tradition and guided by his sense of what is appropriate to the problem.
      
    • The establishment of theory by experiment, frequently by constructing special apparatus, is the most striking feature of Ptolemy's "Optics".
      
    • Ptolemy, whose intention was to develop a comprehensive theory of celestial phenomena, had no access to the methods of data evaluation using arithmetical means with which modern astronomers can derive from a set of varying measurement results, the one representative value needed to test a hypothesis.
      

42. Shoda biography
    
    • This was an exciting period to study at Tokyo University for Takagi had published his famous paper on class field theory in 1920.
      
    • Takagi lectured on group theory, representation theory, Galois theory, and algebraic number theory.
      
    • During his first year of graduate studies he read works on the theory of group representations by Frobenius and Schur.
      
    • After a year in Berlin, Shoda went to Gottingen where he joined Emmy Noether's school, attending her lectures on hypercomplex systems and representation theory.
      
    • There, near Noether, he witnessed the remarkable process of creation of great mathematical ideas and theory, and youthful Shoda buried himself in enthusiastic pursuit of mathematics in a wonderful creative atmosphere generated by the many young, able mathematicians who had come from all over the world to Gottingen, attracted by Emmy Noether.
      
    • The twelfth printing of the book was published in Tokyo in 1971 with the chapter headings: Basic concepts; Field theory; Galois theory; Elimination theory; General ideal theory; Valuation theory.
      
    • The second chapter is on the theory of free systems, including the fundamental theorem and the theorem of change of generators (of Tietze).
      
    • A theory of independence is given, making use of a certain notion of valuation so as to take care of algebraic and linear dependence, the latter being distinguished in that an element is (linearly) dependent on a set of elements if and only if it is contained in the subsystem generated by the set.
      
    • Structural theory of abstract ring-systems is developed, under chain conditions, including (generalized) Peirce decompositions and Wedderburn's theorem; for the latter the notion of matrices is also generalized.
      
    • The last chapter gives the theory of representing (primitive) algebraic systems as systems of endomorphisms of some other systems called representation systems.
      

43. Van der Pol biography
    
    • In his theoretical work van der Pol gave a quantitative proof that, if one neglected the influence of a reflecting layer, experiment and theory were in flagrant disagreement in the case of long-distance propagation.
      
    • even in mathematics, his papers covered number theory, special functions, operational calculus and nonlinear differential equations.
      
    • 35 (1960), 367-376.',4)">4], [Application of asymptotic methods in the theory of nonlinear differential equations (Russian) (Akad.
      
    • We consider the first attempts at solving nonlinear problems of the theory of oscillations.
      
    • The first observation of the inapplicability of the linear theory to problems of this type was made in radio-engineering by van der Pol.
      
    • We mentioned above that van der Pol was interested in number theory.
      
    • Of van der Pol's papers on the theory of numbers [An electro-mechanical investigation of the Riemann Zeta function in the critical strip (1947)] is perhaps the best known.
      
    • In it he combined his knowledge of radio technology and number theory to advantage.
      
    • The branch of number theory, however, which lay closest to his heart was the theory and applications of theta-functions.
      
    • As another example of his work on number theory we mention the paper The primes in k(p) (1951).
      
    • He became a member and then fellow and life member of the Institute of Radio Engineers (U.S.A.) in 1920; he was Vice-President of it in 1934, and was awarded its Medal of Honour in 1935 for contributions to circuit theory.
      
    • His summary ['The non-linear theory of electric oscillations' (1934)] of work on non-linear oscillations up to 1934 was a quite masterly account of the theory up to that time.
      
    • In fact van der Pol corresponded with Nikolai Mitrofanovich Krylov about the theory of nonlinear oscillations; a letter sent by van der Pol to Krylov is published in [Application of asymptotic methods in the theory of nonlinear differential equations (Russian) (Akad.
      

44. Kolchin biography
    
    • Despite serving his country during the war, Kolchin was still able to publish papers such as On the basis theorem for differential systems (1942) and begin his fundamental work on the Galois theory of differential fields in the three part paper Extensions of differential fields.
      
    • Other papers around this time were Algebraic matric groups (1946) and The Picard-Vessiot theory of homogeneous linear ordinary differential equations (1946).
      
    • Although the articles in this volume are in the main devoted to commutative algebra, linear algebraic group theory, and differential algebra, the diversity of subjects covered - complex analysis, algebraic K-theory, logic, stochastic matrices, differential geometry, ..
      
    • His deep and abiding interest has always been in the application of the powerful and clarifying techniques of algebra to problems in the theory of differential equations.
      
    • It was in the course of applying the Ritt theory to the classical Picard-Vessiot theory that he became one of the pioneers of linear algebraic group theory.
      
    • In this volume we celebrate the influence that Kolchin's work on the Galois theory of differential fields has had on the development of differential algebra and linear algebraic group theory.
      
    • This book gives the foundations of a theory of differential algebraic groups.
      
    • It is intended that such a theory bear to algebraic groups the same relation that the theory of differential equations bears to the theory of algebraic equations.
      
    • But in fact the foundational aspects of the theory are better served by an abstract view of algebraic varieties as sets with an additional structure, where the latter is usually a sheaf of integral domains, but could be taken as a collection of fields with specialization relations (the ring of sections over a set is replaced by its quotient field and the restriction maps endue the specializations).
      
    • With these objects the theory is developed.
      

45. Galileo biography
    
    • Galileo spent three years holding this post at the university of Pisa and during this time he wrote De Motu a series of essays on the theory of motion which he never published.
      
    • We mentioned above an error in Galileo's theory of motion as he set it out in De Motu around 1590.
      
    • In fact he had returned to work on the theory of motion in 1602 and over the following two years, through his study of inclined planes and the pendulum, he had formulated the correct law of falling bodies and had worked out that a projectile follows a parabolic path.
      
    • In fact it was his theory of falling bodies which was the most significant in this respect, for opponents of a moving Earth argued that if the Earth rotated and a body was dropped from a tower it should fall behind the tower as the Earth rotated while it fell.
      
    • At a meeting in the Medici palace in Florence in December 1613 with the Grand Duke Cosimo II and his mother the Grand Duchess Christina of Lorraine, Castelli was asked to explain the apparent contradictions between the Copernican theory and Holy Scripture.
      
    • He seems at this time to have seen little reason for the Church to be concerned regarding the Copernican theory.
      
    • The point at issue was whether Copernicus had simply put forward a mathematical theory which enabled the calculation of the positions of the heavenly bodies to be made more simply or whether he was proposing a physical reality.
      
    • At this time Bellarmine viewed the theory as an elegant mathematical one which did not threaten the established Christian belief regarding the structure of the universe.
      
    • In this Galileo stated quite clearly that for him the Copernican theory is not just a mathematical calculating tool, but is a physical reality:- .
      
    • Pope Paul V ordered Bellarmine to have the Sacred Congregation of the Index decide on the Copernican theory.
      
    • Pope Urban VIII invited Galileo to papal audiences on six occasions and led Galileo to believe that the Catholic Church would not make an issue of the Copernican theory.
      
    • The climax of the book is an argument by Salviati that the Earth moves which was based on Galileo's theory of the tides.
      
    • Galileo's theory of the tides was entirely false despite being postulated after Kepler had already put forward the correct explanation.
      
    • It was unfortunate, given the remarkable truths the Dialogue supported, that the argument which Galileo thought to give the strongest proof of Copernicus's theory should be incorrect.
      
    • The truth of the Copernican theory was not an issue therefore; it was taken as a fact at the trial that this theory was false.
      

46. Chernikov biography
    
    • By 1936 Chernikov had no doubt that he wished to study group theory and so he applied to the University of Moscow to undertake research as an external student under Kurosh's supervision.
      
    • He also studied finiteness type conditions that had already been seen to have great importance in ring theory, namely finiteness type conditions which did not allow infinite chains of subgroups of a specified type.
      
    • By 1938 he already had two papers published on generalisations of results from finite group theory to infinite group theory, in particular generalising Frobenius's theorem to infinite groups.
      
    • He then wrote a beautiful survey article Finiteness conditions in the general theory of groups which was published in 1959 and contained many of Chernikov's own results and those of others.
      
    • algebraists, the study of infinite groups with finiteness conditions, enriched group theory with many new concepts, ideas and profound results, and also widened considerably the basis of group theory, extending it by new detailed investigations of infinite groups of specific form.
      
    • One thing was clear, Chernikov was not just looking round for results which he could prove, he was developing a systematic theory in the way that is the hallmark of top quality mathematicians.
      
    • The other algebraists mentioned in the quote above who began to help Chernikov in building his theory included O J Schmidt, Malcev, Baer, Kurosh, Hall, and others.
      
    • This was the study of the theory of linear inequalities, an area of great practical significance because of its connection with the theory of linear programming.
      
    • The practical importance of convenient algorithms for the solution of systems of linear inequalities and their connection with the theory of linear programming is well known.
      
    • In 1968 Chernikov wrote an important book Linear inequalities which gives sets out Chernikov's algebraic theory.
      
    • the basis of this theory lies in the principle of boundary solutions; all its results are deduced from it by means of only a few finite methods..
      
    • Polyhedrally closed systems of linear inequalities are an effective means in the analysis of problems of the theory of approximation of functions, in linear programming (in particular in questions of duality), and in control theory.
      

47. Al-Farisi biography
    
    • He made two major contributions to mathematics, one on light, the other on number theory.
      
    • His work on light, colour and the rainbow is discussed in [Dictionary of Scientific Biography (New York 1970-1990).',1)">1] but no mention of his work on number theory (nor mention of any other work at all by al-Farisi) occurs in that article written by R Rashed.
      
    • On the other hand his contributions to number theory are discussed the references [The development of Arabic mathematics : between arithmetic and algebra (London, 1994).',2)">2], [Entre arithmetique et algebre: Recherches sur l\'histoire des mathematiques arabes (Paris, 1984).',3)">3], [Historia Math.
      
    • The most important part of this work by al-Farisi is his theory of the rainbow.
      
    • Ibn al-Haytham had indeed proposed a theory, but al-Farisi considered both this theory and another proposed by Avicenna before giving his own.
      
    • The theory proposed by al-Farisi was the first mathematically satisfactory explanation of the rainbow.
      
    • It was a theory which did not allow for a possible experimental verification.
      
    • There have been arguments between modern scholars as to whether al-Farisi's theory of the rainbow was due to him or whether it was a theory proposed by his teacher al-Shirazi.
      
    • the discovery of the theory should presumably be ascribed to [al-Shirazi], its elaboration to [al-Farisi].
      
    • Rashed discusses the claims of Boyer and others that the innovation in the theory of the rainbow was from al-Shirazi, but gives sound arguments for his claim that ascribing the theory to al-Shirazi is unconvincing.
      
    • Al-Farisi made a number of important contributions to number theory.
      
    • Al-Farisi's most impressive work in number theory is on amicable numbers.
      
    • Rather he produced a major new approach to a whole area of number theory, introducing ideas concerning factorisation and combinatorial methods.
      

48. Markov biography
    
    • This thesis was outstanding [The St Petersburg school of number theory (American Mathematical Socity, Providence, RI, 2005).',4)">4]:- .
      
    • This work, very highly esteemed by Chebyshev, represents one of the finest achievements of the St Petersburg school of number theory, and perhaps even of all Russian mathematics.
      
    • Markov's early work was mainly in number theory and analysis, algebraic continued fractions, limits of integrals, approximation theory and the convergence of series.
      
    • After 1900 Markov applied the method of continued fractions, pioneered by his teacher Pafnuty Chebyshev, to probability theory [The St Petersburg school of number theory (American Mathematical Socity, Providence, RI, 2005).',4)">4]:- .
      
    • Markov was the most elegant spokesman for Chebyshev's ideas and directions of research in probability theory.
      
    • Especially remarkable is his research relating to the theorem of Jacob Bernoulli known as the Law of Large Numbers, to two fundamental theorems of probability theory due to Chebyshev, and to the method of least squares.
      
    • This work founded a completely new branch of probability theory and launched the theory of stochastic processes.
      
    • The foundation of a general theory was provided during the 1930s by Andrei Kolmogorov.
      
    • Sergi Bernstein, who continued to develop the theory of Markov chains, wrote (see for example [The St Petersburg school of number theory (American Mathematical Socity, Providence, RI, 2005).',4)">4]):- .
      
    • A A Markov's classic course on the computation of probabilities, and his original memoirs, models of accuracy and clarity of exposition, contributed to a very large extent to the transformation of the theory of probability into one of the most perfected areas of mathematics, and to the wide dissemination of Chebyshev's methods and directions of research.
      
    • His profound analysis in the spirit of Chebyshev of the dependencies among observed random phenomena allowed Markov to extend probability theory in an essential way through the introduction and investigation of dependent random quantities.
      
    • It is worth pointing out, however, that although Markov developed his theory of Markov chains as a purely mathematical work without considering physical applications, he did apply the ideas to chains of two states, namely vowels and consonants, in literary texts.
      

49. Skolem biography
    
    • [Skolem] conducted the regular graduate courses in algebra and number theory, and rather infrequently lectured on mathematical logic.
      
    • Skolem was remarkably productive publishing around 180 papers on topics such as Diophantine equations, mathematical logic, group theory, lattice theory and set theory.
      
    • He did some early work in lattice theory.
      
    • It states that if a theory within first-order predicate calculus has a model then it has a countable model.
      
    • He made refinements to Zermelo's axiomatic set theory, publishing work in 1922 and 1929.
      
    • Here he applied the Lowenheim-Skolem theorem to show what became known as Skolem's paradox: If the Zermelo's axiomatic system for set theory is consistent then it must be satisfiable within a countable domain.
      
    • Skolem is commonly portrayed as arguing that certain otherwise well understood concepts are suspect simply because they cannot be characterized in a first-order language; in particular that, since all first-order formalizations of set theory (if consistent) have countable models, the concept of uncountability is flawed.
      
    • I see Skolem as arguing that all the evidence that has been given for the existence of uncountable sets is inconclusive, and the reason why he insists on considering countable models is that axiomatisation was put forward at the time as the only way to secure set theory, and what sets are and which sets exist was claimed to be determined by the axioms and their models (much as what Euclidean geometry is about was claimed to be determined by Hilbert's axioms and their models).
      
    • In 1923 Skolem also developed a theory of recursive functions as a means of avoiding the so-called paradoxes of the infinite in his paper Begrundung der elementaren Arithmetik durch die rekurrierende Denkweise ohne Anwendung scheinbarer Veranderlichen mit unendlichem Ausdehnugsbereich.
      
    • In it he developed number theory using two systems, one to define objects by primitive recursion, the other system to prove properties of the objects defined by the first system.
      
    • With these he defined prime numbers and developed a considerable amount of number theory.
      
    • If one has to single out one most intriguing item, it would probably be his work on nonstandard models of set theory and number theory.
      
    • Skolem: Abstract Set Theory .
      

50. Renyi biography
    
    • The main topic of discussion was number theory ..
      
    • Renyi went to Russia as a postdoctoral student and, between October 1946 and June 1947, worked with Yuri Vladimirovich Linnik on the theory of numbers, in particular working on the Goldbach conjecture [19]:- .
      
    • at present one of the strongest methods of analytical number theory.
      
    • Other papers published early in his career include: On a Tauberian theorem of O Szasz (1946); Integral formulae in the theory of convex curves (1947); On the minimal number of terms of the square of a polynomial (1947); On some new applications of the method of Academician I M Vinogradov (1947); (with Yu V Linnik) On certain hypotheses in the theory of Dirichlet characters (Russian) (1947).
      
    • Renyi worked on probability theory which was to be his main research topic throughout his life, but his interests were broad and also covered statistics, information theory, combinatorics, graph theory, number theory and analysis.
      
    • In the hands of writers like Linnik, Erdos and Renyi, the theory of numbers is not clearly distinguished from the theory of probability.
      
    • Probability theory and its applications had been neglected in the curriculum of Hungarian universities until very recently when the author started to lecture regularly on these topics.
      
    • It is thus the first modern Hungarian text book on probability theory and offers an excellent introduction into this field.
      
    • A French edition appeared in 1966 and an English edition, containing three new sections, was published as Probability theory in 1970.
      
    • After his sudden death, material was found for a book on which he was working Diary on information theory.
      
    • a pure mathematician of massive achievements and towering stature in the classical fields of number theory and analysis.
      

51. Adams Frank biography
    
    • After taking his first degree he started graduate work at Cambridge with Besicovitch on geometric measure theory.
      
    • His work turned towards K-theory, the generalised cohomology theory on vector bundles.
      
    • Using this theory he solved another important conjecture, this one being about vector fields on spheres.
      
    • He continued to produce work of outstanding depth and originality, and during his first few years at Manchester he wrote a series of papers On the groups J(X) which were highly influential in homotopy theory.
      
    • His research continued to be of fundamental importance in the homotopy theory of classifying spaces of topological groups, finite H-spaces and equivariant homotopy theory.
      
    • These books are of major importance, and include Stable homotopy theory (1964), Lectures on Lie groups (1969), Algebraic topology: a student's guide (1972), Stable homotopy theory and generalized homology (1974), Localisation and completion (1975), and Infinite loop spaces (1978).
      
    • Stable homotopy theory (1964) is a short 74 page book which is based on six lectures Adams gave at the University of California at Berkeley in 1961.
      
    • This book covers in a concise manner the fine structure and representation theory of compact Lie groups, with emphasis on the classical groups.
      
    • The exposition of the book is aimed at the reader who has some understanding of algebraic topology and would like to understand the aspects of the theory of compact Lie groups that are relevant to algebraic topology.
      
    • Stable homotopy theory and generalized homology (1974) comprises of three lecture courses, one on the algebra of stable operations in complex cobordism delivered in 1967, the second on complex cobordism theory delivered in 1970, and the third on stable homotopy and generalized homology theories delivered in 1971.
      
    • Over the past few years, various topologists have been heard to complain about the lengthy and technical nature of infinite loop space theory.
      
    • in recognition of his solution of several outstanding problems of algebraic topology and of the methods he invented for this purpose which have proved of prime importance in the theory of that subject.
      

52. Kronecker biography
    
    • Back in Berlin he worked on his doctoral thesis on algebraic number theory under Dirichlet's supervision.
      
    • students to hear that Kronecker was questioned at his oral on a wide range of topics including the theory of probability as applied to astronomical observations, the theory of definite integrals, series and differential equations, as well as on Greek, and the history of philosophy.
      
    • These were on number theory, elliptic functions and algebra, but, more importantly, he explored the interconnections between these topics.
      
    • The topics on which he lectured were very much related to his research: number theory, the theory of equations, the theory of determinants, and the theory of integrals.
      
    • We have already indicated that Kronecker's primary contributions were in the theory of equations and higher algebra, with his major contributions in elliptic functions, the theory of algebraic equations, and the theory of algebraic numbers.
      
    • In that year he argued against the theory of irrational numbers used by Dedekind, Cantor and Heine giving the arguments by which he opposed:- .
      
    • Of course, given his belief that only finitely constructible mathematical objects existed, he was completely opposed to Cantor's developing ideas in set theory.
      
    • History Topics: The beginnings of set theory .
      
    • History Topics: A history of group theory .
      
    • History Topics: The development of Ring Theory .
      

53. Turan biography
    
    • We first met at the University of Budapest in September 1930 and immediately discovered our common interest in number theory.
      
    • One was A problem in the elementary theory of numbers which appeared in the American Mathematical Monthly.
      
    • Rather it was the method of proof which, although it does not use probabilistic terminology, in fact became one of the foundations of probabilistic number theory.
      
    • Their importance first of all is that they lead to interesting deep problems of a completely new type; they have quite unexpectedly surprising consequences in many branches of mathematics - differential equations, numerical algebra, and various branches of function theory.
      
    • Number Theory 13 (3) (1981), 271-278.',3)">3]:- .
      
    • Their results in this field were published in nearly 20 papers and were called comparative number theory by the authors.
      
    • Number Theory 13 (3) (1981), 271-278.',3)">3]:- .
      
    • Another remarkable fact is that extremal graph theory, an area which Turan founded, was one of the "best ideas" that he had while in the labour camps.
      
    • In 1949 he was appointed to the Chair of Algebra and Number Theory at Eotvos Lorand University of Budapest, a position he held until his death.
      
    • From 1955 he was Head of the Complex Function Theory Department in the Mathematical Institute of the Hungarian Academy of Sciences.
      
    • As regards the latter, Turan found new approaches to such topics as quasi-analytic classes, Fabry's gap theorem and the theory of lacunary series, amongst others.
      
    • We mention, however, his work on statistical group theory, much of which was undertaken jointly with Erdos.
      
    • Of course conjugacy classes of the symmetric group Sn on n letters are characterized by partitions of n, so the connection with number theory is clear.
      
    • outstanding in analytic number theory but not a good manager of a department.
      
    • Theory 86 (3) (1996), 253-254.',4)">4].
      
    • But this is only a part of the editorial work Turan undertook, being on the editorial boards of Acta Arithmetica, Archiv fur Mathematik, Analysis Mathematica, Compositio Mathematica, Journal of Number Theory, and essentially all Hungarian mathematical journals.
      

54. Van Lint biography
    
    • In the thesis, he developed a theory of generalized Hecke operators, being automorphic forms of not necessarily integral dimension, as defined by Klaus Wohlfahrt in his 1955 dissertation at Munster University.
      
    • This theory allowed him to derive, in a systematic way, many known results and some new ones.
      
    • He then became interested in number theory publishing papers in that topic such as (jointly with N G de Bruijn) On the number of integers ≥ x whose prime factors divide n (1962), (jointly with H E Richert) On primes in arithmetic progressions (1965), and (with P Erdos) On the number of positive integers ≤ x and free of prime factors > y (1966).
      
    • In 1971 he published the book Coding theory.
      
    • In 1975 he published Graph theory, coding theory and block designs written jointly with Peter Cameron.
      
    • The material has been well adapted to the chosen format, and the result is a brisk and entertaining introduction to some areas of common ground between graph theory, coding theory and design theory.
      
    • In 1982 van Lint published Introduction to coding theory.
      
    • This is a concise, self-contained and neat introduction to the subject of coding theory suitable for students of mathematics.
      
    • To some extent, it is an updated version of 'Coding theory', by the same author, but the scope is wider.
      
    • The next, written jointly with Gerard van der Geer, was Introduction to coding theory and algebraic geometry.
      
    • The book was based on lectures given in the seminar on "Coding Theory and Algebraic Geometry" held at Schloss Mickeln, Dusseldorf, 16-21 November 1987.
      
    • Journal of Combinatorial Theory A (1976-2004); Discrete Mathematics (1972-2004); .
      
    • IEEE Transactions Information Theory (1986-1989); .
      

55. Fresnel biography
    
    • During this period his work on optics convinced him of the validity of the wave theory of light which was, at that time, totally discarded in favour of the corpuscular theory.
      
    • By applying mathematical analysis to his work Fresnel removed many of the objections to the wave theory of light.
      
    • He neither knew of the wave theories that had been postulated by Huygens, Euler and Young, nor did he know of the latest developments in the corpuscular theory supported by the majority of scientists.
      
    • He published his first paper in October 1815 on his wave theory of light and made a first attempt to explain the phenomenon of diffraction.
      
    • At this stage he had carried out fairly similar investigations that Thomas Young had carried out between 1797 and 1799 in Cambridge, but Fresnel next moved forward to a new understanding by developing a theory based on a new mathematical formulation.
      
    • It was a great chance for Fresnel to put his revolutionary work before the world and he was very confident of his theory since his mathematical deductions from the one simple hypothesis led to results which he had verified experimentally giving a highly accurate agreement between theory and experimental evidence.
      
    • It was a committee which was not well disposed to the wave theory of light, most believing in the corpuscular model.
      
    • Indeed the bright spot was seen to be there exactly as Fresnel's theory predicted.
      
    • Fresnel was awarded the Grand Prix and his work was a strong argument for a wave theory of light.
      
    • However polarisation of light produced by reflection still provided a strong argument in favour of the corpuscular theory, since no explanation from a wave theory had ever been made.
      
    • Fresnel and Arago, now very confident that they could explain this effect with Fresnel's theory, undertook further work on polarisation and Fresnel discovered what was later called circularly polarised light.
      
    • Although Fresnel had made many converts to the wave theory of light, even from the most ardent of those previously believing in the corpuscular theory, his assertion that light is a transverse wave was a step too far for most.
      

56. Borel biography
    
    • the theory of measure, Borel's theory of divergent series, his theory of non-analytic continuation and the theory of quasi-analytic functions all derive from ideas which make their first appearance in this paper.
      
    • In 1909 Borel was appointed to a chair of Theory of Functions created specially for him at the Sorbonne and he went on to hold this professorship until 1941.
      
    • In [Enseignement mathematique 11 (1965), 1-95.',8)">8] Borel's mathematical work is divided into the following topics: Arithmetic; Numerical series; Set theory; Measure of sets; Rarefaction of a set of measure zero; Real functions of real variables; Complex functions of complex variables; Differential equations; Geometry; Calculus of probabilities; and Mathematical physics.
      
    • Borel created the first effective theory of the measure of sets of points.
      
    • This work, along with that of two other French mathematicians, Rene Baire and Henri Lebesgue, marked the beginning of the modern theory of functions of a real variable.
      
    • Borel, although not the first to define the sum of a divergent series, was the first to develop a systematic theory for a divergent series which he did in 1899.
      
    • After 1905 he became interested in the theory of probability.
      
    • He stressed the important and practical value of probability theory.
      
    • He preferred to elucidate these applications instead of looking for an axiomatization of probability theory.
      
    • In addition, between 1921 and 1927, Borel published a series of papers on game theory and became the first to define games of strategy.
      
    • a number of valuable contributions to the knowledge of Einstein's theory of relativity.
      
    • History Topics: The beginnings of set theory .
      
    • Paul Walker (A history of Game Theory) .
      

57. Eudoxus biography
    
    • Other topics that it is probable that he learnt about from Archytas include number theory and the theory of music.
      
    • Eudoxus made important contributions to the theory of proportion, where he made a definition allowing possibly irrational lengths to be compared in a similar way to the method of cross multiplying used today.
      
    • The theory developed by Eudoxus is set out in Euclid's Elements Book V.
      
    • It is difficult to exaggerate the significance of the theory, for it amounts to a rigorous definition of real number.
      
    • Number theory was allowed to advance again, after the paralysis imposed on it by the Pythagorean discovery of irrationals, to the inestimable benefit of all subsequent mathematics.
      
    • corresponds exactly to the modern theory of irrationals due to Dedekind, and that it is word for word the same as Weierstrass's definition of equal numbers.
      
    • analyses, first, the historical significance of the theory of proportions contained in Book V of Euclid's "Elements" and attributed to Eudoxus.
      
    • It then demonstrates the radical originality, relative to this theory, of the definition of real numbers on the basis of the set of rationals proposed by Dedekind.
      
    • Two conclusions: (1) there are not in Book V of the "Elements" the gaps perceived by Dedekind; (2) one cannot properly speak of an 'influence' of Eudoxus's ideas on Dedekind's theory.
      
    • This work developed directly out of his work on the theory of proportion since he was now able to compare irrational numbers.
      
    • Eudoxus was able to make Antiphon's theory into a rigorous one, applying his methods to give rigorous proofs of the theorems, first stated by Democritus, that .
      
    • We have still to discuss Eudoxus's planetary theory, perhaps the work for which he is most famous, which he published in the book On velocities which is now lost.
      
    • not only do we not have evidence for numerical data in the construction of Eudoxus's homocentric spheres but it would also be difficult how his theory could have survived a comparison with observational parameters.
      
    • Perhaps it is just too modern a way of thinking to wonder how Eudoxus could have developed such an intricate theory without testing it out with observational data.
      

58. Ehrenfest biography
    
    • Ehrenfest attended Boltzmann's lectures on the mechanical theory of heat during 1899-1900.
      
    • There he took Max Abraham's course on the electromagnetic theory of light and also attended courses by Stark, Walther Nernst, Schwarzschild and Zermelo.
      
    • In 1905 Ehrenfest published a paper on Planck's theory of black-body radiation.
      
    • An important paper was published by Ehrenfest in 1911 in Annalen der Physik on the essential features of quantum theory.
      
    • On his travels he learnt that Poincare had written a paper on quantum theory which gave similar results to those in his Annalen der Physik paper.
      
    • Ehrenfest's arguments were based both on Newton's celestial mechanics and also on Einstein's relativity theory.
      
    • He worked on quantum theory applying it to rotating bodies.
      
    • The modern theory of nonequilibrium thermodynamics brings together the molecular, collisional ideas of Boltzmann with the statistical ideas of Ehrenfest's to give a nonlinear, statistical theory.
      
    • Ehrenfest's graduate lectures consisted of a two-year course: Maxwell theory, ending with the theory of electrons and some relativity, one year; and statistical mechanics, ending with atomic structure and quantum theory the other.
      
    • When they met in Leiden [The lessons of the quantum theory (Amsterdam, 1986), 325.',7)">7]:- .
      
    • [Ehrenfest] and Bohr had much to talk about together -- from the current problems of quantum theory to the Icelandic sagas, from the stages of a child's development to the difference between genuine physicists and the other.
      
    • Ehrenfest was unhappy at the disagreement between Bohr and Einstein over quantum theory.
      

59. Amitsur biography
    
    • the editors divide Amitsur's work into four main areas: general ring theory, structure theory of rings with polynomial identities, combinatorial theory of PI-algebras and theory of division algebras.
      
    • Vesselin Drensky, reviewing the work, writes about the results on general ring theory:- .
      
    • Embarking on his mathematical career at the time when ring theorists were searching for a general structure theory in the spirit of invariant theory, representation theory and theory of finite-dimensional algebras, Amitsur breathed life into the new theory and developed a body of theorems which were to provide inspiration for a generation of ring theorists.
      
    • The results include the general theory of radicals, radicals of polynomial rings and related rings, algebras over nondenumerable fields, rings of quotients and Morita equivalence, elementary conditions on algebras (e.g.
      
    • For the section on the structure theory of rings satisfying a polynomial identity, Drensky explains that it contains:- .
      
    • The second volume contains a section on the combinatorial theory of PI-algebras which:- .
      
    • It continues with the remarkable result that every PI-algebra satisfies a power of the standard identity, the primeness property of the T-ideal of the polynomial identities of matrices, some striking properties of Capelli identities which led to the construction of new central polynomials which simplifies and sometimes extends various results in structure theory of PI-algebras and is closed with results on sequences of codimensions and cocharacters of PI-algebras.
      
    • The group of papers devoted to the theory of division algebras contains some of Amitsur's most influential results.
      

60. Lax Peter biography
    
    • In 1957 he published an extremely important paper Asymptotic solutions of oscillating initial value problems where the beginnings of the theory of Fourier integral operators appears.
      
    • for his groundbreaking contributions to the theory and application of partial differential equations and to the computation of their solutions.
      
    • In the 1950s and 1960s, Lax laid the foundations for the modern theory of nonlinear equations of this type (hyperbolic systems).
      
    • Scattering theory is concerned with the change in a wave as it goes around an obstacle.
      
    • Together with Phillips, Lax developed a broad theory of scattering and described the long-term behaviour of solutions (specifically, the decay of energy).
      
    • Their work also turned out to be important in fields of mathematics apparently very distant from differential equations, such as number theory.
      
    • He collaborated with Ralph S Phillips in writing Scattering theory published in 1967.
      
    • This is a well-organized treatment of scattering theory for the time evolution of systems of hyperbolic type.
      
    • In this monograph, written more than twenty years ago, we based our scattering theory on the wave equation rather than the Schrodinger equation.
      
    • Following up on a hint in Gelfand's address to the 1962 Stockholm International Congress, they showed that the Lax-Phillips scattering theory, applied to the wave equation appropriate to hyperbolic space, is a natural tool in the theory of automorphic functions.
      
    • In fact the use of scattering theory for automorphic functions was studied by Lax and Phillips in Scattering theory for automorphic functions (1976).
      
    • However, Lax published other books between these two texts on scattering theory.
      
    • SIAM published Lax's Hyperbolic systems of conservation laws and the mathematical theory of shock waves in their Conference Series in Applied Mathematics in 1973.
      

61. Zaanen biography
    
    • In this thesis he studied the theory of the Sturm-Liouville two boundary value problem [Positivity 9 (3) (2005), 269-272.',1)">1]:- .
      
    • his study of the theory of linear integral equations.
      
    • Inspired by Marshall H Stone's book 'Linear Transformations in Hilbert space' he was able to extend the classical results of the theory of the linear integral transformations to the domain of the Orlicz spaces of measurable functions.
      
    • During this period Zaanen made essential contributions to several parts of mathematics, mainly in functional analysis, integration theory and Riesz space theory.
      
    • Next came An introduction to the theory of integration (1958).
      
    • This book is a presentation in textbook form of the modern approach to the theory of integration, not however after the manner of Bourbaki, but based on measure theory of abstract sets in the Caratheodory manner, combined with the general integral concepts due in the first place to P J Daniell, and elaborated and extended by M H Stone.
      
    • In 1967 Zaanen published a completely revised and extended edition of An introduction to the theory of integration under the title Integration.
      
    • This book is Volume I of a comprehensive two-volume account of the theory of Riesz spaces, or vector lattices.
      
    • from 1963 to 1965) did much to bring the theory to its present state of development.
      
    • Links with pure lattice theory and ring theory are also explored.
      
    • In 1989 Zaanen published Continuity, integration and Fourier theory.
      
    • His final book Introduction to operator theory in Riesz spaces was published in 1997.
      

62. Laplace biography
    
    • Not only had he made major contributions to difference equations and differential equations but he had examined applications to mathematical astronomy and to the theory of probability, two major topics which he would work on throughout his life.
      
    • Although Laplace soon returned to his study of mathematical astronomy, this work with Lavoisier marked the beginning of a third important area of research for Laplace, namely his work in physics particularly on the theory of heat which he worked on towards the end of his career.
      
    • after a general introduction concerning the principles of probability theory, one finds a discussion of a host of applications, including those to games of chance, natural philosophy, the moral sciences, testimony, judicial decisions and mortality.
      
    • The Exposition consisted of five books: the first was on the apparent motions of the celestial bodies, the motion of the sea, and also atmospheric refraction; the second was on the actual motion of the celestial bodies; the third was on force and momentum; the fourth was on the theory of universal gravitation and included an account of the motion of the sea and the shape of the Earth; the final book gave an historical account of astronomy and included his famous nebular hypothesis.
      
    • In it Laplace included a study of the shape of the Earth which included a discussion of data obtained from several different expeditions, and Laplace applied his theory of errors to the results.
      
    • Another topic studied here by Laplace was the theory of the tides but Airy, giving his own results nearly 50 years later, wrote:- .
      
    • It would be useless to offer this theory in the same shape in which Laplace has given it; for that part of the Mecanique Celeste which contains the theory of tides is perhaps on the whole more obscure than any other part..
      
    • The first book studies generating functions and also approximations to various expressions occurring in probability theory.
      
    • I have sought to establish that the phenomena of nature can be reduced in the last analysis to actions at a distance between molecule and molecule, and that the consideration of these actions must serve as the basis of the mathematical theory of these phenomena.
      
    • It is worth remarking that it was a new approach, not because theories of molecules were new, but rather because it was applied to a much wider range of problems than any previous theory and, typically of Laplace, it was much more mathematical than any previous theories.
      
    • After the publication of the fourth volume of the Mecanique Celeste, Laplace continued to apply his ideas of physics to other problems such as capillary action (1806-07), double refraction (1809), the velocity of sound (1816), the theory of heat, in particular the shape and rotation of the cooling Earth (1817-1820), and elastic fluids (1821).
      
    • Arago, who had been a staunch member of the Society, began to favour the wave theory of light as proposed by Fresnel around 1815 which was directly opposed to the corpuscular theory which Laplace supported and developed.
      
    • Many of Laplace's other physical theories were attacked, for instance his caloric theory of heat was at odds with the work of Petit and of Fourier.
      

63. Kleene biography
    
    • Kleene received a doctorate from Princeton for his thesis entitled A Theory of Positive Integers in Formal Logic in 1934.
      
    • Our object is to demonstrate empirically that the system is adequate for the theory of positive integers, by exhibiting a construction of a significant portion of the theory within the system.
      
    • By carrying out the construction on the basis of a certain subset of Church's formal axioms, we show that this portion at least of the theory of positive integers can be deduced from logic without the use of the notions of negation, class, and description.
      
    • Kleene's research was on the theory of algorithms and recursive function theory, an area which he created and retained an interest in throughout his life.
      
    • He developed the field of recursion theory with Church, Godel, Turing and others.
      
    • In this lecture he spoke about how his interpretation of intuitionistic number theory by means of a "realization" might extend to intuitionistic set theory.
      
    • Kleene's work on recursion theory helped to provide the foundations of theoretical computer science.
      
    • Kleene developed a diverse array of topics in computability: the arithmetical hierarchy, degrees of computability, computable ordinals and hyperarithmetic theory, finite automata and regular sets with enormous consequences for computer science, computability on higher types, recursive realizability for intuitionistic arithmetic with consequences for philosphy and for program correctness in computer science.
      
    • for three important papers which formed the basis for later developments in generalized recursion theory and descriptive set theory "Arithmetical predicates and function quantifiers", "On the forms of the predicates in the theory of constructive ordinals (second paper)", and "Hierarchies of number-theoretic predicates".
      
    • For his leadership in the theory of recursion and effective computability and for developing it into a deep and broad field of mathematical research.
      

64. Dilworth biography
    
    • He worked in lattice theory and it would not be an exaggeration to say that he was one of the main factors in the subject moving from being merely a tool of other disciplines to an important subject in its own right.
      
    • He began his studies in the 1930s by reading the first contributions to lattice theory which were by Dedekind.
      
    • By the time Dilworth began his research, the motivation behind much of lattice theory was to develop methods to attack problems in group theory.
      
    • The theory of groups provided much of the motivation and many of the technical ideas in the early development of lattice theory.
      
    • Indeed, it was the hope of many of the early researchers that lattice-theoretic methods would lead to the solution of some of the important problems in group theory.
      
    • Two decades later, it seems to be a fair judgement that, while this hope has not been realised, lattice theory has provided a useful framework for the formulation of certain topics in the theory of groups ..
      
    • Dilworth then goes on to explain where the main thrust in developing lattice theory subsequently come from and one has to say that, although he modestly does not say so, he played the major role in this development himself:- .
      
    • On the other hand, the fundamental problems of lattice theory have, for the most part, not come from this source but have arisen from attempts to answer the intrinsically natural questions concerning lattices and partially ordered sets; namely, questions concerning the decompositions, representations, imbedding, and free structure of such systems ..
      
    • The main topics in lattice theory to which Dilworth contributed are: Chain partitions in ordered sets, in particular his chain decomposition theorem for partially ordered sets; Uniquely complemented lattices; Lattices with unique irreducible decompositions; Modular and distributive lattices, in particular his covering theorem for modular lattices; Geometric and semimodular lattices; and Multiplicative lattices, where he studied, among other topics, abstract ideal theory, and the representation and embedding theorems for Noether lattices and r-lattices.
      
    • One important aspect of Dilworth's research was that he always attacked the big problems in lattice theory.
      
    • the construction of a set of structure invariants for certain classes of Boolean algebras, the characterisation of the lattice of congruence relations of a lattice, the imbedding of finite lattices in finite partition lattices, the word problem for free modular lattices, and a construction of a dimension theory for continuous, non-complemented, modular lattices, have an intrinsic interest independent of the problems associated with other algebraic systems.
      

65. Auslander biography
    
    • Remaining at Columbia he worked for his doctorate on group theory under the supervision of R L Taylor.
      
    • In 1975 he visited Mexico setting up a research group there on the representation theory of Artin algebras.
      
    • A visit to China in 1986 saw him help establish a successful research group on representation theory.
      
    • The authors of [Representation theory of algebras (Providence, RI, 1996), 1-15.',4)">4] write about his contributions to the representation theory of algebras:- .
      
    • [Maurice Auslander's] contributions to the modern representation theory of algebras as well as to other fields of mathematics were deep and influential.
      
    • When Maurice Auslander entered representation theory he was already a widely known mathematician with important contributions in commutative and homological algebra.
      
    • While on the theme of representation theory of algebras, Dieter Happel reviewing [M Auslander, Selected works of Maurice Auslander Part 2 (Providence, RI, 1999).',2)">2] writes:- .
      
    • the famous "Queen Mary Notes", which were written at a very early stage of modern representation theory of Artin algebras, and also early papers on the use of functors.
      
    • They show clearly the insight and influence of Auslander on the directions and developments of representation theory of Artin algebras.
      
    • (1) Homological dimension and local rings, (2) Ramification theory, (3) Functors, (4) Almost split sequences and Artin algebras, (5) Some topics in representation theory, (6) Lattices over general orders, (7) Tilting theory and homologically finite subcategories, (8) Almost split sequences and commutative rings, (9) Grothendieck groups and Cohen-Macaulay approximations, and (10) Relative theory and syzygy modules ..
      
    • For [Maurice], the difference between pragmatics and theory was only theoretical: his complete comprehension of complex situations, and only that, guided his actions.
      

66. Hall biography
    
    • Hall's interest in group theory came from Burnside's book which he was encouraged to read by Arthur Berry, the Assistant Tutor in Mathematics at King's College.
      
    • I began with Berry's encouragement to study the works of William Burnside, especially his magnificent treatise on the "Theory of Groups" and some of his later papers.
      
    • Fortunately for mathematics, and particularly group theory, he was not successful.
      
    • Despite its deficiencies, it shows that already Hall was way ahead of his time in his approach to group theory and certainly nobody at Cambridge could have been in a position to properly evaluate the work.
      
    • He took up this post in January 1927 and his first published papers are on the theory of correlation.
      
    • I asked Burnside's advice on topics of group theory which would be worth investigation and received a postcard in reply containing valuable suggestions as to worth-while problems.
      
    • Returning to Cambridge in September 1927 to take up the Fellowship at King's he made an important discovery in group theory, generalising the Sylow theorems for finite soluble groups to prove what are now called Hall's theorems.
      
    • Hall certainly made 'a bit of an effort' for in 1932 he wrote what is perhaps his most famous paper A contribution to the theory of groups of prime power order.
      
    • It is a beautiful paper which is one of the fundamental sources of modern group theory.
      
    • In it, in addition to its main aims of developing the theory of regular p-groups, Hall introduces the commutator calculus, commutator collection, and the connection between p-groups and Lie rings.
      
    • In 1955 he was one of the main speakers at the Edinburgh Mathematical Colloquium in St Andrews where he gave five lectures on Symmetric Functions in the Theory of Groups.
      
    • The subject I have in mind is symmetric functions, in relation to various branches of the theory of groups.
      
    • In particular he spoke about partitions and their connection to representation theory:- .
      
    • The paper has indeed proved highly influential and much of the rapid development of group theory in the 1960s was built on this foundation.
      
    • The ideas of these papers continue to be one of the main areas of group theory research.
      

67. Carleman biography
    
    • International success came, but his spectral theory was overshadowed by the abstract theory and he had also bad luck with his mean ergodic theorem.
      
    • Before his professorship in Lund he published about thirty papers, the majority treating of the problems in the theory of integral equations, and the theory of real and complex functions, where he gave extraordinary evidence of originality, penetration and capacity to use various methods of analysis.
      
    • Names such as Carleman inequality, Carleman theorems (Denjoy-Carleman theorem on quasi-analytic classes of functions, Carleman theorem on conditions of well-definedness of moment problems, Carleman theorem on uniform approximation by entire functions, Carleman theorem on approximation of analytic functions by polynomials in the mean), Carleman singularity of orthogonal system, integral equation of Carleman type, Carleman operator, Carleman kernel, Carleman method of reducing an integral equation to a boundary value problem in the theory of analytic functions, Jensen-Carleman formula in complex analysis, Carleman continuum, Carleman linearization or Carleman embedding technique, Carleman polynomials, Carleman estimate in the unique continuation problem for solutions of partial differential equations and Carleman system in the kinetic theory of gas are well-known in mathematics (see [Encyclopaedia of Mathematics 2 (Kluwer 1988), 25-26.
      
    • XII.17], [Nonlinear dynamical systems and Carleman linearization (World Scientific, 1991).',12)">12], [Carleman\'s formulas in complex analysis : theory and applications (Kluwer, 1993).
      
    • Different generalizations as well as some applications of these formulae to various problems of mathematics (problems of analytic continuation in the theory of functions), in theoretical and mathematical physics, in extrapolation and interpolation of signals having a finite Fourier spectrum, and results obtained by computer simulation on the elimination of noise in a given frequency band, are presented in the book [Carleman\'s formulas in complex analysis : theory and applications (Kluwer, 1993).
      
    • ',13)">13], which looks like an encyclopaedia on the theory and applications of the Carleman-type ideas and methods.
      
    • Carleman is also one of the authors of a mean ergodic theorem (see [Ergodic theory in the 1930\'s : a study in international mathematical activity (manuscript, Jan 2000).',17)">17], where more is written about priority questions).
      
    • Results on unique continuation for solutions to partial differential equations are important in many areas of applied mathematics, in particular in control theory and inverse problems.
      
    • In June 1947 Carleman participated in a CNRS meeting in Nancy and presented his theory there.
      
    • Carleman lectured at the Sorbonne in 1937 on Boltzmann's equation, which appears in the kinetic theory of gas, and published several papers on this subject.
      
    • Also his last book Mathematical problems of the kinetic theory of gas which deals with the mathematical aspects of the Boltzmann transport equation was published, after his death, in 1957 with some additional material submitted by L Carleson and O Frostman.
      

68. Luzin biography
    
    • At Moscow University Luzin studied under Bugaev, learning from him the theory of functions which was to influence greatly the direction his research would eventually take.
      
    • However in April 1908 he wrote of the joy he was finding in number theory (see [Historia Mathematica 25 (1998), 332-339.',9)">9]):- .
      
    • He worked for a year with Egorov and they went on to publish joint papers on function theory which mark the beginnings of the Moscow school of function theory.
      
    • The influence of Luzin's dissertation on the future development of the theory of functions cannot be overestimated.
      
    • Luzin's main contributions are in the area of foundations of mathematics and measure theory.
      
    • In the theory of boundary properties of analytic functions he proved an important result in 1919 on the invariance of sets of boundary points under conformal mappings.
      
    • From 1917 onwards, Luzin studied descriptive set theory.
      
    • The aim of set theory is a question of great importance: can we regard a line atomistically as a set of points: incidentally this question is not new, but goes back to the Greeks.
      
    • Much of Luzin's work on set theory involved the study of effective sets, that is sets which can be constructed without the axiom of choice.
      
    • But whereas the French had analysed set-theoretical constructions carried out with the help of the Axiom of Choice, Luzin went considerably further and considered difficulties arising within the theory of effective sets.
      
    • The study of effective sets that he embarked upon was pursued intensively for more than two decades and led to the solution of many important problems of set theory ..
      
    • Luzin's school was at its peak during the years 1922 to 1926, but then Luzin concentrated on writing his second monograph on the theory of functions and spent less time with the young mathematicians in the school.
      
    • From 1935 he headed the Department of the Theory of Functions of Real Variables at the Steklov Institute.
      
    • In 1931 Luzin himself turned to a new area when he began to study differential equations and their application to geometry and to control theory.
      

69. Ribenboim biography
    
    • During 1951 up to the spring of 1952 he attended lecture courses on Lie groups by Delsarte, algebraic numbers by Dieudonne and the theory of distributions by Laurent Schwartz.
      
    • He studied ideal theory and valuation theory with Krull between August 1953 and August 1956.
      
    • clearly written introduction to the theory of abelian ordered groups, assuming only an elementary knowledge of abelian group theory and topology.
      
    • This is a text suitable for an unorthodox course in algebraic number theory.
      
    • Instead of the customary material on ideal factorization and unit theorems, the reader will find such topics as p-adic logarithms, the Witt ring, infinite Galois theory, ordering of number fields and diophantine dimension (including Terjanian's celebrated counterexample).
      
    • The first of these is devoted to ramification theory in Galois extensions and the second to a proof of the theorem by Kronecker and Heinrich Weber on the abelian extensions of the field of rational numbers.
      
    • Many people will think of Ribenboim as a writer of superb number theory books.
      
    • Further wonderful books of number theory have excited many students and turned them on to mathematics.
      
    • Further number theory books are Catalan's Conjecture published in 1994, The new book of prime number records (1995) and Fermat's last theorem for amateurs (1999).
      
    • More recent books by Ribenboim are: The theory of classical valuations (1999); My numbers, my friends (2000); Classical theory of algebraic numbers (2001) In the Preface to the first of these Ribenboim explains the what he intends to study:- .
      
    • Ostrowski, Hasse, Schmidt, and others developed this theory.
      
    • Its coverage is vast: although much of the material in the lectures can be found in number theory textbooks, Ribenboim has integrated and consolidated so much related material from the literature that each lecture sparkles from its new treatment.
      

70. Arnold biography
    
    • His work on Hamiltonian dynamics, which includes cocreation of KAM (Kolmogorov- Arnold- Moser) theory and the discovery of "Arnold diffusion", made him world famous at an early age.
      
    • Arnold's contributions to the theory of singularities complement Thom's catastrophe theory and have transformed this field.
      
    • Arnold has also made innumerable and fundamental contributions to the theory of differential equations, symplectic geometry, real algebraic geometry, the calculus of variations, hydrodynamics, and magneto- hydrodynamics.
      
    • The areas are Dynamical Systems, Differential Equations, Hydrodynamics, Magnetohydrodynamics, Classical and Celestial Mechanics, Geometry, Topology, Algebraic Geometry, Symplectic Geometry, and Singularity Theory.
      
    • He published Problemes ergodiques de la mecanique classique (with A Avez) (1967), Ordinary differential equations (Russian) (1971), Mathematical methods of classical mechanics (Russian) (1974), Supplementary chapters to the theory of ordinary differential equations (Russian) (1978), Singularity theory (1981), Singularities of differentiable mappings (Russian) (with A N Varchenko and S M Gusein-Zade) (1982), Catastrophe theory (1984), Huygens and Barrow, Newton and Hooke (Russian) (1989), Contact geometry and wave propagation (1989), Singularities of caustics and wave fronts (1990), The theory of singularities and its applications (1991), Topological invariants of plane curves and caustics (1994), Lectures on partial differential equations (Russian) (1997), Topological methods in hydrodynamics (with B A Khesin) (1998), and Arnold problems (Russian) (2000).
      
    • In recognition of his basic contribution to the stability theory of Dynamical Systems, his pioneering work on singularity theory and seminal contributions to analysis and geometry.
      
    • for his deep and influential work in a multitude of areas of mathematics, including dynamical systems, differential equations, and singularity theory.
      
    • The face of modern mathematics would be unrecognisable without his work in dynamical systems, classical and celestial mechanics, singularity theory, topology, real and complex algebraic geometry, symplectic and contact geometry, hydrodynamics, variation calculus, differential geometry, potential theory, mathematical physics, superposition theory, etc.
      

71. Sierpinski biography
    
    • In 1903 the Department of Mathematics and Physics offered a prize for the best essay from a student on Voronoy's contribution to number theory.
      
    • I was awarded a gold medal by the university for work in a competition on the theory of numbers.
      
    • In fact it was in 1907 that Sierpinski first became interested in set theory.
      
    • Sierpinski began to study set theory and in 1909 he gave the first ever lecture course devoted entirely to set theory.
      
    • These books were The theory of irrational numbers (1910), Outline of Set Theory (1912) and The theory of numbers (1912).
      
    • Sierpinski edited the journal which specialised in papers on set theory.
      
    • From this period Sierpinski worked mostly in the area of set theory but also on point set topology and functions of a real variable.
      
    • In set theory he made important contributions to the axiom of choice and to the continuum hypothesis.
      
    • He was an assistant professor at Warsaw University, one of the leading experts in the world in the theory of the integral..
      
    • He was an assistant professor at Warsaw University and a distinguished author of works on set theory.
      
    • It was a great loss for Polish mathematics which was developing favourably in some fields such as set theory and topology ..
      
    • He retired in 1960 as professor at the University of Warsaw but he continued to give a seminar on the theory of numbers at the Polish Academy of Sciences up to 1967.
      

72. Tarski biography
    
    • In this paper he investigated set theory questions, and in fact set theory would be a continuing research interest for Tarski throughout his life.
      
    • Tarski's first major results were published in 1924 when he began building on the set theory results obtained by Cantor, Zermelo and Dedekind.
      
    • for the thesis that Alfred Tarski's original definition of truth, together with its later elaboration in model theory, is an explication of the classical correspondence theory of truth.
      
    • Tarski made important contributions in many areas of mathematics: set theory, measure theory, topology, geometry, classical and universal algebra, algebraic logic, various branches of formal logic and metamathematics.
      
    • He produced axioms for 'logical consequence', worked on deductive systems, the algebra of logic and the theory of definability.
      
    • Metamathematics, introduced by Hilbert in 1922 meaning "proof theory" as a part of his programme to establish the consistency of arithmetic, was transformed by Tarski when he introduced semantic methods leading to his development of model theory with its combination of semantic and syntactic relations.
      
    • He worked on model theory, mathematical decision problems and with universal algebra.
      
    • In A decision method for elementary algebra and geometry Tarski showed that the first-order theory of the real numbers under addition and multiplication is decidable which is in contrast, in a way which is really surprising to non-experts, to the results of Godel and Church who showed that the first-order theory of the natural numbers under addition and multiplication is undecidable.
      
    • In Undecidable theories Tarski showed that group theory, lattices, abstract projective geometry, closure algebras and others mathematical systems are undecidable.
      
    • In Ordinal algebras Tarski defines an algebra which captures the properties of the additive theory of order types.
      

73. Plemelj biography
    
    • He was taught analysis by von Escherich while Gegenbauer (appointed a professor in Vienna in 1893) and Mertens (appointed to Vienna in 1894) taught him algebra and number theory.
      
    • An important mathematical event occurred while he was at Gottingen, for that was the year in which Holmgren lectured on Fredholm's theory of integral equations at Gottingen.
      
    • Hilbert immediately saw the he importance of Fredholm's theory and work at Gottingen on this topic began immediately.
      
    • The contributions he made to integral equations and potential theory were brought together in a work he published in 1911 for which he was awarded the Prince Jablonowski Prize.
      
    • The equations are today important in a number of different fields, including neutron transport theory where a singular integral equation is encountered.
      
    • Plemelj's methods for solving the Riemann's problem were further developed by Nikolai Ivanovich Mushelisvili into the theory of singular integral equations.
      
    • Also important are Plemelj's contributions to the theory of analytic functions which he developed while investigation the problem of uniformization of algebraic functions [Drustvo Matematikov, Fizikov in Astronomov SR Slovenije (Ljubljana, 1973).',2)">2]:- .
      
    • Within the theory of differential equations he worked mostly on equations of the Fuchs type and on Klein's theorems.
      
    • This was a period during which he received many honours, for his book on potential theory not only led to him receiving the Prince Jablonowski Prize, which we mentioned above, but also the Richard Lieben Prize in 1912.
      
    • He used to hold a general course of mathematics and a three-year cycle of lectures on differential equations, the theory of analytic functions, and algebra including number theory.
      
    • They were The theory of analytic functions (1953), Differential and integral equations.
      
    • The theory and the application (1960), and Algebra and the number theory (1962).
      

74. Lesniewski biography
    
    • Although he had presented his first ideas on a new theory of classes which would avoid the paradoxes while he was in Lwow, it was during his time in Moscow that Lesniewski published his formal theory called mereology.
      
    • We give some further technical details of this theory below.
      
    • Led by Janiszewski this school was particularly interested in set theory, and the foundations of mathematics.
      
    • it is my intention to present, if possible, all Polish mathematicians working in the field of set theory, to which the journal is devoted.
      
    • His theories overcame the paradoxes of Russell in set theory.
      
    • His mathematical work concentrates on set theory, where his concern is the nature of a set.
      
    • The three major logical systems which Lesniewski developed were: Protothetic, a theory of propositions and propositional functors, similar in power to a theory of propositional types, providing an extended propositional calculus with quantified functional variables; Ontology, which is an axiomatised theory of common names based on protothetic which may be characterised as a cross between traditional term logic and modern type theory, containing, besides singular terms, also empty and plural terms and a host of other interesting features; and Mereology, which is an axiomatic extension of ontology for a theory of classes quite different from set theory providing a formal theory of part and whole similar to the calculus of individuals.
      

75. Andrews biography
    
    • He already knew that he wanted to undertake research in number theory but at the time he began his course it was prime numbers which fascinated him.
      
    • He took a graduate course on number theory given by Hans Rademacher and there he met the notion of a partition.
      
    • Andrews had published three papers by the time he had completed his thesis work: An asymptotic expression for the number of solutions of a general class of Diophantine equations (1961); A lower bound for the volume of strictly convex bodies with many boundary lattice points (1963); and On estimates in number theory (1963).
      
    • The year 1970 saw the publication of Andrews' first book Number theory.
      
    • This book is highly suitable for a course in elementary number theory.
      
    • About the same time as Andrews was making his major discovery of what is now called 'Ramanujan's lost notebook', his second book The theory of partitions was published.
      
    • The combinatorial and formal power series aspects of the subject have usually been treated in books on elementary number theory or combinatorial analysis.
      
    • Asymptotic problems concerning partitions have been considered in books on analytic or additive number theory.
      
    • My research centres on the theory of partitions and related areas.
      
    • In addition, I have written more than 250 scientific papers, and several books on number theory and the theory of partitions, as well as edited the collected papers of Percy A MacMahon.
      
    • He has served on the editorial boards of many journals: Discrete Mathematics, the Journal of Combinatorial Theory (A); The Ramanujan Journal; Integers: The Electronic Journal of Combinatorial Number Theory; the Proceedings of the Jangjeon Mathematical Society; Advanced Studies in Contemporary Mathematics; the Acharya Nagarjuna International Journal of Mathematics and Information Technology; and the International Journal of Number Theory.
      

76. Fenyo biography
    
    • After studying for his doctorate he submitted his thesis On the theory of mean values (Hungarian) in 1945.
      
    • The book opens with a discussion of elementary set theory, Lebesgue integration and Stieltjes integration and then goes on to the first major topic, the operator calculus, following the ideas of Mikusinski and others.
      
    • The authors prove Titchmarsh's theorem, which is central to the theory, and include a number of worked examples.
      
    • The next major topic is the theory of distributions.
      
    • The last topic in the book is the theory of non-linear ordinary differential equations, beginning with questions of existence consequences and stability.
      
    • The structure of the solutions is then examined, including singular points and limit cycles, and the book concludes with an account of the elementary theory of non-linear oscillations.
      
    • The unusual feature of this volume is the unified content: linear algebra, graph theory and network theory, with heavy reliance on linear algebra methods throughout.
      
    • The first section, comprising over half the book, deals with the theory of linear operators.
      
    • The matters discussed are metric and normed spaces with particular reference to Hilbert spaces, Hahn-Banach theory, operators (including inverse, dual and compact operators) and eigenvalues and eigenvectors.
      
    • Amongst the topics covered are Volterra integral equations and their relation with ordinary differential equations, Fredholm equations, self-conjugate and non-self-conjugate integral operators, and the associated eigenvalue theory.
      
    • These three volumes complete the encyclopaedic work (roughly 1700 pages) by Fenyo and Stolle on the theory and application of linear integral equations.
      
    • Their thesis is that the classical theory of linear integral equations produced many ideas for the later development of the theory of linear operators, and in turn functional analysis has helped the further development of integral equations.
      

77. Moser Jurgen biography
    
    • The difficulty that Moser had no money was overcome and he began to study the spectral theory of differential equations with Rellich as his advisor.
      
    • In particular Moser acquired an interest in astronomy and number theory through Siegel.
      
    • Moser worked in ordinary differential equations, partial differential equations, spectral theory, celestial mechanics, and stability theory.
      
    • Writing about his contributions to dynamics, the authors of [Ergodic Theory Dynam.
      
    • When combined with the work of Arnold this led to what is today called KAM Theory.
      
    • Based on initial ideas by Kolmogorov, presented in his famous address to the International Congress in 1954, this theory provided a stunning new approach to stability problems in celestial mechanics.
      
    • Integrable Hamiltonian systems and spectral theory (1983) arises from a course of lectures which Moser gave at the Scuola Normale Superiore in Pisa in 1981.
      
    • Here Moser examines inverse spectral theory for the one-dimensional Schrodinger equation with the aim, as he writes in the introduction, of showing that:- .
      
    • They aimed to write an introductory text with complete proofs using examples from physics and celestial mechanics to illustrate the theory.
      
    • The missing final two chapters would have been on KAM theory and unstable hyperbolic solutions.
      
    • for his contributions to the theory of Hamiltonian dynamical systems, especially his proof of the stability of periodic solutions of Hamiltonian systems having two degrees of freedom and his specific applications of the ideas in connection with this work.
      
    • He delivered a lecture Dynamical systems - past and present which contains an historical review of KAM theory followed by applications to particle accelerators.
      
    • The Stormer problem concerning charged particles in the Earth's magnetic field is discussed as is Hill's lunar theory.
      

78. Levi-Civita biography
    
    • By putting together Ricci-Curbastro's algorithm with some results from Lie's theory of transformation groups, Levi-Civita extended the theory of absolute invariants to more general cases than those considered by Ricci-Curbastro.
      
    • He is best known, however, for his work on the absolute differential calculus and with its applications to the theory of relativity.
      
    • In 1900 he published, jointly with Ricci-Curbastro, the theory of tensors in Methodes de calcul differential absolu et leures applications in a form which was used by Einstein 15 years later.
      
    • Weyl was to take up Levi-Civita's ideas and make them into a unified theory of gravitation and electromagnetism.
      
    • Levi-Civita's work was of extreme importance in the theory of relativity, and he produced a series of papers elegantly treating the problem of a static gravitational field.
      
    • This excellent monograph on the n-body problem in the general theory of relativity was prepared about ten years ago, but its appearance now is none the less timely for those who have worried themselves with one or another aspect of the problem.
      
    • He also wrote on the theory of systems of ordinary and partial differential equations.
      
    • In [Italian mathematics between the two world wars (Pitagora, Bologna, 1987), 125-141.',18)">18] the authors argue that Levi-Civita was interested in the theory of stability and qualitative analysis of ordinary differential equations for three reasons: his interest in geometry and geometric models; his interest in classical mechanics and celestial mechanics, in particular, the three-body problem; and his interest in stability of movement in the domain of analytic mechanics.
      
    • He added to the theory of Cauchy and Kovalevskaya and wrote up this work in an excellent book written in 1931.
      
    • Their results include the conception of the localized induction approximation for the induced velocity of thin vortex filaments, the derivation of the intrinsic equations of motion, the asymptotic potential theory applied to vortex tubes, the derivation of stationary solutions in the shape of helical vortices and loop-generated vortex configurations, and the stability analysis of circular vortex filaments.
      
    • Levi-Civita's work on asymptotic potential for slender tubes is at the core of the mathematical formulation of potential theory and capacity theory.
      
    • In 1933 Levi-Civita contributed to Dirac's equations of quantum theory.
      

79. Jeans biography
    
    • Although he would not return again to pure mathematics, Jeans wrote a paper on the theory of numbers while an undergraduate.
      
    • During this period of forced rest due to the tuberculosis, Jeans worked on his first major text The dynamical theory of gases.
      
    • the theory of the equipartition of energy and Maxwell's law, and the chapters in which he ..
      
    • Pure mathematicians will know what I mean when I describe the effect of the impact of Jeans' statistical mechanics on a young man's mind as comparable with the impact of a first introduction to the theory of functions of a complex variable.
      
    • The dynamical theory of gases is far more than an account of Jeans' own research.
      
    • We should also note that Jeans' paper was written after the Michelson-Morley experiment disproved the existence of the ether, and in the same year that Einstein published the special theory of relativity.
      
    • He published The Mathematical Theory of Electricity and Magnetism in 1908 while still in the United States.
      
    • Certainly Jeans continued to produce a remarkable output, and he wrote an excellent report on Radiation and Quantum Theory for the Physical Society in 1914.
      
    • Although World War I prevented Jeans' report from being widely read in Britain until after 1918, it then had a major impact on having quantum theory and the Bohr theory of the atom accepted by the British scientific community.
      
    • Jeans favoured, incorrectly as it turned out, the theory that the energy was the result of contraction while Eddington, correctly of course, believed it resulted from a slow process of annihilation of matter.
      
    • Instead he proposed a tidal theory based on a star passing close to the Sun and pulling matter out which condensed into the planets.
      
    • His technical books, other than those mentioned above, are Astronomy and Cosmogony (1928), and Introduction to the Kinetic Theory of Gases (1940).
      
    • Although Jeans never published original contributions to quantum theory, he showed in such popular books that he had kept up with the developments in this area.
      

80. Freundlich biography
    
    • Freundlich was awarded a doctorate by the University of Gottingen for a thesis on analytic function theory in 1910.
      
    • At this time Einstein was working on the general theory of relativity and, although he did not have the details of the theory worked out, he was beginning to understand some of its consequences.
      
    • Already there was evidence that the orbit of Mercury did not fit that predicted by Newton's theory of gravitation and in 1911 Einstein asked Freundlich to make accurate observations of Mercury's orbit.
      
    • Freundlich worked with Einstein in 1911 attempting to make the measurements of Mercury's orbit required to confirm the general theory of relativity.
      
    • It is important to realise how daring this publication by Freundlich was, for it claimed that Newton's theory of gravitation, so long held as one of the greatest achievements of the human mind, was wrong.
      
    • the first among fellow-scientists who has taken pains to put the theory to the test.
      
    • Freundlich was interested in measuring the deflection in a light ray passing close to the sun since again Einstein's incomplete theory of relativity suggested that this test could be used to check the validity of the theory and show that Newton's theory was incorrect.
      
    • He wrote his first book in 1916 following Einstein's publication of the general theory of relativity.
      
    • Freundlich's book Grundlagen der Einsteinschen Gravitationstheorie discussed the ways that the general theory of relativity could be tested by astronomical observations.
      
    • His occasional inability to comprehend these ideas had the salutary effect of making Einstein seek to simplify their mathematical formulation, for if one of Felix Klein's pupils could not make sense of his equations who could? Through his intimate contact with Einstein, Freundlich was the first to become thoroughly acquainted with the fundamental principles of the new gravitational theory and, as Einstein himself remarks in the foreword of Freundlich's book, he was particularly well qualified as its exponent because he had been the first to attempt to put it to the test.
      
    • However, one to Sumatra in 1929 was completely successful but the value which Freundlich found for the deflection of light was more than that predicted by Einstein's theory.
      

81. Mostowski biography
    
    • He entered Warsaw University after graduating from the Stefan Batory Gymnasium and it was at this time that he became especially interested in the foundations of mathematics, particularly mathematical logic and set theory.
      
    • However this subject did not please him so he attended the mathematics courses instead and undertook research on recursion theory and the axiom of choice.
      
    • He was an assistant professor at Warsaw University and a distinguished author of works on set theory.
      
    • Many of Mostowski's wartime results - on the hierarchy of projective sets, on arithmetically definable sets of natural numbers, and on consequences of the axiom of constructibility in descriptive set theory - were lost when his apartment was destroyed during the uprising.
      
    • The theory of models, especially models for set theory and for arithmetic; model products and their theories; families of models as topological spaces; models with indiscernible elements and the Ehrenfeucht-Mostowski theorem.
      
    • The Fraenkel-Mostowski permutation method and its application to proving the independence of the well-ordering principle from the linear-ordering principle in the theory of sets with atoms (urelements).
      
    • a popular, and at the same time completely rigorous, presentation of Godel's ideas: his theory reaching the conclusion that in every formal system of mathematics, satisfying certain very general conditions, there exist statements such that neither their truth nor falsity can be established within the system.
      
    • In 1952 Mostowski published a monograph Theory of sets written jointly with Kuratowski.
      
    • The book is extremely readable, due to a system of development of set-theory which combines the two approaches: the "naive" method followed by Cantor himself, and the formalistic treatment developed on the axiomatic method.
      
    • This intuition will continue to be necessary as no one system of axioms now known can claim to express the full intuitive freedom of construction available to "naive" set-theory.
      
    • It is this combination of methodologies that makes the book unique as a monograph on set-theory.
      
    • The topics include completeness, incompleteness, decidability, and undecidability theorems; computability, recursive functions, hierarchies, and functionals; intuitionistic logic and its interpretations; constructive mathematics, foundations of set theory, including Cohen's independence proofs; and finally model theory, ending in a special chapter on direct and reduced products.
      

82. Lyapin biography
    
    • At this time Lyapin's research was on the theory of groups and he published his first papers on this topic in 1936-37.
      
    • The volume under review is a survey of the algebraic theory of semigroups as of the year 1960 (topological semigroups are not considered except in the bibliography).
      
    • As the author points out, semigroups were considered early in the development of the theory of groups, but were put aside because of the inadequacy of the then available algebraic techniques.
      
    • The author clearly, and perhaps even a little defensively, states his belief in the importance of semigroups, maintaining that while group theory is the abstract form of the theory of one-to-one mappings of a set onto itself, semigroup theory is the abstract form of the theory of single-valued mappings of a set into itself.
      
    • Analysis, algebra, geometry, and topology being rich in examples of the latter, their abstract theory deserves recognition.
      
    • The aim of this monograph is to survey the main results concerning partial groupoids (i.e., partial algebras with one partial binary operation) and to investigate the relation of this theory to the theory of (complete) groupoids.
      
    • At the same time, in view of the wealth of material on partial groupoids (there are more than 140 references), the monograph will also be of interest as a reference book for specialists in the theory of partial algebras.
      
    • An English translation was published in 1997 under the title The theory of partial algebraic operations.
      
    • In this restrictive sense, the book can also be considered as an introduction to the theory of partial operations and partial algebras.
      
    • Exercises in group theory .
      

83. Mazya biography
    
    • Maz'ya lists his mathematical interests as: linear and non-linear PDEs; asymptotic and numerical methods for PDEs, including homogenization and boundary elements; spectral theory; harmonic analysis; approximation theory; wavelets; elasticity theory; function spaces; ill-posed problems; non-linear potential theory; fluid mechanics; and the history of mathematics.
      
    • He published (with Ju S Burago) Certain questions of potential theory and function theory for regions with irregular boundaries (Russian) in 1967.
      
    • The book is in two parts, the first is on the higher-dimensional potential theory and the solution of the boundary problems for regions with irregular boundaries while the second part is on the space of functions whose derivatives are measures.
      
    • In 1979 and 1981 he published his two part work Imbedding theorems for Sobolev spaces and On the theory of Sobolev spaces.
      
    • In 1985, together with Tatyana O Shaposhnikova, Maz'ya published Theory of multipliers in spaces of differentiable functions which was based on results discovered by the authors in 1979-80 and published in a number of papers.
      
    • In 1997 (with Vladimir Kozlov) Maz'ya published Theory of a higher-order Sturm-Liouville equation which Eastham summarises by writing that:- .
      
    • the authors have identified a special type of higher-order analogue of the hyperbolic Sturm-Liouville equation and they have developed a coherent theory based on the Green's function.
      
    • The exposition is very clear and detailed, illustrated by many examples of applications of the stated theory.
      
    • All the proofs are complete and rely on undergraduate university courses on real and complex analysis and some basic facts of functional analysis and of the theory of partial differential equations.
      
    • For example we list a few recent works without detailing the co-authors: Spectral problems associated with corner singularities of solutions to elliptic equations (2000); Asymptotic theory of elliptic boundary value problems in singularly perturbed domains (2000); Spectral problems associated with corner singularities of solutions to elliptic equations (2001); and Linear water waves (2002).
      
    • In 2009 he published, jointly with Tatyana O Shaposhnikova, the major volume Theory of Sobolev multipliers.
      
    • in recognition of his contributions to the theory of differential equations.
      

84. Clausius biography
    
    • Clausius's first paper on the mechanical theory of heat was published in 1850.
      
    • To understand the significance of Clausius's paper we should say a few words about the theory of heat which existed at this time.
      
    • This theory, called the caloric theory, was based on two axioms, namely that the heat in the universe is conserved and that the heat in a substance is a function of the state of the substance.
      
    • Laplace, Poisson, Sadi Carnot and Clapeyron had all developed the subject using this caloric theory as a basis.
      
    • However, in his 1850 paper, Clausius states clearly that the assumptions of the caloric theory are false and he gives two laws of thermodynamics to replace the incorrect assumptions.
      
    • The acceptance of the First Law of Thermodynamics showed immediately that both of the axioms of the caloric theory are false.
      
    • Still without giving the concept a name Clausius formulated, in a memoir of 1854, the rudiments of the theory of the concept of the measure of transformation equivalence he later called entropy.
      
    • A more bitter dispute between Tait and Clausius began in 1872 when Maxwell published Theory of Heat.
      
    • Clausius stated that the British were trying to claim more than they deserved for the theory of heat which, Clausius said, he alone was the discoverer.
      
    • We must also not give the impression that he only worked on thermodynamics for, after 1875, he concentrated on electrodynamic theory.
      
    • His theory was in fairly good agreement with most experimental results but, being based on absolute velocities, resulted in a charge at rest on the earth being subjected to a force due to the motion of the earth.
      
    • Despite the difficulties in the theory it played an important role in the development of electrodynamic theory.
      

85. Fredholm biography
    
    • Fredholm is best remembered for his work on integral equations and spectral theory.
      
    • The answer is immediate: potential theory ..
      
    • Two years later in Stockholm a lecture about the 'principal solutions' of Roux and their connections with Volterra's equation led to a vivid discussion Finally, after a long silence Fredholm spoke and remarked in his usual slow drawl: in potential theory there is also such an equation.
      
    • In 1900 a preliminary report on his theory of Fredholm integral equations was published as Sur une nouvelle methode pour la resolution du probleme de Dirichlet.
      
    • Of course Riemann, Schwarz, Carl Neumann, and Poincare had all solved problems which now came under Fredholm's general case of an integral equation; this was an indication of how powerful his theory was.
      
    • Fredholm's contributions quickly became well known to the world of mathematics when Holmgren lectured on Fredholm's theory at Gottingen in 1901.
      
    • Hilbert immediately saw the he importance of Fredholm's theory, and during the first quarter of the 20th century the theory of integral equations became a major research topic.
      
    • Fredholm published a fuller version of his theory of integral equations in Sur une classe d'equations fonctionelle which appeared in Acta Mathematica in 1903.
      
    • Hilbert extended Fredholm's work to include a complete eigenvalue theory for the Fredholm integral equation.
      
    • This work led directly to the theory of Hilbert spaces.
      
    • But the qualitative insight that the theory gave could also be achieved in a simpler way.
      
    • Fredholm was not what is usually called a brilliant speaker He talked slowly in a monotone voice and it could happen that he got embroiled in computational mistakes at the blackboard But this had little importance In fact, his lectures revealed an unusual mastery of his subject and he had the ability of communicating to his students a feeling for the unity and logic of physical theory which is so apparent in his own written work.
      
    • Fredholm received many honours for his mathematical contributions, including the V A Wallmarks Prize for the theory of differential equations in 1903, the Poncelet Prize from the French Academy of Sciences in 1908, and an honorary doctorate from the University of Leipzig in 1909.
      

86. Dickson biography
    
    • He worked on finite fields and extended the theory of linear associative algebras initiated by Wedderburn and Cartan.
      
    • He proved many interesting results in number theory, using results of Vinogradov to deduce the ideal Waring theorem in his investigations of additive number theory.
      
    • In 1901 his famous book Linear groups with an exposition of the Galois field theory was published.
      
    • presenting parts of the theory without the difficult calculations given in the published papers.
      
    • Dickson presented a unified, complete, and general theory of the classical linear groups - not merely over the prime field GF(p) as Jordan had done - but over the general finite field GF(pn), and he did this against the backdrop of a well-developed theory of these underlying fields.
      
    • Dickson published 17 books in addition to Linear groups with an exposition of the Galois field theory.
      
    • The 3-volume History of the Theory of Numbers (1919-23) is another famous work still much consulted today.
      
    • Dickson published Modern Elementary Theory of Numbers in 1939.
      
    • The first four chapters of this book furnish a brief but satisfactory introduction to the usual elementary topics of number theory, including a short account of binary quadratic forms.
      
    • 24 (1) (1997), 7-24.',17)">17] asks the question of how Dickson could attract 67 doctoral students into research in algebra and number theory when he was recognised as not particularly good at classroom teaching [Historia Math.
      
    • Dickson: Theory of Equations .
      

87. Kaluznin biography
    
    • (In fact, in later years, Lev Arkad'evich could still recall great lectures he had heard there on world history, Roman law, etc.) During this time he did some research in Galois theory.
      
    • Generalisations of Galois theory.
      
    • As a researcher, Kaluznin is best known for his work in group theory and in particular permutation groups.
      
    • These results have been included in many textbooks on group theory.
      
    • This theorem is widely used in the theory of group varieties, combinatorial group theory, and permutation group theory.
      
    • Kaluznin's other significant contributions to group theory include his work on stable automorphism groups, the structure of the variety of n-abelian groups, a classification of metabelian groups, work on locally normal groups of higher categories, and characterisations of the maximal subgroups of the symmetric and alternating groups.
      
    • Another area of algebra which had always attracted Kaluznin's interest was Galois theory.
      
    • His first papers in this area were devoted to Galois theory of normal extensions of fields.
      
    • The ideas in this work were later employed by Jacobson in his study of the Galois theory of arbitrary finite extensions of fields.
      
    • Developing further the methods of abstract Galois theory which had been initiated by Krasner, Kaluznin and his students were able to establish a Galois correspondence between Post algebras and Krasner algebras.
      
    • Though Lev Arkad'evich had never considered himself to be an expert in mathematical linguistics, automata theory, and applications of computers in algebra, his mathematical interests were broad and he did not hesitate to do research in these areas outside of his main strengths.
      

88. Sokhotsky biography
    
    • He submitted his Master's dissertation The theory of integral residues with some applications to the University of St Petersburg as part of the requirement for a Master's Degree in 1867 and, after defending his thesis in the following year, he was awarded the degree.
      
    • In this thesis Sokhotsky discussed the Cauchy integral and the theory of analytic functions, which he called "single-valued".
      
    • The courses he gave in 1869-70 included the first course on the theory of functions of a complex variable to be taught in that university.
      
    • His doctoral dissertation On definite integrals and functions with applications to expansion of series was an early investigation of the theory of singular integral equations.
      
    • It investigated in detail Cauchy-type integrals which played an important role in boundary value problems in the theory of functions of a complex variable.
      
    • One of the first to approach problems of the theory of singular integral equations, Sokhotsky in this work considered important boundary properties of integrals of the type of Cauchy and, essentially, arrived at the so-called formulas of I Plemelj (1908).
      
    • His lectures, especially on higher algebra, the theory of numbers, and the theory of definite integrals, were extremely successful.
      
    • He also wrote a number of papers on the theory of elliptic functions and on theta functions.
      
    • His work is important in the development of the theory of functions, in particular having applications in the theory of hypergeometric series and differential equations.
      
    • Other topics which Sokhotsky studied included Zolotarev's theory of divisibility of algebraic numbers in The application of the principle of the greatest divisor to the theory of divisibility of algebraic numbers (1898).
      

89. Eddington biography
    
    • Eddington made important contributions to the theory of general relativity.
      
    • He became interested in this theory, particularly since it provided an explanation for the previously noticed, but unexplained, advance of the perihelion of Mercury.
      
    • Its aim was to verify the bending of light passing close to the sun which was predicted by relativity theory.
      
    • The results from the Africa expedition provided the first confirmation of Einstein's theory that gravity will bend the path of light when it passes near a massive star.
      
    • He used these lectures as a basis for his book Mathematical Theory of Relativity which was published in 1923.
      
    • In addition to his work in relativity theory Eddington also did important work on the internal structure of stars.
      
    • He discovered the mass-luminosity relationship for stars, he calculated the abundance of hydrogen, and he produced a theory to explain the pulsation of Cepheid variable stars.
      
    • Jeans, however, favoured the theory that the energy was the result of contraction.
      
    • Eddington had a fascination with the fundamental constants of nature and produced some surprising numerical coincidences most of which were published after his death in Fundamental Theory (1946), a book prepared for publication by Whittaker.
      
    • In [Eddington\'s search for a fundamental theory : a key to the universe (Cambridge, 1994).',9)">9] Kilmister delves deeply into the ideas which led Eddington to the theories he put forward in Fundamental Theory in attempting to unite quantum mechanics and general relativity.
      
    • Mathematical Theory of Relativity Preface .
      
    • Mathematical Theory of Relativity Introduction .
      

90. Kolmogorov biography
    
    • Kolmogorov's results from his work by the Lake were published in 1931 and mark the beginning of diffusion theory.
      
    • His monograph on probability theory Grundbegriffe der Wahrscheinlichkeitsrechnung published in 1933 built up probability theory in a rigorous way from fundamental axioms in a way comparable with Euclid's treatment of geometry.
      
    • After mentioning the highly significant paper Analytic methods in probability theory which Kolmogorov published in 1938 laying the foundations of the theory of Markov random processes, they continue to describe:- .
      
    • his ideas in set-theoretic topology, approximation theory, the theory of turbulent flow, functional analysis, the foundations of geometry, and the history and methodology of mathematics.
      
    • He thus demonstrated the vital role of probability theory in physics.
      
    • These are on the theory of dynamical systems with applications to Hamiltonian dynamics.
      
    • These papers mark the beginning of KAM-theory, which is named after Kolmogorov, Arnold and Moser.
      
    • Kolmogorov addressed the International Congress of Mathematicians in Amsterdam in 1954 on this topic with his important talk General theory of dynamical systems and classical mechanics.
      
    • 22 (1) (1990), 31-100.',10)">10] notes Kolmogorov's major part in setting up the theory to answer the probability part of Hilbert's Sixth Problem "to treat ..
      
    • by means of axioms those physical sciences in which mathematics plays an important part; in the first rank are the theory of probability and mechanics" in his 1933 monograph Grundbegriffe der Wahrscheinlichkeitsrechnung.
      

91. Huygens biography
    
    • By 1656 Huygens was able to confirm his ring theory to Boulliau and the results were reported to the Paris group.
      
    • However by 1665 even Fabri was persuaded to accept Huygens' ring theory as improving telescopes confirmed his observations.
      
    • In the Horologium Oscillatorium sive de motu pendulorum (1673) he described the theory of pendulum motion.
      
    • Circular motion was a topic which Huygens took up at this time but he also continued to think about Descartes' theory of gravity based on vortices.
      
    • He seems to have shown signs of being unhappy with Descartes' theory around this time but he still addressed the Academie on this topic in 1669 although after his address Roberval and Mariotte argued strongly, and correctly, against Descartes' theory and this may have influenced Huygens.
      
    • He, quite wrongly, criticised Newton's theory of light, in particular his theory of colour.
      
    • In that year his Traite de la lumiere appeared, in it Huygens argued in favour of a wave theory of light.
      
    • It is not known what discussions went on between Huygens and Newton but we do know that Huygens had a great admiration for Newton but at the same time did not believe the theory of universal gravitation which he said .
      
    • In some sense of course Huygens was right, how can one believe that two distant masses attract one another when there is nothing between them, nothing in Newton's theory explains how one mass can possible even know the other mass is there.
      
    • The theory of the pendulum .
      
    • The theory of collisions .
      

92. Ledermann biography
    
    • Ledermann was able to use his expert knowledge of matrix theory to put the work of this group onto a sound mathematical footing.
      
    • As well as matrix theory he was involved in using statistical methods and he retained this interest in his later research publications.
      
    • As well as work in matrix theory which we have commented on above, Ledermann was especially known for his work in homology theory, group theory, and number theory.
      
    • This is evident, even in his work in what is usually thought of as one of the most abstract of topics, homology theory.
      
    • First I [EFR] will make some comments on Ledermann's book Introduction to the Theory of Finite Groups (1949).
      
    • It is the book from which I learnt group theory and, although it was not the only influence on my choice of research topic, it was a major factor in my decision to work on group theory problems for my doctoral dissertation.
      
    • The topics covered in the book look fairly standard but one has to remember that in the 1940s there were few group theory texts and the concept of standard material for such courses did not exist.
      
    • Other books which Ledermann has written for undergraduates include Complex numbers (1960), Integral calculus (1964), Multiple integrals (1966), Introduction to group theory (1973), and Introduction to group characters (1977).
      
    • This last volume, which still shows Schur's influence, strikes a good balance between the abstract approach to representation theory emphasising modules, and the concrete approach built around matrices.
      

93. Lehmer Derrick biography
    
    • DNL was a professor of mathematics at Berkeley who was interested in number theory and mechanical computation.
      
    • He attended school in Berkeley but it was his father who had the greatest influence on him, and even at a very young age he became involved in his father's ideas in number theory and particularly his interest in constructing machines to assist with number theory calculations.
      
    • He was now highly involved with his father's ideas so on the one hand he studied physics courses, while on the other hand he helped his father both with the number theory computations he was undertaking and with the mechanical ideas that he was developing to help him make these calculations.
      
    • His dissertation, which was supervised by Tamarkin, was An Extended Theory of Lucas' Functions.
      
    • Although the computer worked most of the time computing trajectories for ballistics problems, on some weekends the Lehmers used it to solve certain number theory problems using it as an electronic sieve [Acta Arith.
      
    • The chapter headings are: Lucas' functions; Tests for primality; Continued fractions; Bernoulli numbers and polynomials; Diophantine equations; Numerical functions; Matrices; Power residues; Analytic number theory; Partitions; Modular forms; Cyclotomy; Combinatorics; Sieves; Equation solving; Computing techniques; and Miscellaneous.
      
    • His most famous monograph was Guide to Tables in the Theory of Numbers.
      
    • A descriptive account is given of existing tables in the theory of numbers; this is set forth in such a way as to indicate clearly what each table contains.
      
    • Prolific in research, you have made far-reaching contributions to number theory.
      
    • With great energy and enthusiasm, you demonstrated how, in both theory and practice, computers can be an invaluable tool in testing conjectures.
      
    • He was a pioneer in the application of mechanical methods, including digital computers, to the solution of problems in number theory and he talked about some of the methods used to factorise numbers including: factor tables, trial division, Legendre's method, factor stencils, the continued fraction method, Fermat's method, methods based on quadratic forms, and Shanks' method.
      
    • As a lecturer [Lehmer] was much appreciated not only for his classical scholarship in mathematics and number theory, but also for his dry sense of humour and wit.
      

94. Bohr Niels biography
    
    • He received his Master's degree from the University of Copenhagen in 1909 and his doctorate in May 1911 for a thesis entitled Studies on the electron theory of metals.
      
    • It does not seem possible at the present stage of the development of the electron theory to explain the magnetic properties of bodies from this theory.
      
    • In the last years he has worked out a theory of the structure of atoms, which seems to be quite a bit more firmly founded than anything which has existed up to now.
      
    • On 24 July 1912, with his paper still unfinished, Bohr left Rutherford's group in Manchester and returned to Copenhagen to continue to develop his new theory of the atom, completing the work in 1913.
      
    • The same year he published three papers of fundamental importance on the theory of atomic structure.
      
    • He talked of atomic stability and electrodynamic theory giving an account of the origins of quantum theory, the hydrogen spectrum, explaining the relationships between the elements.
      
    • His explanation covered the absorption and excitation of spectral lines and the correspondence principle which he had set out in three papers On the quantum theory of spectra between 1918 and 1922.
      
    • as a fundamentally new interpretation of the foundations of quantum theory.
      
    • It was Bohr's view of quantum theory which was eventually to become accepted.
      
    • Bohr's other major contributions, in addition to quantum theory, include his theoretical description of the periodic table of elements around 1920, his theory of the atomic nucleus being a compound structure in 1936, and his understanding of uranium fission in terms of the isotope 235 in 1939.
      

95. Tait biography
    
    • It was the physical insight which Hamilton's quaternion differential calculus then gave which impressed Tait and he began to work hard developing a physical theory.
      
    • The idea led Tait, Thomson and Maxwell to begin to work on knot theory since the basic building blocks, in Thomson's vortex atom theory, would be the rings knotted in three dimensions.
      
    • By Helmholtz's theory of a perfect fluid, these knotted rings, although they could be distorted, would retain the 'same knot' as a circular knotted piece of string that can be moved around yet the form of the knot remains an invariant.
      
    • Soon they discovered Listing's 1847 contributions to knot theory.
      
    • Tait, although at first unconvinced by Thomson's vortex atom theory, began to include the theory in his lecture courses at Edinburgh in the early 1870s and he gave popular lectures describing the theory.
      
    • Without any rigorous theory, which would have been well beyond nineteenth century mathematics, Tait began to classify knots using his mathematical and geometrical intuition.
      
    • His proof is fallacious and, sadly, he did not relate colouring of graphs to the knot theory he had considered a few years earlier.
      
    • Thomson suggested that he work on the kinetic theory of gases and between 1886 and 1892 Tait published more than 20 papers on the topic.
      
    • A more bitter dispute between Tait and Clausius began in 1872 when Maxwell published his Theory of Heat.
      
    • Clausius stated that the British were trying to claim more than they deserved for the theory of heat which, given Tait's writing, was a fair comment.
      

96. Quillen biography
    
    • Frank Adams had formulated a conjecture in homotopy theory which Quillen worked on.
      
    • Quillen approached the Adams conjecture with two quite distinct approaches, namely using techniques from algebraic geometry and also using techniques from the modular representation theory of groups .
      
    • The techniques using modular representation theory of groups were used by Quillen to great effect in later work on cohomology of groups and algebraic K-theory.
      
    • He received the award as the principal architect of the higher algebraic K-theory in 1972, a new tool that successfully used geometric and topological methods and ideas to formulate and solve major problems in algebra, particularly ring theory and module theory.
      
    • Algebraic K-theory is an extension of ideas of Grothendieck to commutative rings.
      
    • Grothendieck's ideas were used by Atiyah and Hirzebruch when they created topological K-theory.
      
    • Clearly Quillen's year spent in Paris under Grothendieck's influence and at Princeton working with Atiyah were important factors in Quillen's development of algebraic K-theory.
      
    • he borrowed techniques from homotopy theory, and in a completely novel way.
      
    • Higher algebraic K-theory is effectively built there from first principles and, in 63 pages, reaches a state of maturity that one normally expects from the efforts of several mathematicians over several years.
      
    • Mathematical talent tends to express itself either in problem solving or in theory building.
      
    • In 2000 the journal K-Theory issued a special part dedicated to Quillen on the occasion of his sixtieth birthday.
      
    • In the same year he gave the Erdos Colloquium at the University of Florida on Module theory for nonunital rings.
      
    • On 22 May 2006 the 39th K-theory Day at Oxford was set up to celebrate Quillen's 65th birthday.
      
    • Jacek Brodzki lectured on Analysis and geometry on discrete groups, Mathai Varghese lectured on T-duality and non-commutative geometry, Joachim Cuntz lectured on K-theory for locally convex algebras, and Eric Friedlander closed the proceedings with the lecture Dan and me: looking back at some of Dan's remarkable mathematics.
      

97. Northcott biography
    
    • Sometimes he tried to reconstruct proofs of results that he had learnt as a student; at other; he attempted to build up a theory of integration for functions with values in a Banach space.
      
    • He recorded his results about this theory in a notebook that he kept in his gas-mask case.
      
    • Two papers he published in 1949, An inequality in the theory of arithmetic on algebraic varieties and A further inequality in the theory of arithmetic on algebraic varieties developed Weil's ideas.
      
    • The first was Ideal theory (1953).
      
    • This well-written book provides a self-contained treatment of certain portions of the modern theory of ideals in Noetherian rings, including the elements of the theory of local rings.
      
    • No previous knowledge whatsoever of ring theory is assumed, and beginners to the subject will find here a very readable account.
      
    • the book will encourage many who would not otherwise have done so to study ideal theory and algebraic geometry.
      
    • Here is Northcott's Ideal Theory Preface.
      
    • for a first course of homological algebra, assuming only a knowledge of the most elementary parts of the theory of modules.
      
    • It focuses on the construction of the tensor, exterior and symmetric algebras of a module over a commutative ring and, by bringing out some of their relationships, develops the theory of several associated structures.
      
    • Northcott: "Ideal theory" .
      

98. Thom biography
    
    • Rene Thom is known for his development of catastrophe theory, a mathematical treatment of continuous action producing a discontinuous result.
      
    • The foundations of the theory of cobordism, for which Thom later received a Fields Medal, already appear in his doctoral thesis.
      
    • It is as the inventor of catastrophe theory that Thom is best known but his earlier work had made him well known before he worked on catastrophe theory.
      
    • His work on topology, in particular on characteristic classes, cobordism theory and the Thom transversality theorem led to his being awarded a Fields medal in 1958.
      
    • Hopf, who awarded the Fields Medal to Thom in Edinburgh, pointed in his presentation address to the importance of Thom's theory:- .
      
    • That made me leave the strictly mathematical world and tackle more general notions, like the theory of morphogenesis, a subject which interested me more and led me towards a very general form of 'philosophical' biology.
      
    • Thom's theory is an attempt to describe, in a way that is impossible using differential calculus, those situations in which gradually changing forces lead to so-called catastrophes, or abrupt changes.
      
    • The theory has widespread application in the physical and biological sciences and in the social sciences.
      
    • Presented by Thom in Structural Stability and Morphogenesis (1972), the theory has since been developed by many mathematicians.
      
    • However, writing in [Fields Medallists Lectures (Singapore, 1997), 71-76.',6)">6], Thom explains why the theory which was marked by enormous popular success has fallen from favour:- .
      
    • It is a fact that catastrophe theory is dead.
      
    • For as soon as it became clear that the theory did not permit quantitative prediction, all good minds ..
      

99. Zermelo biography
    
    • Zermelo began to work on the problems of set theory, in particular taking up Hilbert's idea to head towards a resolution of the problem of the continuum hypothesis.
      
    • In 1902 Zermelo published his first work on set theory which was on the addition of transfinite cardinals.
      
    • Although Zermelo certainly gained fame for his proof of the well ordering property, set theory at this time was in the rather unusual position that many mathematicians rejected the type of proofs that Zermelo had discovered.
      
    • Zermelo made other fundamental contributions to axiomatic set theory which were partly a consequence of the criticism of his first major contribution to the subject and partly because set theory began to become an important research topic at Gottingen.
      
    • The set theory paradoxes first appeared around 1903 with the publication of Russell's paradox.
      
    • Rather it prompted him to make the first attempt to axiomatise set theory and he began this task in 1905.
      
    • The resulting system, with ten axioms, is now the most commonly used one for axiomatic set theory.
      
    • It enables the contradictions of set theory to be eliminated yet the results of classical set theory excluding the paradoxes can be derived.
      
    • His health was poor but his position was helped by the award of a prize of 5000 marks for his major contributions to set theory.
      
    • History Topics: The beginnings of set theory .
      
    • Other Web sitesPaul Walker (A history of Game Theory) .
      

100. Gelfond biography
     
     • During 1929-30 he taught mathematics at Moscow Technological College but already he had published some important papers: The arithmetic properties of entire functions (1929); Transcendental numbers (1929); and An outline of the history and the present state of the theory of transcendental numbers (1930).
       
     • After his return to Russia, Gelfond taught mathematics from 1931 at Moscow State University where he held chairs of analysis, theory of numbers and the history of mathematics.
       
     • In addition to his important work in the number theory of transcendental numbers, Gelfond made significant contributions to the theory of interpolation and the approximation of functions of a complex variable.
       
     • to show the contemporary state of the theory of transcendental numbers, to exhibit the fundamental methods of this theory, to present the historical course of development of these methods, and to show the connections which exist between this theory and other problems in the theory of numbers.
       
     • This book is very much in the spirit of the modern Russian school concerned with the so-called constructive theory of functions, approximative methods for the solution of differential equations, and so forth.
       
     • In 1962 Gelfond published the book Elementary methods in the analytic theory of numbers written jointly with Linnik.
       
     • The book covers a great variety of topics in number theory, and the unifying feature is that all are treated by methods conventionally called elementary.
       
     • In broad terms this means that problems are attacked by direct methods within the framework of the problems themselves, without the use of extraneous disciplines such as the theory of functions of a complex variable, Fourier analysis, trigonometric sums.
       
     • The chapter titles of this book are: Residues; Singular points and series representations of a function; Expansion of a function in a series and properties of the gamma function; Some functional identities and asymptotic estimates; and Laplace transformation and some problems which are solved by the use of residue theory.
       

101. Wigner biography
     
     • Wigner's thesis contains the first theory of the rates of association and dissociation of molecules.
       
     • He also told me that this had to do with group theory and that I should read a book on group theory and then work it out and tell him.
       
     • However, he asked von Neumann for advice on the mathematical difficulties and was told to read about the theory of group characters in Schur's papers.
       
     • He also studied the representation theory of the symmetric group due to Frobenius and Burnside.
       
     • The theory, as von Neumann suggested, was exactly what he needed to develop a theory of the spectrum of atoms with n electrons.
       
     • He then began the work for which he is famous, namely applying group theory to quantum mechanics.
       
     • Wigner returned to Berlin after the year in Gottingen where he lectured on quantum mechanics, worked on writing his famous text Group theory and its application to the quantum mechanics of atomic spectra and continued his research.
       
     • In fact Wigner's book on the applications of group theory to quantum mechanics was not the first to appear, since Weyl published his a little before Wigner.
       
     • He has given a general theory of nuclear reactions and has made decisive contributions to the practical use of nuclear energy.
       
     • discrete symmetries and superselection rules in quantum mechanics, symmetry implications for atomic and molecular spectra, natural line-width theory, contrast of microscopic and macroscopic physics and of general relativity and quantum mechanics, explanation of why symmetry yields more information for quantum than for classical mechanics, philosophical questions such as what nature laws should be, limits on causality, and whether quantum mechanics could in principle explain life.
       
     • His important works include Nuclear Structure (1958) with L Eisenbud, The Physical Theory of Neutron Chain Reactors (1958) with A Weinberg, Dispersion Relations and Their Connection with Causality (1964), and Symmetries and Reflections (1967).
       

102. Brouwer biography
     
     • In particular Brouwer attacked Hilbert's fifth problem concerning the theory of continuous groups.
       
     • However, after studying Schonflies' report on set theory, he wrote to Hilbert:- .
       
     • Brouwer was elected to the Royal Academy of Sciences in 1912 and, in the same year, was appointed extraordinary professor of set theory, function theory and axiomatics at the University of Amsterdam; he would hold the post until he retired in 1951.
       
     • As is mentioned in this quotation, Brouwer was a major contributor to the theory of topology and he is considered by many to be its founder.
       
     • Controversy surrounded Cantor's general set theory because of the set-theoretic paradoxes or contradictions.
       
     • Point set theory was widely applied in analysis and somewhat less widely applied in geometry, but it did not have the character of a unified theory.
       
     • In 1918 he published a set theory developed without using the Principle of the Excluded Middle Founding Set Theory Independently of the Principle of the Excluded Middle.
       
     • Part One, General Set Theory.
       
     • Also in 1920 he published Intuitionistic Set Theory, then in 1927 he developed a theory of functions On the Domains of Definition of Functions without the use of the Principle of the Excluded Middle.
       

103. Ramsey biography
     
     • As well as starting up the new area of mathematics now called 'Ramsey theory', which we say more about below, he wrote on the foundations of mathematics, economics and philosophy.
       
     • His second simplification is to suggest simplifying Russell's theory of types by regarding certain semantic paradoxes as linguistic.
       
     • He accepted Russell's solution to remove the logical paradoxes of set theory arising from, for example, "the set of all sets which are not members of themselves".
       
     • This examines methods for determining the consistency of a logical formula and it includes some theorems on combinatorics which have led to the study of a whole new area of mathematics called Ramsey theory.
       
     • Harary describes this birth of Ramsey theory in [J.
       
     • Graph Theory 7 (1) (1983), 1-7.',8)">8] where he writes the following:- .
       
     • The celebrated paper of Ramsey [in 1930] has stimulated an enormous study in both graph theory ..
       
     • Most certainly 'Ramsey theory' is now an established and growing branch of combinatorics.
       
     • Graph Theory 7 (1) (1983), 9-13.',9)">9], it is now known that there is a more direct proof than that given by Ramsey, while the general case of the decision problem cannot be solved.
       
     • Graph Theory 7 (1) (1983), 9-13.',9)">9]:- .
       
     • Ramsey made a systematic attempt to base the mathematical theory of probability on the notion of partial belief.
       
     • In economics, Ramsey wrote two papers A contribution to the theory of taxation and A mathematical theory of saving.
       

104. Chung biography
     
     • I pulled out one of my magnetic subjects, Ramsey theory, that is guaranteed to get graduate students hooked on combinatorics because it is very pretty stuff.
       
     • I gave her a book and told her to read the chapter on Ramsey theory.
       
     • What she wrote was incredible! In just one week, from a cold start, she had a major result in Ramsey theory.
       
     • She had found her first original results in Ramsey theory and it led to the publication of her first paper On the Ramsey numbers N(3, 3, ..
       
     • Also in 1975 Chung published her first joint paper with Ron Graham On multicolor Ramsey numbers for complete bipartite graphs which appeared in the Journal of Combinatorial Theory.
       
     • Her interests are wide and among her nearly 200 publications there are contributions to spectral graph theory, extremal graphs, graph labelling, graph decompositions, random graphs, graph algorithms, parallel structures and various applications of graph theory in Internet computing, communication networks, software reliability, and discrete geometry.
       
     • In 1997 the American Mathematical Society published a major book Spectral graph theory by Chung.
       
     • the underlying mathematics of spectral graph theory through all its connections to the pure and applied, the continuous and discrete, can be viewed as a single unified subject.
       
     • Spectral graph theory studies how the spectrum of the Laplacian of a graph is related to its combinatorial properties.
       
     • This is Erdos on graphs and in it many of the problems and conjectures in graph theory made by Paul Erdos are listed.
       
     • It was based on an article Chung published on the same topic in the previous year in the Journal of Graph Theory.
       
     • Thus this text will be an important reference volume for the graph theory researcher.
       
     • She has made other substantial contributions to the mathematical community with her editorial work, serving on the editorial boards of Mathematics Research Letters, Random Structures and Algorithms, SIAM Journal on Discrete Mathematics, the Journal of Combinatorial Designs, SIAM Review, the Journal of Graph Theory, Annals of Applied Mathematics, the Journal of Combinatorial Optimization, Annals of Combinatorics, the Taiwanese Journal of Mathematics, the Journal of Computer and System Sciences, and Mathematical Systems Theory.
       
     • In addition she has served as Co-Editor-in-Chief of Advances in Applied Mathematics and the Electronic Journal of Combinatorics, and as Editor-in-Chief of Internet Mathematics and the Journal of Graph Theory.
       
     • In turn, these "massive graphs" shed insights and lead to new directions for random graph theory.
       

105. Feit biography
     
     • He had developed a strong interest in group theory and was advised to go to the University of Michigan at Ann Arbor to study for his doctorate under Richard Brauer.
       
     • Feit attended Brauer's seminar which was on the modular representations of finite groups and also took an informal reading course from Brauer on class field theory.
       
     • Feit graduated with a doctorate in 1955, awarded for the thesis Topics in the Theory of Group Characters.
       
     • Adrian Albert, chairmen of the Chicago Mathematics Department, decided to facilitate the ongoing work by organising a 'Finite Group Theory Year' in 1960-61.
       
     • a moment in the evolution of finite group theory analogous to the emergence of fish onto dry land.
       
     • Although he published around 100 other papers, his name will always be most closely associated with this one result, described by Zelmanov as "easily the best single theorem in group theory." However, his other contributions on finite group theory, character theory, and modular representation theory, are also impressive.
       
     • Towards the end of his career he added an interest in Galois theory to this list of interests.
       
     • He addressed the International Congresses of Mathematicians in Nice in 1970 on The Current Situation in the Theory of Finite Simple Groups.
       
     • His retirement from Yale in October 2003 was marked with the holding of a 'Conference on Groups, Representations and Galois Theory' in his honour.
       

106. Democritus biography
     
     • Democritus of Abdera is best known for his atomic theory but he was also an excellent geometer.
       
     • The second source is in the work of Epicurus but, in contrast to Aristotle, Epicurus is a strong believer in Democritus's atomic theory.
       
     • Certainly Democritus was not the first to propose an atomic theory.
       
     • In fact traces of an atomic theory go back further than this, perhaps to the Pythagorean notion of the regular solids playing a fundamental role in the makeup of the universe.
       
     • This was a remarkable theory which attempted to explain the whole of physics based on a small number of ideas and also brought mathematics into a fundamental physical role since the whole of the structure proposed by Democritus was quantitative and subject to mathematical laws.
       
     • Another fundamental idea in Democritus's theory is that nature behaves like a machine, it is nothing more than a highly complex mechanism.
       
     • Where do qualities such as warmth, colour, and taste fit into the atomic theory? To Democritus atoms differ only in quantity, and all qualitative differences are only apparent and result from impressions of an observer caused by differing configurations of atoms.
       
     • In his theory atoms are eternal and so is motion.
       
     • There was no place in his theory for divine intervention.
       
     • Democritus built an ethical theory on top of his atomist philosophy.
       
     • Heath points out that if Democritus carried over his atomic theory to geometrical lines then there is no dilemma for him since his cone is indeed stepped with atom sized steps.
       
     • 65 (2) (1981), 105-116.',10)">10], have come to the opposite conclusion, believing that Democritus made contributions to problems of applied mathematics but, because of his atomic theory, he could not deal with the infinitesimal questions arising.
       

107. Broglie biography
     
     • De Broglie's doctoral thesis Recherches sur la theorie des quanta (Researches on the quantum theory) of 1924 put forward this theory of electron waves, based on the work of Einstein and Planck.
       
     • It proposed the theory for which he is best known, namely the particle-wave duality theory that matter has the properties of both particles and waves.
       
     • But these two systems of physics could not remain detached from each other: they had to be united by the formulation of a theory of exchanges of energy between matter and radiation.
       
     • And I realised that, on the one hand, the Hamilton-Jacobi theory pointed somewhat in that direction, for it can be applied to particles and, in addition, it represents a geometrical optics; on the other hand, in quantum phenomena one obtains quantum numbers, which are rarely found in mechanics but occur very frequently in wave phenomena and in all problems dealing with wave motion.
       
     • De Broglie's theory of electron matter waves was later used by Schrodinger, Dirac and others to develop wave mechanics.
       
     • Among publications on many topics he published work on Dirac's theory of the electron, on the new theory of light, on Uhlenbeck's theory of spin, and on applications of wave mechanics to nuclear physics.
       
     • The last three mentioned books were published in English translations as Non-linear Wave Mechanics: A Causal Interpretation (1960), Introduction to the Vigier Theory of elementary particles (1963), and The Current Interpretation of Wave Mechanics: A Critical Study (1964).
       
     • The central question in de Broglie's life was whether the statistical nature of atomic physics reflects an ignorance of the underlying theory or whether statistics is all that can be known.
       

108. Jordan biography
     
     • Volumes 1 and 2 contain Jordan's papers on finite groups, Volume 3 contains his papers on linear and multilinear algebra and on the theory of numbers, while Volume 4 contains papers on the topology of polyhedra, differential equations, and mechanics.
       
     • Jordan was particularly interested in the theory of finite groups.
       
     • In fact this is not really an accurate statement, for it would be reasonable to argue that before Jordan began his research in this area there was no theory of finite groups.
       
     • Serret, Bertrand and Hermite had attended Liouville's lectures on Galois theory and had begun to contribute to the topic but it was Jordan who was the first to formulate the direction the subject would take.
       
     • To give an illustration of the way he tried to build up groups theory we will say a little about his contributions to finite soluble groups.
       
     • His work on group theory done between 1860 and 1870 was written up into a major text Traite des substitutions et des equations algebraique which he published in 1870.
       
     • This treatise gave a comprehensive study of Galois theory as well as providing the first ever group theory book.
       
     • His book brought permutation groups into a central role in mathematics and, until Burnside wrote his famous group theory text nearly 30 years later, this work provided the foundation on which the whole subject was built.
       
     • It would also be fair to say that group theory was one of the major areas of mathematical research for 100 years following Jordan's fundamental publication.
       
     • The publication of Traite des substitutions et des equations algebraique did not mark the end of Jordan's contribution to group theory.
       
     • History Topics: A history of group theory .
       

109. Cramer Harald biography
     
     • He began to produce a series of papers on analytic number theory, and he addressed the Scandinavian Congress of Mathematicians in 1922 on Contributions to the analytic theory of numbers detailing his work on the topic up to that time.
       
     • Cramer's work in prime numbers is put into the context of the history of prime number theory from Eratosthenes to the mid 1990s in [Harald Cramer Symposium, Stockholm, 1993, Scand.
       
     • It was not only through his work on number theory that Cramer was led towards probability theory.
       
     • In 1927 he published an elementary text in Swedish Probability theory and some of its applications.
       
     • In this classic of statistical mathematical theory, Harald Cramer joins the two major lines of development in the field: while British and American statisticians were developing the science of statistical inference, French and Russian probabilists transformed the classical calculus of probability into a rigorous and pure mathematical theory.
       
     • One finds treated such fields as number theory, function theory, mathematical statistics, probability and stochastic processes, demography, insurance risk theory, functional analysis and the history of mathematics.
       
     • Such highlights as the probabilistic method in the study of asymptotic properties of prime numbers, the spectral analysis of stationary processes, the mathematical foundation of inference and the fundamental work on risk theory all add up to a brilliant career as a scientist.
       

110. Sommerfeld biography
     
     • Hilbert, Hurwitz and Lindemann all lectured to Sommerfeld and, after attending a course by Hilbert on the theory of ideal numbers, he felt that abstract pure mathematics was the right subject for him.
       
     • As indicated, the direction of Sommerfeld's research was immediately influenced by Klein who at this time was heavily involved in applying the theory of functions of a complex variable, and other pure mathematics, to a range of physical topics from astronomy to dynamics.
       
     • Sommerfeld's first work under Klein's supervision was an impressive piece of work on the mathematical theory of diffraction.
       
     • His work on this topic contains important theory of partial differential equations.
       
     • In March 1895 Sommerfeld presented his habilitation thesis The mathematical theory of diffraction to Gottingen and became a privatdozent in mathematics.
       
     • The lectures Klein gave in 1895-96 on the spinning top led to Klein and Sommerfeld starting a joint project to write a four volume text on the theory of gyroscopes.
       
     • This would eventually be published in 1909-1910, the first two volumes dealing with the mathematical theory, while the final two volumes deal with applications to geophysics, astronomy and technology.
       
     • From 1911 his main area of interest became quantum theory.
       
     • This replaced an earlier theory due to Lorentz in 1905 based on classical physics.
       
     • He was able to explain features which were unexplained by the earlier classical theory.
       
     • [He] was at the forefront of the work in electromagnetic theory, relativity and quantum theory and he was the great systematizer and teacher who inspired many of the most creative physicists in the first thirty years of this century.
       

111. Cartan biography
     
     • He added greatly to the theory of continuous groups which had been initiated by Lie.
       
     • He turned to the theory of associative algebras and investigated the structure for these algebras over the real and complex field.
       
     • His work is a striking synthesis of Lie theory, classical geometry, differential geometry and topology which was to be found in all Cartan's work.
       
     • He applied Grassmann algebra to the theory of exterior differential forms.
       
     • He developed this theory between 1894 and 1904 and applied his theory of exterior differential forms to a wide variety of problems in differential geometry, dynamics and relativity.
       
     • He examined a space acted on by an arbitrary Lie group of transformations, developing a theory of moving frames which generalises the kinematical theory of Darboux.
       
     • Cartan further contributed to geometry with his theory of symmetric spaces which have their origins in papers he wrote in 1926.
       
     • one of the few instances in which the initiator of a mathematical theory was also the one who brought it to completion.
       
     • Cartan discovered the theory of spinors in 1913.
       
     • This was due partly to his extreme modesty and partly to the fact that in France the main trend of mathematical research after 1900 was in the field of function theory, but chiefly to his extraordinary originality.
       

112. Mazur Barry biography
     
     • Mazur received four prizes from the American Mathematical Society, namely the Veblen Prize for geometry in 1966, the Cole Prize for number theory in 1982, the Chauvenet Prize for exposition in 1994, and the Steele Prize for seminal contribution to research in 2000.
       
     • Mazur began his research career in geometric topology but has become one of the world's leading experts in number theory after working in algebraic geometry.
       
     • I came to number theory through the route of algebraic geometry and before that, topology.
       
     • His move towards number theory, and some of his remarkable contributions to that topic, were detailed in the citation for the Steele Prize which was awarded:- .
       
     • The proof of the Main Conjecture of Iwasawa theory by Mazur and Andrew Wiles, in "Class fields of abelian extensions of Q" (1984).
       
     • This was awarded for his work on the arithmetic theory of elliptic curves.
       
     • Of course his deep work in number theory is based on the concept of 'number'.
       
     • But what is the field of number theory like? Mazur gives his views in poetic fashion in Number Theory as Gadfly.
       
     • He writes that number theory:- .
       
     • number theory swarms with bugs, waiting to bite the tempted flower-lovers who, once bitten, are inspired to excesses of effort! .
       
     • AMS Cole Prize in Number Theory1982 .
       

113. Thomason biography
     
     • His supervisor at Princeton was John Moore and he wrote a dissertation on category theory in which he produced results which were to become fundamental tools in topology.
       
     • While working on conjectures of Quillen-Lichtenbaum connecting K-theory to etale cohomology Thomason produced what was first thought to be a remarkable proof.
       
     • Thomason then developed material which he had studied for his doctorate considering the homotopy theory of the category of small categories and the homotopy theory of the category of small symmetric monoidal categories.
       
     • We have already mentioned Thomason's results on the conjectures of Quillen-Lichtenbaum connecting K-theory to etale cohomology which he achieved during 1980-83.
       
     • A reviewer of his paper Algebraic K-theory and etale cohomology (1985) wrote:- .
       
     • This paper is one of the most important papers in algebraic K-theory since a paper by D Quillen ..
       
     • The author pushes the applications of stable homotopy and homotopical algebra to algebraic K-theory and algebraic geometry further than anyone else and his methods have exerted considerable influence on other workers in the field.
       
     • Thomason's work during the next three years was on equivariant algebraic K-theory.
       
     • He worked on the algebraic K-theory of algebraic group actions on schemes.
       
     • one of the most important and powerful tools in algebraic K-theory.
       
     • Few have had the simultaneous grasp of topology, algebraic geometry and K-theory that Thomason did.
       

114. Heisenberg biography
     
     • In fact by this time he had become interested in number theory and he read Kronecker's work and tried to find a proof of Fermat's Last Theorem.
       
     • He had read Weyl and also Bachmann's text which gave a complete survey of number theory and this was to be his intended research topic for his doctorate.
       
     • He avoided courses by Lindemann, however, so his mathematical interests moved from number theory to geometry.
       
     • However Pauli, who was at that time working on his major survey of the theory of relativity, advised him against doing research in that topic.
       
     • On atomic structure, however, Pauli explained, much needed to be done since theory and experiment did not agree.
       
     • There he worked with Born on atomic theory, writing a joint paper with him on helium.
       
     • As the fundamental factor of Heisenberg's theory can be put forward the rule set out by him with reference to the relationship between the position coordinate and the velocity of an electron, by which rule Planck's constant is introduced into the quantum-mechanics calculations as a determining factor.
       
     • It should also be mentioned that Heisenberg, when he applied his theory to molecules consisting of two similar atoms, found among other things that the hydrogen molecule must exist in two different forms which should appear in some given ratio to each other.
       
     • To those of us who participated in the development of atomic theory, the five years following the Solvay Conference in Brussels in 1927 looked so wonderful that we often spoke of them as the golden age of atomic physics.
       
     • Heisenberg published The Physical Principles of Quantum Theory in 1928.
       
     • These papers opened the way for others to apply quantum theory to the atomic nucleus.
       
     • Relativity and quantum theory were classed as "Jewish" and as a consequence Heisenberg's appointment to Munich was blocked.
       

115. Larmor biography
     
     • Larmor's contributions came at a time when there were major revolutions in physics with the passing of classical physics to be replaced by quantum theory and relativity.
       
     • He published three papers all entitled A dynamical theory of the electric and luminiferous medium between 1894 and 1897.
       
     • These papers presented his theory of the electron, which of course gained further weight in 1897 when J J Thomson experimentally identified the electron.
       
     • Between 1873 and 1894 British and American physicists were proponents of a theory which they almost all learned directly from J C Maxwell's book Treatise on electricity and magnetism (1873).
       
     • After 1897 only a few among them, including Heaviside, still adhered to that theory.
       
     • During these three years (1894-97) the most basic principles of Maxwell's theory of electromagnetism were abandoned, and the entire subject was reconstructed on a new foundation - the electron - by Joseph Larmor in consultation with George FitzGerald.
       
     • Warwick in [Archive for History of Exact Science 43 (1) (1991), 29-91.',14)">14] explains in detail how Larmor developed his theory.
       
     • as he made contact with other Maxwellians beyond Cambridge - especially with George FitzGerald - he came increasingly to make electromagnetic theory fundamental to his work.
       
     • His book of 1900, Aether and matter, Cambridge University Press, Cambridge, 1900, helped to establish a research school that guided the development of mathematical electromagnetic theory in Cambridge until the end of World War I.
       
     • It was difficult to ascertain how much he appreciated the new developments (especially quantum theory), because he was accustomed to adopt a pose which exaggerated his aloofness.
       
     • He wavered much over Einstein's theory of gravitation.
       
     • In the end he rejected, not only the curvature of space, but even the standpoint of the earlier special theory of relativity.
       

116. Eisenstein biography
     
     • In 1842 he bought a French translation of Gauss's Disquisitiones arithmeticae and, like Dirichlet, he became fascinated by the number theory which he read there.
       
     • He worked on the theory of forms with the aim of generalising the results obtained by Gauss in Disquisitiones arithmeticae for the theory of quadratic forms.
       
     • In his work on this topic Eisenstein used Kummer's theory of ideals.
       
     • The work of both Kummer and Eisenstein, and the rivalry which existed between the two in their work published in 1850 on the higher reciprocity laws, is discussed in [Number theory related to Fermat\'s last theorem (Boston, Mass., 1982), 31-43.',7)">7].
       
     • These two topics on which Eisenstein worked were both strongly motivated by Gauss's Disquisitiones arithmeticae and the paper [Algebraic number theory (Boston, MA, 1989), 463-469.',13)">13] discusses the copy of this work which Eisenstein owned from his days at school which is now in the mathematical library in Giessen.
       
     • In the paper [Algebraic number theory (Boston, MA, 1989), 463-469.',13)">13] Weil examines the annotations in the book made by Eisenstein and conjectures that Riemann received ideas in conversations with Eisenstein which led to his famous paper on the zeta function.
       
     • The third topic to which Eisenstein made a major contribution was the theory of elliptic functions.
       
     • Eisenstein, having laid the foundations for a theory of elliptic functions, was able to carry out much of his design for the building itself, and to indicate how he wished it completed.
       
     • developed his own independent analytic theory of elliptic functions, based on the technique of summing certain conditionally convergent series.
       
     • particularly concerning the transformation theory of theta-functions ..
       
     • Moreover, this case provides not merely an illuminating introduction to his theory, but also the simplest proofs for a series of results, originally discussed by Euler ..
       

117. Brauer biography
     
     • He entered the Technische Hochschule of Charlottenburg in February 1919 where he studied for a term before, having realised that his talents were in theory rather than practice, he transferred to the University of Berlin.
       
     • He gave the impression of developing the theory right there and then.
       
     • They had been a fellow students in one of Schur's courses on number theory.
       
     • This was the time when Brauer made his fundamental contribution to the algebraic theory of simple algebras.
       
     • a theory of central division algebras over a given perfect field, and showed that the isomorphism classes of these algebras can be used to form a commutative group whose properties gave great insight into the structure of simple algebras.
       
     • This work was to provide a background for the work of Paul Dirac in his exposition of the theory of the spinning electron within the framework of quantum mechanics.
       
     • Brauer carried Frobenius's theory of ordinary characters (where the characteristic of the field does not divide the order of the group) to the case of modular characters (where the characteristic does divide the group order).
       
     • He also studied applications to number theory.
       
     • Instead we spent many hours exploring examples of the representation theory ideas that were evolving in his mind.
       
     • It was in joint work with Nesbitt, published in 1937, that Brauer introduced the theory of blocks.
       
     • This he used to obtain results on finite groups, particularly finite simple groups, and the theory of blocks would play a big part in much of Brauer's later work.
       
     • Alperin also spoke of Brauer's thirteen years in Toronto (see [Pioneers of representation theory : Frobenius, Burnside, Schur, and Brauer (Providence, Rhode Island, 1999).',3)">3]):- .
       

118. Kelly Max biography
     
     • Kelly's first paper Single-space axioms for homology theory was based on the work of his doctoral thesis and published in 1959.
       
     • Eilenberg and Steenrod give a set of axioms for the homology theory of pairs of spaces and their maps, and prove that these axioms are categorical on triangular pairs.
       
     • Here we give a set of axioms for the homology theory of single spaces and their maps, that is, for absolute rather that relative homology.
       
     • Max has been a prominent figure in category theory for nearly four decades.
       
     • While categories merited only passing mention in Max's doctoral work, the subjects were very close to the kind of mathematics that led Eilenberg and Mac Lane to initiate category theory.
       
     • Shaun Wylie, well known for his book with Peter Hilton on homology theory, was Max's supervisor at Cambridge.
       
     • However, very soon, Max was led to general considerations within category theory itself, which were then applied to the problems at hand.
       
     • In this early work we can find the germ of many of Max's later keen interests: enriched category theory, coherence, and higher-dimensional universal algebra.
       
     • This all culminated in the meticulous Eilenberg-Kelly contribution to the 1965 LaJolla Conference whose Proceedings many of us see as representative of category theory reaching maturity as a subject in its own right.
       
     • We should also single out for mention the important book Basic concepts of enriched category theory which Kelly published in 1982.
       
     • In the interview reported in [Sydney Morning Herald (22 April 2003).',2)">2], Kelly tried to explain the importance of category theory in a way that non-mathematicians would understand.
       
     • Category theory sheds light on the relations between various aspects of mathematics and in doing so it brings unity and simplicity.
       

119. Fomin biography
     
     • It was not long before Fomin had proved some new results in the theory of infinite abelian groups, examining conditions for such a group to be the direct product of a periodic subgroup and a torsion free subgroup.
       
     • Kolmogorov suggested problems in the theory of dynamical systems for Fomin to investigate, but Fomin was also advised by Aleksandrov to look at some problems in point-set topology and he also began to work in this area.
       
     • Aleksandrov and Urysohn had made a conjecture in 1923 concerning necessary and sufficient conditions for a Hausdorff space to be compact and this was not proved until 1935 when M H Stone gave an exceedingly complicated proof using representation theory of Boolean algebras.
       
     • In 1964 Fomin became professor in the Department of the Theory of Functions and Functional Analysis and two years later he was appointed as a professor in the Department of General Control Problems.
       
     • One of the areas with which he is particularly associated is ergodic theory.
       
     • Fomin wrote a couple of papers with Gelfand and in the first of these, also published in 1950, they apply the theory of infinite dimensional representations of Lie groups to the theory of dynamical systems.
       
     • In this area he examined the theory of differentiable measures in infinite dimensional spaces and the theory of distributions.
       
     • He worked with a number of collaborators from 1973 on the writing of a monograph on measure theory and differential equations.
       
     • Some of the mathematical interests of Sergei Vasilovich were always close to some of mine (measure and ergodic theory); he supervised the translation of a couple of my books into Russian.
       
     • He wrote Elements of the theory of functions and functional analysis in two volumes.
       

120. Zeeman biography
     
     • In the following year I spend a sabbatical with Thom at the Institut des Hautes Etudes Scientifiques in Paris, where I learnt all about catastrophe theory.
       
     • Perhaps he is best known for his work on catastrophe theory for, although this theory was due initially to Rene Thom, it was Zeeman who brought it before the general public giving widespread publicity to applications of what was before that time thought of as pure mathematics.
       
     • In particular Zeeman pioneered the applications of catastrophe theory in the biological and behavioural sciences, as well as the physical sciences.
       
     • Among the books which Zeeman has published are the texts Catastrophe theory (1977), Geometry and perspective (1987) and Gyroscopes and boomerangs (1989).
       
     • One of his many memorable quotes, from his Catastrophe theory text, says much about mathematical philosophy:- .
       
     • A shorter introduction to catastrophe theory than his 1977 book was given by Zeeman in his beautifully written survey article Bifurcation and catastrophe theory [Contemp.
       
     • The article introduces catastrophe theory in a unified way giving both elementary and non-elementary aspects.
       
     • And amongst my applications of catastrophe theory I particularly liked buckling, capsizing, embryology, evolution, psychology, anorexia, animal behaviour, ideologies, committee behaviour, economics and drama.
       
     • I wanted to get my hands dirty, and make predictions, and get the experimentalists to test them, because I knew that the scientific community would never take a theory seriously unless it was capable of being tested experimentally.
       
     • Classical CD collections occupy the few shelves not crammed with tomes on Catastrophe Theory and other pet subjects.
       

121. Hilbert biography
     
     • Hilbert's first work was on invariant theory and, in 1888, he proved his famous Basis Theorem.
       
     • However Gordan was the expert on invariant theory for Mathematische Annalen and he found Hilbert's revolutionary approach difficult to appreciate.
       
     • At the time Klein received these two letters from Hilbert and Gordan, Hilbert was an assistant lecturer while Gordan was the recognised leading world expert on invariant theory and also a close friend of Klein's.
       
     • In 1893 while still at Konigsberg Hilbert began a work Zahlbericht on algebraic number theory.
       
     • The ideas of the present day subject of 'Class field theory' are all contained in this work.
       
     • but also fashioned new concepts that shaped the course of research on algebraic number theory for many years to come.
       
     • Making use of his results on integral equations, Hilbert contributed to the development of mathematical physics by his important memoirs on kinetic gas theory and the theory of radiations.
       
     • In the printed version of his paper, Hilbert added a reference to Einstein's conclusive paper and a concession to the latter's priority: "The differential equations of gravitation that result are, as it seems to me, in agreement with the magnificent theory of general relativity established by Einstein in his later papers".
       
     • In 1934 and 1939 two volumes of Grundlagen der Mathematik were published which were intended to lead to a 'proof theory', a direct check for the consistency of mathematics.
       
     • History Topics: The beginnings of set theory .
       
     • History Topics: The development of Ring Theory .
       

122. Einstein biography
     
     • This seemed to contradict classical electromagnetic theory, based on Maxwell's equations and the laws of thermodynamics which assumed that electromagnetic energy consisted of waves which could contain any small amount of energy.
       
     • Einstein's second 1905 paper proposed what is today called the special theory of relativity.
       
     • He based his new theory on a reinterpretation of the classical principle of relativity, namely that the laws of physics had to have the same form in any frame of reference.
       
     • As a second fundamental hypothesis, Einstein assumed that the speed of light remained constant in all frames of reference, as required by Maxwell's theory.
       
     • Einstein was not the first to propose all the components of special theory of relativity.
       
     • He made important contributions to quantum theory, but he sought to extend the special theory of relativity to phenomena involving acceleration.
       
     • This would be highly significant as it would lead to the first experimental evidence in favour of Einstein's theory.
       
     • Einstein called his new work the general theory of relativity.
       
     • After a number of false starts Einstein published, late in 1915, the definitive version of general theory.
       
     • Revolution in science - New theory of the Universe - Newtonian ideas overthrown.
       
     • Niels Bohr and Einstein were to carry on a debate on quantum theory which began at the Solvay Conference in 1927.
       

123. Hirzebruch biography
     
     • After serving as a Scientific Assistant at the University of Erlangen during 1951-52, he spent the two years 1952-54 at the Institute for Advanced Study in Princeton in the United States working with Armand Borel, Kunihiko Kodaira, and D C Spencer on topics such as sheaf theory, vector bundles, characteristic classes and Thom cobordism.
       
     • Garben-und Cohomologietheorie (1957), written with G Scheja, sets out the cohomology theory of sheaves.
       
     • In 1974 Hirzebruch, jointly with D Zagier, published The Atiyah-Singer theorem and elementary number theory.
       
     • Basically, the role of topology in number theory has progressed beyond the local methods such as p-adic theory to global methods such as intersection numbers of homology classes.
       
     • In particular develops enough of the theory of algebraic surfaces to allow an investigation of Hilbert modular surfaces.
       
     • for outstanding work combining topology, algebraic and differential geometry, and algebraic number theory; and for his stimulation of mathematical cooperation and research.
       
     • the proportionality theorem for complex homogeneous manifolds and (with Armand Borel) the general theory of characteristic classes of homogeneous spaces of compact Lie groups, .
       
     • complex K-theory and its spectral sequence and various geometrical applications (with M F Atiyah), .
       
     • the 'topological' proof of the Dedekind reciprocity theorem through 4-manifold theory and other fascinating relations between differential topology and algebraic number theory .
       
     • His ideas and discoveries - particularly in connection with Riemann-Roch theorems, characteristic classes, and K-theory - have contributed to the instigation of one of the most important developments in mathematics in the second half of the 20th century.
       

124. Blum biography
     
     • After her thesis, perhaps her next important piece of work was Towards a Mathematical Theory of Inductive Inference, Information and Control which she published jointly with her husband Manuel Blum.
       
     • Then, in 1988, she became a member of the Theory Group of the International Computer Science Institute in Berkeley.
       
     • Beginning in the late 1980s and continuing throughout the 1990s Blum, with several co-authors, has developed new directions in the theory of computation and complexity.
       
     • An important first contribution was Blum's 1989 paper Lectures on a theory of computation and complexity over the reals (or an arbitrary ring) which extended the theories of computation and computational complexity from the standard discrete situation to study how these ideas can be developed in continuous domains such as the real number system.
       
     • They proved that the Mandelbrot set is undecidable, a question which Turing theory does not allow one to even formulate.
       
     • In this book they argue that classical complexity theory, based on the Turing model, is inadequate for studying many problems and algorithms in modern scientific computing; the book then develops a complexity theory which can be applied to these areas.
       
     • The classical theory of computation had its origin in the work of logicians - of Godel, Turing, ..
       
     • Especially striking is the interplay of various mathematical disciplines such as algebraic number theory, algebraic geometry, logic, and numerical analysis, to mention a few.
       
     • The theory of computer science deals with counting numbers but not real numbers.
       
     • We've developed a parallel theory ..
       
     • Continuity is the mathematics of calculus and physics but there's never been a theory of computation that deals with this continuum.
       

125. Grassmann biography
     
     • In the Foreword of his Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (Linear Extension Theory, a new branch of mathematics) (1844) Grassmann described how he was led to these ideas starting around 1832.
       
     • He had to submit an essay on the theory of the tides as part of the examination.
       
     • He took the basic theory from Laplace's Mechanique celeste and from Lagrange's Mechanique analytique but he realised that he was able to apply the vector methods which he had been developing since 1832 (described in the preface to Die Lineale Ausdehnungslehre) to produce an original and simplified approach.
       
     • His essay Theorie der Ebbe und Flut was 200 pages long and introduced for the first time an analysis based on vectors, including vector addition and subtraction, vector differentiation, and vector function theory.
       
     • On the other hand it had shown Grassmann that his theory was widely applicable and he decided to spend as much time as he could spare on further developing his ideas on vector spaces.
       
     • He then develops the theory of linear independence in a way which is astonishingly similar to the presentation one finds in modern linear algebra texts.
       
     • Space theory may serve here as an example..
       
     • One cannot here go further than up to three independent directions (rules of change), while in the pure theory of extension their quantity can increase up to infinity.
       
     • Clifford algebras are used today in the theory of quadratic forms and in relativistic quantum mechanics.
       
     • Grassmann, however, went on to apply his new concepts to other situations, feeling that once people saw how the theory could be applied they would take it seriously.
       
     • But Grassmann also studied problems in physics, in particular publishing a theory of the mixing of colours in 1853 which contradicted that proposed by Helmholtz.
       
     • By the middle of the following year, however, he had returned to mathematics and his theory of extension deciding that rather than write a second volume, as he had originally intended, he would completely rewrite the work in an attempt to have its significance recognised.
       

126. Lopatynsky biography
     
     • His treatment of this special case of the algebraic theory of algebraic differential equations yields a well-rounded ideal theory of linear differential operators; in many respects it differs essentially from the treatment due to Ritt (for instance, ideals and sums of integral manifolds are defined differently).
       
     • He submitted his doctoral dissertation, dealing with an algebraic theory of rings of differential operators, to Moscow University, and he defended his thesis in 1946.
       
     • Lopatynsky's research continued to impress as he continued to prove major results in the theory of systems of linear differential equations of the elliptic type.
       
     • Lopatynsky's contributions to the theory of differential equations are particularly important, with important contributions to the theory of linear and nonlinear partial differential equations.
       
     • He worked on the general theory of boundary value problems for linear systems of partial differential equations of elliptic type, finding general methods of solving boundary value problems.
       
     • His work opened up broad possibilities for applying Morse theory to variational elliptic problems.
       
     • In 1980 Lopatynsky published an important book Introduction to the Contemporary Theory of Partial Differential Equations.
       
     • This book makes the reader familiar with the basic notions and facts of algebra, topology, and functional analysis, and gives a general idea how to apply these notions to the theory of differential equations.
       
     • We emphasize the relation of the theory of differential equations to other areas of mathematics.
       
     • In addition a book Theory of general boundary value problems was published in 1984 containing a selection of 35 of his papers from the total of 59.
       

127. Weierstrass biography
     
     • He came to understand the necessary methods in elliptic function theory by studying transcripts of lectures by Gudermann.
       
     • The concepts on which Weierstrass based his theory of functions of a complex variable in later years after 1857 are found explicitly in his unpublished works written in Munster from 1841 through 1842, while still under the influence of Gudermann.
       
     • This paper did not give the full theory of inversion of hyperelliptic integrals that Weierstrass had developed but rather gave a preliminary description of his methods involving representing abelian functions as constantly converging power series.
       
     • Weierstrass published a full version of his theory of inversion of hyperelliptic integrals in his next paper Theorie der Abelschen Functionen in Crelle's Journal in 1856.
       
     • The topics of his lectures included:- the application of Fourier series and integrals to mathematical physics (1856/57), an introduction to the theory of analytic functions (where he set out results he had obtained in 1841 but never published), the theory of elliptic functions (his main research topic), and applications to problems in geometry and mechanics.
       
     • In his 1863/64 course on The general theory of analytic functions Weierstrass began to formulate his theory of the real numbers.
       
     • Introduction to the theory of analytic functions, .
       
     • Known as the father of modern analysis, Weierstrass devised tests for the convergence of series and contributed to the theory of periodic functions, functions of real variables, elliptic functions, Abelian functions, converging infinite products, and the calculus of variations.
       
     • He also advanced the theory of bilinear and quadratic forms.
       
     • History Topics: The beginnings of set theory .
       

128. Bott biography
     
     • He explained how his "conversion" to mathematics took place via an attempt to move into medicine (see for example [The founders of index theory: reminiscences of Atiyah, Bott, Hirzebruch, and Singer (International Press, Somerville, MA, 2003).',22)">22]):- .
       
     • Bott was awarded a PhD for his thesis Electrical Network Theory in 1947 and he remained at Carnegie Tech undertaking research until 1949.
       
     • The aim of the invitation was so that Bott could write a book on network theory but in fact the experience took his interests away from network theory and towards topology [17:- .
       
     • Samelson was a real master of geometry and Lie group theory.
       
     • Most of them concern K-theory, index theory of operators and Lefschetz fixed point theory for elliptic complexes.
       
     • About half of the papers in this volume were written in collaboration with M Atiyah; their time span is the golden decade of index theory, 1964 - 73.
       
     • The underlying topic is Morse theory, in particular, the equivariant one.
       
     • I personally find the papers "Marston Morse and his mathematical works" and "Lectures on Morse theory, old and new" truly exceptional.
       
     • For example in [The founders of index theory: reminiscences of Atiyah, Bott, Hirzebruch, and Singer (International Press, Somerville, MA, 2003).',22)">22] Tu writes about Bott as a lecturer:- .
       

129. Calugareanu biography
     
     • Calugareanu continued to chair the Department of the Theory of Functions at Babes-Bolyai University.
       
     • Calugareanu worked in a number of different mathematical areas such as the theory of functions of one complex variable, geometry, algebra, and topology.
       
     • His early work, including the work of his doctoral dissertation, were on the theory of functions of one complex variable and here he particularly extended the work of Dimitrie Pompeiu.
       
     • His remarkable contributions to the theory of meromorphic functions and univalent functions mean that he is considered as the founder of the school of the geometric theory of univalent functions at Cluj.
       
     • Through their work in the theory of polygenic functions, both Pompeiu and Calugareanu established themselves as genuine founders of the theory of generalized analytic functions, a branch of complex analysis that developed considerably in the period 1930-1970.
       
     • During the same period of rapid development of aerodynamics, studies connected with the theory of conformal representation became of great interest.
       
     • Calugareanu investigated necessary and sufficient conditions for univalence of functions holomorphic in a disc and obtained results in the theory of conformal representation of multiply connected domains.
       
     • Returning to Weierstrass's point of view on analyticity and the definition of analytic functions via Taylor series elements, Calugareanu created the theory of invariants and covariants of analytic continuation.
       
     • In differential topology, starting from an invariant introduced by Gauss, Calugareanu discovered a system of invariants which found applications in knot theory and molecular biology.
       
     • In 1963 Calugareanu published Elements of the theory of functions of a complex variable in Romanian.
       

130. Vagner biography
     
     • He was particularly attracted by the theory of relativity and he now asked Igor Tamm, a professor in Moscow, if he would supervise his doctoral studies in that topic.
       
     • Although Tamm was very interested in the theory of relativity, he was not allowed to have students in this field.
       
     • He published a major 70 page paper General affine and central projective geometry of a hypersurface in a central affine space and its application to the geometrical theory of Caratheodory's transformations in the calculus of variations (Russian) in 1952.
       
     • The next sections deal with the affine and central-projective normals of a hypersurface and the general theory of hypersurfaces in a central affine En under transformations of the affine and central-projective group.
       
     • The theory is applied to affine hyperspheres (all normals through one point) and hyperquadrics (Darboux tensor vanishes).
       
     • The paper ends with the general theory of curves under the same groups.
       
     • He also published On the theory of partial transformations (Russian) in 1952 and then the major 90 page paper The theory of generalized heaps and generalized groups (Russian) in the following year.
       
     • Now the theory of inverse semigroups is one of the most vital and important parts of semigroup theory.
       
     • It was from him that I heard first about Freudian psychology (a taboo topic; S Freud's books were on the index and could not be borrowed in the libraries), K Menger's dimension theory (luckily, Menger's works might be read and I did that), the Vienna Circle of philosophers, R M Rilke's poetry, German expressionist fiction, Schopenhauer's philosophy, comparative studies of Indo-European languages, and many many other things.
       

131. Reidemeister biography
     
     • His doctoral thesis was on algebraic number theory, the particular problem having been suggested by Hecke, and the resulting publication Relativklassenzahl gewisser relativ- quadratischer Zahlkoper appeared in 1921.
       
     • In [The History of Combinatorial Group Theory: A Case Study in the History of Ideas (New York - Heidelberg - Berlin, 1982), 91-92.',4)">4] it is noted that:- .
       
     • Of all of the 71 papers listed in Reidemeister's obituary by Artzy ([Jahresberichte der Deutschen Mathematiker-Vereinigung 74 (1972), 96-104.',2)">2]), this is the only one which deals with number theory.
       
     • Here he became a colleague of Wirtinger who interested Reidemeister in knot theory.
       
     • Reidemeister worked on the foundations of geometry and he wrote an important book on knot theory Knoten und Gruppen (1926).
       
     • He established a geometry and topology based on group theory without the concept of a limit.
       
     • As is remarked in [The History of Combinatorial Group Theory: A Case Study in the History of Ideas (New York - Heidelberg - Berlin, 1982), 91-92.',4)">4]:- .
       
     • Reidemeister had an important influence on group theory, partly through his work on knots and groups, partly through his influence on Schreier.
       
     • Talking of this influence on group theory, Chandler and Magnus write in [The History of Combinatorial Group Theory: A Case Study in the History of Ideas (New York - Heidelberg - Berlin, 1982), 91-92.',4)">4]:- .
       
     • His influence on combinatorial group theory is largely that of a pioneer.
       

132. Cohen Wim biography
     
     • This doctoral thesis unravelled an inconsistency in classical shell theory; the results of the thesis provided the basis for a sound strength calculation of wide blade ship propellers.
       
     • Using the theory of Markov chains plus some interesting combinatorial manipulations, the author gives explicit formula for the equilibrium probability that a new arrival will meet j busy trunks and the probability that there are j busy trunks at some arbitrary time.
       
     • Later papers during this period include Derived Markov chains (1962) and Applications of derived Markov chains in queueing theory (1963) and in these and other appers Cohen made deep studies of stochastic processes and queueing theory.
       
     • She created an ideal atmosphere for research and scholarship, which enabled Cohen to undertake the immense task of writing 'The Single Server Queue.' This monumental work, which firmly established Cohen as the world's leading queueing theorist, is still considered to be a standard work in the area of mathematical queueing theory.
       
     • It set a standard of mathematical rigour which greatly enhanced the maturing of queueing theory as a mathematical discipline.
       
     • This book is a synthesis of the theory of queueing systems with a single server, including some new results of the author's.
       
     • Like its predecessor, the book is a "must have" for the serious student of queueing theory.
       
     • Particularly noteworthy publications while in this Chair included On regenerative processes in queueing theory (1976) which provided (writes Shaler Stidham, Jr):- .
       
     • a thorough survey of applications of the theory of regenerative processes to the analysis of queueing systems.
       
     • It is an excellent introduction to the power of the regenerative-process approach to queueing theory, especially when it comes to providing simple, intuitively based arguments for well-known results in a general setting.
       

133. Rayleigh biography
     
     • Stokes inspired Rayleigh with his lectures which combined theory and practice in a novel way with many physical experiments being carried out during the lectures.
       
     • His first paper was inspired by reading Maxwell's 1865 paper on electromagnetic theory.
       
     • Rayleigh's theory of scattering, published in 1871, was the first correct explanation of why the sky is blue.
       
     • Rather remarkably he began writing a major text The Theory of Sound while on the trip.
       
     • he applied the wave theory of light to the mathematical investigation of the resolving power of prisms and diffraction gratings; thus he showed that the resolving power of a grating is determined by the total number of lines in the grating multiplied by the order of the spectrum, and not by the closeness of the lines.
       
     • In 1879 Rayleigh wrote a paper on travelling waves, this theory has now developed into the theory of solitons.
       
     • The preface of [Rayleigh-wave theory and application, London, July 15-17, 1985 (Berlin, 1985)',4)">4] explains why Rayleigh-wave theory, introduced by him in 1885 in a paper in the Proceedings of the London Mathematical Society, has proved so important:- .
       
     • [There were] two domains in fluid mechanics in which Lord Rayleigh made explicit use of hydrodynamic similarity: the theory of aerodynamic drag and the treatment of the Aeolian tones.
       
     • [There was a] great impact of Rayleigh's ideas on the development of hydrodynamic similarity theory and applications during his lifetime and beyond.
       

134. Langlands biography
     
     • extraordinary vision that has brought the theory of group representations into a revolutionary new relationship with the theory of automorphic forms and number theory.
       
     • Then, over the next couple of years, he produced deep results on Eisenstein series and went on to apply Eisenstein series to prove a number theory conjecture due to Weil.
       
     • a collection of far-reaching and uncannily accurate conjectures relating number theory, automorphic forms, and representation theory.
       
     • He received the Cole Prize in Number Theory from the American Mathematical Society in 1982 for his pioneering work on automorphic forms Eisenstein series, and product formulae.
       
     • path-blazing work and extraordinary insights in the fields of number theory, automorphic forms, and group representation.
       
     • [Langlands'] astounding insight has provided a whole generation of mathematicians working in automorphic forms and representation theory with a seemingly unlimited expanse of deep, interesting, and above all approachable problems to work away on.
       
     • AMS Cole Prize in Number Theory1982 .
       
     • AMS Cole Prize for number theory1982 .
       

135. Bellman biography
     
     • Results from his dissertation appeared in the book Stability theory of differential equations which he published in 1953.
       
     • While there he began to ponder whether he should remain at Stanford where he was working on the topic he loved most, namely analytic number theory, or whether he should take up a position at RAND in Los Angeles.
       
     • Bellman's first publication on dynamic programming appeared in 1952 and his first book on the topic An introduction to the theory of dynamic programming was published by the RAND Corporation in 1953.
       
     • These include, in addition to those already mentioned: A Survey of the Theory of the Boundedness, Stability, and Asymptotic Behavior of Solutions of Linear and Nonlinear Differential and Difference Equations (1949); A survey of the mathematical theory of time-lag, retarded control, and hereditary processes (1954); Dynamic programming of continuous processes (1954); Dynamic programming (1957); Some aspects of the mathematical theory of control processes (1958); Introduction to matrix analysis (1960); A brief introduction to theta functions (1961); An introduction to inequalities (1961); Adaptive control processes: A guided tour (1961); Inequalities (1961); Applied dynamic programming (1962); Differential-difference equations (1963); Perturbation techniques in mathematics, physics, and engineering (1964); and Dynamic programming and modern control theory (1965).
       
     • However he also wrote Analytic number theory (1980), Mathematical methods in medicine (1983), and The Laplace transform (1984).
       
     • He was elected to Fellowship in the American Academy of Arts and Sciences in 1975 and, in the following years, he received the John von Neumann Theory Award, another joint award this time by the Institute of Management Sciences and the Operations Research Society of America.
       
     • For contributions to decision processes and control system theory, particularly the creation and application of dynamic programming.
       
     • Richard Bellman is a towering figure among the contributors to modern control theory and systems analysis.
       

136. Schutzenberger biography
     
     • In 1953 he was awarded a doctorate in mathematics for his work on the mathematical theory of communications, developing ideas that Shannon had presented in his famous 1948 paper A Mathematical Theory of Communication.
       
     • In the theory of Young tableaux he discovered the jeu de taquin that became the basis for so many later investigations; he illuminated the Schensted correspondence between pairs of permutations and tableaux by revealing many facets of its fine structure; he created the subject of context-free languages and explored some of its many consequences, as well as finding many results in the theory of combinatorial words and languages of other kinds; with Foata he developed a general theory of counting families of unlabeled combinatoral objects by factoring them into primes, etc.
       
     • For example his theory of formal languages has spawned many fine successes in the enumeration of various kinds of polyominoes and cellular structures and his work on factorisation of families of unlabeled objects has been responsible for methods of selecting such objects at random.
       
     • Semigroups appear in his works essentially in two forms: 1) as free monoids, for example in coding and language theories and often in combinatorial designs; 2) as finite semigroups, for example in transition semigroups of automata or in the theory of pseudo-cvarieties for the study of varieties of languages.
       
     • Schutzenberger's works contributed strongly to giving semigroup theory its letters of nobility by taking it out of its self-contemplation and putting it at work in areas of mainstream mathematics.
       
     • It was in this area that Schutzenberger introduced what have been called Schutzenberger groups and Schutzenberger representations by Clifford and Preston in their influential book The Algebraic Theory of Semigroups (1961).
       
     • He found there an incredible interplay between algebra through the use of finite semigroups, probability theory and combinatorics.
       
     • Later he published a series of results on variable-length codes all of them reported in our book with Jean Berstel (Theory of Codes, Academic Press, 1984).
       

137. Mackey biography
     
     • Early in his career Mackey worked on the duality theory of locally convex spaces publishing papers which include On infinite dimensional linear spaces (1943), On convex topological linear spaces (1943), Equivalence of a problem in measure theory to a problem in the theory of vector lattices (1944), (with Shizuo Kakutani) Ring and lattice characterization of complex Hilbert space (1946), On convex topological linear spaces (1946).
       
     • In 1955 Mackey gave a course of lectures on the theory of group representations at the University of Chicago in the summer of 1955.
       
     • A set of mimeographed notes was produced at the time and these later were incorporated as the first half of his book The theory of unitary group representations published in 1976.
       
     • In 1967 Mackey published Lectures on the theory of functions of a complex variable which was based on an undergraduate course he gave at Harvard during the academic year 1959-60.
       
     • This happened when Unitary group representations in physics, probability, and number theory was published in 1978.
       
     • It is well known that the author's approach to representation theory has proved very successful in finding the unitary representations, and their decompositions, for the groups which come up frequently both in (pure) mathematics and in physics, as well as in mathematical physics.
       
     • But the author's theory is surprisingly versatile, with applications in number theory, harmonic analysis, ergodic theory, quantum mechanics, and statistical mechanics, and these applications are worked out in detail..
       

138. Straus biography
     
     • In 1946 Straus and Einstein published A generalization of the relativistic theory of gravitation II in the Annals of Mathematics.
       
     • Straus submitted his doctoral thesis on Einstein's unified field theory to Columbia University and was awarded his doctorate in 1948.
       
     • He published Some results in Einstein's unified field theory in 1949 which was based on his thesis.
       
     • Straus had been working with Einstein on mathematical physics yet while at Princeton he had developed an interest in number theory from Artin, Erdos, Selberg and Siegel.
       
     • geometry, convexity, combinatorics, group theory and linear algebra.
       
     • His joint work with Motzkin, which we referred to above, was on graph theory.
       
     • This work provided one of the foundations of extremal graph theory.
       
     • Their work was again basic in developing a new theory, this time it was Euclidean Ramsey theory.
       
     • Another combinatorics paper was On a problem in the theory of partitions (1962).
       
     • He could solve crossword puzzles in ink in English (his third language) using only the horizontal clues, and in the next minute discourse profoundly on relativity theory, European history, or theology.
       

139. Deligne biography
     
     • He also collaborated with David Mumford on a new description of the moduli spaces for curves: this work has been much used in later developments arising from string theory.
       
     • Andre Weil gave for the first time a theory of varieties defined by equations with coefficients in an arbitrary field, in his Foundations of Algebraic Geometry (1946).
       
     • This work brought together algebraic geometry and algebraic number theory and it led to Deligne being awarded a Fields Medal at the International Congress of Mathematicians in Helsinki in 1978.
       
     • The areas on which he has worked, in addition to algebraic geometry, are Hilbert's 21st problem, Hodge theory, theory of moduli, modular forms, Galois representations, L-series and the Langlands conjectures, and representations of algebraic groups.
       
     • Through an unparalleled blend of penetrating insights, fearless technical mastery and dazzling ingenuity, Deligne has singlehandedly brought about a new understanding of the cohomology of varieties, both classical and in finite characteristic, with numerous applications to deep problems in geometry and number theory.
       
     • for major contributions to several important domains of mathematics (like algebraic geometry, algebraic and analytic number theory, group theory, topology, Grothendieck theory of motives), enriching them with new and powerful tools and with magnificent results such as his spectacular proof of the "Riemann hypothesis over finite fields" (Weil conjectures).
       
     • A remarkable feature of Pierre Deligne's thinking is that, when confronted with a new problem or a new theory, he understands and, so to speak, makes his own its basic principles at a tremendous speed, and is immediately able to discuss the problem or use the theory as a completely familiar object.
       

140. Polkinghorne biography
     
     • A new definition is given for the "normal product" of a set of field operators in the Heisenberg representation of quantum field theory.
       
     • Dyson, reviewed Temporally ordered graphs in quantum field theory:- .
       
     • A graphical representation of the perturbation-theory expansions of quantum field theory is defined, in which the vertices (points at which the interaction operates) are given a definite order in time.
       
     • The equivalence of the Feynman method of setting up a quantum field theory with the usual canonical formalism is here proved.
       
     • Then he published General dispersion relations in 1956 and Causal products in quantum field theory in the following year.
       
     • Also in a joint paper in 1957 he published Cauchy's problem in quantum field theory which explores the relation between the classical and quantum versions of field theories.
       
     • We give the titles of a few further papers which were fundamental in the development of a mathematical theory of elementary particles: On Schwinger's variational principle (1957), On the strong interactions (1957), Causal amplitudes and the Yang-Feldman formalism (1957), Generalized retarded products (1958), Higher order spinor Lagragians (1958), Unstable states and the separable potential model (1959), and The analytic properties of perturbation theory (1960, 1960, 1961).
       
     • As well as three papers on analytic properties in perturbation theory, he lectured on that topic at the 1961 Brandeis Summer Institute in Theoretical Physics and these lectures were published in the following year.
       
     • There are also a number of popular books on mathematical physics such as The Particle Play (1979), The Quantum World (1984) and Rochester Roundabout (1989) and Quantum Theory : A very short introduction (2002).
       

141. Weil biography
     
     • After the summer vacation he went to Rome and then on to Gottingen where he produced his first substantial piece of mathematical research on the theory of algebraic curves.
       
     • He developed for his thesis the ideas on the theory of algebraic curves which he had begun to study at Gottingen.
       
     • Weil's research was in number theory, algebraic geometry and group theory.
       
     • Beginning in the 1940s, Weil started the rapid advance of algebraic geometry and number theory by laying the foundations for abstract algebraic geometry and the modern theory of abelian varieties.
       
     • His work on algebraic curves has influenced a wide variety of areas, including some outside mathematics, such as elementary particle physics and string theory.
       
     • In fact Weil's work in this area was basic to work by mathematicians such as Yau who was awarded a Fields Medal in 1982 for work in three dimensional algebraic geometry which has major applications to quantum field theory.
       
     • Weil's work on bringing together number theory and algebraic geometry was highly fruitful.
       
     • The foundations of many topics studied in depth today were laid by Weil in this work, such as the foundations of the theory of modular forms, automorphic functions and automorphic representations.
       
     • Also bringing these areas together was his work on the geometric theory of the theta function and Kahler geometry.
       

142. Horrocks biography
     
     • He soon discovered that Lansberge's tables were seriously wrong and he realised that the tables were based on a false planetary theory.
       
     • Comparing the theoretical positions with his own observations he realised that these were by far the most accurate tables and that they were founded on a correct planetary theory.
       
     • He therefore accepted Kepler's theory of elliptical orbits for the planets and tested Kepler's laws by direct observation.
       
     • However he rejected Kepler's theory of why the planetary orbits were ellipses, which was based on alternate attraction and repulsion of a planet by the sun.
       
     • Not content with this theory without evidence, he supported it by analogy with the conical pendulum.
       
     • Although he is best known for his observations of the transit of Venus in 1639, Horrocks' most important work was his lunar theory.
       
     • He realised that the moon's orbit was perturbed by the sun (remember that he worked before Newton proposed his theory of universal gravitation) and was able to give a lunar theory which was much better than anything available at the time.
       
     • In fact Horrocks' lunar theory was used for around 100 years, a remarkable achievement.
       
     • 18 (2) (1987), 77-94.',16)">16] where Wilson traces the origin of Horrocks' theory in Kepler's work on the motion of the moon, as transformed and calibrated by further data, in particular critical data concerning the duration of lunar eclipses.
       
     • The final theory includes an explanation of the inequality depending on both elongation and anomaly of the moon by means of a variable eccentricity and an oscillating apse line.
       

143. Novikov Sergi biography
     
     • Sergei's mother, Ludmila Vsevolodovna Keldysh, was also an outstanding mathematician who became a professor of mathematics and made important contributions to set theory and geometric topology.
       
     • We should mention especially Sergei's uncle Mstislav Keldysh who made major contributions to complex function theory, differential equations and applications to aerodynamics.
       
     • V A Uspenskii, a pupil of Kolmogorov, organised a seminar during Novikov's first year as a student in which problems in set theory, mathematical logic, and functions of a real variable were studied.
       
     • Another important paper Some problems in the topology of manifolds connected with the theory of Thom spaces was published by Novikov in 1960.
       
     • Novikov's original motivation was the theory, in the simply connected case, of Browder-Novikov and Wall, which led to the classification of manifolds in high dimensions.
       
     • He studied a wide variety of applications of mathematics such as dynamical systems in the theory of homogeneous cosmological models, the theory of solitons, the spectral theory of linear operators, quantum field theory and string theory.
       
     • In 1982 he became interested in topological problems which arise in the physical theory of metals.
       
     • He constructed a global version of Morse theory on manifolds and loop spaces that had novel applications to quantum field theory (multivalued action functionals).
       
     • These include a systematic study of finite-gap solutions of two-dimensional integrable systems, formulation of the equivalence of the classification of algebraic-geometric solutions of the KP equation with the conformal classification of Riemann surfaces, and work (with Krichever) on "almost commuting" operators that appear in string theory and matrix models ("Krichever-Novikov algebras", now widely used in physics).
       

144. Nash biography
     
     • In September 1948 Nash entered Princeton where he showed an interest in a broad range of pure mathematics: topology, algebraic geometry, game theory and logic were among his interests but he seems to have avoided attending lectures.
       
     • During this period Nash established the mathematical principles of game theory.
       
     • The concept of a Nash equilibrium n-tuple is perhaps the most important idea in noncooperative game theory.
       
     • He was always full of mathematical ideas, not only on game theory, but in geometry and topology as well.
       
     • In the summer of that year he worked for the RAND Corporation where his work on game theory made him a leading expert on the Cold War conflict which dominated RAND's work.
       
     • He worked there from time to time over the next few years as the Corporation tried to apply game theory to military and diplomatic strategy.
       
     • He had already obtained results on manifolds and algebraic varieties before writing his thesis on game theory.
       
     • His famous theorem, that any compact real manifold is diffeomorphic to a component of a real-algebraic variety, was thought of by Nash as a possible result to fall back on if his work on game theory was not considered suitable for a doctoral thesis.
       
     • His research on the theory of real algebraic varieties, Riemannian geometry, parabolic and elliptic equations was, however, extremely deep and significant in the development of all these topics.
       
     • On 4 January he was back at the university and started to teach his game theory course.
       
     • Nash was awarded (jointly with Harsanyi and Selten) the 1994 Nobel Prize in Economic Science for his work on game theory.
       

145. Klein Oskar biography
     
     • He defended his doctorate in 1921 at Stockholm Hogskola and was opposed by Erik Ivar Fredholm the mathematical physicist best known for his work on integral equations and spectral theory.
       
     • His first work in this new arena was a philosophical paper that was a refutation of an objection to relativity theory by Swedish philosophers.
       
     • The problem was that the classical theory only effectively described atoms with a total electron spin of zero.
       
     • This was the beginning of his landmark work on a unified field theory.
       
     • Kaluza, in 1919, sent a paper to Albert Einstein proposing a unification of gravity with Maxwell's theory of light.
       
     • Unfortunately, despite a lot of initial interest in unification, most physicists eventually went on to more promising and experimentally testable research leaving Kaluza-Klein theory to be explored by another generation of physicists nearly half a century later.
       
     • It is interesting to note that this equation appeared exactly as it has been written in David Bohm's 1951 book Quantum Theory but was not called the Klein-Gordon equation.
       
     • But, nonetheless, it was an important point in quantum theory and, along with his unification theory, was to ensure a lasting legacy for Klein and cemented 1926 as a pivotal year in his life.
       
     • Klein's hypothesis was yet another crack at a unified field theory, this time in attempt to unify the strong, weak, and electromagnetic forces.
       
     • J P van der Schaar (Kaluza-Klein Theory) .
       

146. Landau Lev biography
     
     • In fact his first publication appeared in print in the year he graduated, being a paper on quantum theory.
       
     • In 1932, soon after he returned to Leningrad, Landau was appointed as head of the Theory Division of the Ukrainian Technical Institute in Kharkov and he was also appointed to the chair of theoretical physics at the Kharkov Institute of Mechanical Engineering.
       
     • In 1937 Landau went to Moscow to become Head of the Theory Division of the Physical Institute of the USSR Academy of Sciences.
       
     • He worked on atomic collisions, astrophysics, low-temperature physics, atomic and nuclear physics, thermodynamics, quantum electrodynamics, kinetic theory of gases, quantum field theory, and plasma physics.
       
     • The work he did on the theory to explain why liquid helium was super-fluid earned him the 1962 Nobel Prize for Physics.
       
     • Landau devised a theory to explain such behaviour which was published in 1941.
       
     • These include Statistical physics (1938), Mechanics, Field theory, Quantum mechanics, and Theory of elasticity.
       
     • The chapters of the book indicates the main topics of their joint research: Mechanics, theory of fields, quantum mechanics, quantum electrodynamics, classical statistical physics, quantum statistical physics, fluid mechanics, theory of elasticity, electrodynamics of continuous media, and physical kinetics.
       

147. Halmos biography
     
     • This was awarded in 1938 for his thesis on measure-theoretic probability Invariants of Certain Stochastic Transformation: The Mathematical Theory of Gambling Systems.
       
     • Halmos is known for both his outstanding contributions to operator theory, ergodic theory, functional analysis, in particular Hilbert spaces, and for his series of exceptionally well written textbooks.
       
     • These include Finite dimensional vector spaces (1942), Measure theory (1950), Introduction to Hilbert space and theory of spectral multiplicity (1951), Lectures on ergodic theory (1956), Entropy in ergodic theory (1959), Naive set theory (1960), Algebraic logic (1962), A Hilbert space problem book (1967) and Lectures on Boolean algebras (1974).
       
     • The award for a book or substantial survey or research-expository paper is made to Paul R Halmos for his many graduate texts in mathematics, dealing with finite dimensional vector spaces, measure theory, ergodic theory and Hilbert space.
       
     • J B Conway writes in [Paul Halmos : Celebrating 50 years of mathematics (New York, 1991).',1)">1] about Halmos's contributions to operator theory:- .
       

148. Hausdorff biography
     
     • After 1904 Hausdorff began working in the area for which he is famous, namely topology and set theory.
       
     • A year later, in 1914, Hausdorff published his famous text Grundzuge der Mengenlehre which builds on work by Frechet and others to created a theory of topological and metric spaces.
       
     • He succeeded in creating a theory of topological and metric spaces into which the previous results fitted well, and he enriched it with many new notions and theorems.
       
     • He continued to undertake research in topology and set theory but the results could not be published in Germany.
       
     • We also mentioned his work on ordered sets and his masterpiece on set theory and topology Grundzuge der Mengenlehre (1914).
       
     • Within the mathematical work of Hausdorff the two publications devoted explicitly to measure theory occupy a significant place: they are not only important for measure theory but have also contributed fundamentally to its development.
       
     • It is not well known that throughout his life Hausdorff had been interested in various fundamental problems of measure and integration theory and had made important contributions at different times.
       
     • One such lecture course was given on probability theory by Hausdorff in Bonn in the summer of 1923.
       
     • He studied the Gaussian law of errors, limit theorems and problems of moments, and set theory and the strong law of large numbers, which he based on measure theory.
       

149. Abraham Max biography
     
     • During this time Abraham and Einstein disagreed strongly about the theory of relativity in a correspondence discussed in [Einstein and the history of general relativity (Boston, MA, 1989), 160-174.',3)">3] and [Italian mathematics between the two world wars (Bologna, 1987), 143-159.',4)">4].
       
     • Forced to return to Germany at the start of World War I, Abraham worked on the theory of radio transmission.
       
     • Abraham's work is almost all related to Maxwell's theory and he wrote a text which was the standard work on electrodynamics in Germany for a long time.
       
     • His theory of the electron was developed in 1902, and its case strongly agrued in his text, but in 1904 Lorentz and Einstein produced a different theory.
       
     • At first his ideas were supported by experiment, particularly work carried out by Wilhelm Kaufmann, but later work was to favour the theory developed by Lorentz and Einstein.
       
     • At first he objected both to the postulates on which relativity was based and also to the fact that he felt that the experimental evidence did not support the theory.
       
     • By 1912 Abraham, who despite his objections was one of those who best understood relativity theory, was prepared to accept that the theory was logically sound.
       
     • However, he still did not accept that the theory accurately described the physical world.
       
     • He hoped that further astronomical data would support the aether theory and show that relativity was not in fact a good description of the real world.
       

150. Pitt biography
     
     • He was tutored by J C Burkill and attended courses by world-leading mathematicians such as: functions of a complex variable from A E Ingham, almost periodic functions from A S Besicovitch, the theory of functions from J E Littlewood, and divergent series from G H Hardy.
       
     • Coastal Command he used probability theory, and the newly developing operational research, to devise methods for attacking German U-boats.
       
     • As the author states in the preface, the purpose of this book is to provide an introduction to the modern theory of probability and the fundamental ideas and techniques on which it is based, namely, those of measure and integration.
       
     • The first part is devoted to the theory of integration in a sufficiently general context to make it applicable in other branches of analysis.
       
     • Although of unquestioned power and practical utility, the Lebesgue Theory of measure and integration tends to be avoided by mathematicians, due to the difficulty of obtaining detailed proofs of a few crucial theorems.
       
     • In this concise and easy-to-read introduction, the author demonstrates that the day-to-day skills gleaned from Lebesgue Theory far outweigh the effort needed to master it.
       
     • This compact account develops the theory as it applies to abstract spaces, describes its importance to differential and integral calculus, and shows how the theory can be applied to geometry, harmonic analysis, and probability.
       
     • Postgraduates in mathematics and science who need integration and measure theory as a working tool, as well as statisticians and other scientists, will find this practical work invaluable.
       
     • It gives a compact account of the essential theory and some of its applications.
       
     • This approach is far from the original development of the theory and consequently, perhaps, less pedagogical for beginners.
       

151. Doeblin biography
     
     • Wolfang Doeblin is arguably one of the four major contributors to probability theory in the first half of the 20th century up to World War II (the other three are Khinchin, Kolmogorov and Levy).
       
     • He laid many of the cornerstones of the modern theory for Markov chains and processes, to be developed after the war by others.
       
     • In his work on the theory of Markov chains and processes, his main field, we notice major contributions to: Markov chains with general state spaces, Jump Markov processes, the coupling method (innovation), and diffusions.
       
     • The book by Doob [Stochastic Processes (New York, 1953).',3)">3] from 1953 has been crucial for the development of probability theory; for a large part of its contents on Markov chains and processes, Doeblin's work is the base.
       
     • It is essentially Doblin's theory as completed during the quarter of a century following the publication of his papers that is presented here.
       
     • In Doeblin's mine of ideas, the coupling method was paid attention to by very few until the early 1970s; then the time was ripe to explore it, and the method is now a major tool in probability theory, with applications ranging from elementary theory to front research.
       
     • On Doeblin's work concerning sums of independent random variables, Feller [An Introduction to Probability Theory and Its Applications Vol.
       
     • The interest in the theory was stimulated by W.
       
     • The modern theory still carries the imprint..
       
     • Doeblin also contributed to the theory of random chains with complete connection, some of which was used in a paper by him on ergodic properties of continued fractions.
       

152. Jacobson biography
     
     • Having attended a course by Wedderburn on matrices in which he ended by developing his classical structure theory of finite dimensional algebras over finite fields, Jacobson was given the task of studying division algebras which were idealizers of one-sided ideals of polynomial rings.
       
     • Jacobson had heard Emmy Noether lecture on class field theory in Princeton in the spring semester of 1935 and again at the summer meeting of the American Mathematical Society where she made comments on Jacobson's thesis results which he presented there.
       
     • My research at Hopkins during the years 1943-46 was mainly in the area of Galois theory and the general structure of rings.
       
     • Jacobson is well known for his outstanding contributions to ring theory.
       
     • Jacobson discovered a deep structure theory for rings and has given his name to the Jacobson radical, the intersection of the maximal ideals of a ring.
       
     • They include The theory of rings (1943) and the three volume work Lectures in abstract algebra (1951-64) covering basic concepts, linear algebra and the theory of fields and Galois theory.
       
     • includes his account of his structure theory.
       
     • Jacobson Theory of Rings .
       
     • History Topics: The development of Ring Theory .
       

153. Kingman biography
     
     • Queueing theory, which has endured a long period in which people treated one example after another, is finally breaking out of its confinement to independent arrivals, service times, etc.
       
     • It was around this time that he was studying subadditive ergodic theory which in many ways turned out to be the most influential of all his mathematical achievements.
       
     • Kingman succeeded in proving the theorem: he published The ergodic theory of subadditive stochastic processes in the Journal of the Royal Statistical Society in 1968 and An ergodic theorem in the Bulletin of the London Mathematical Society in the following year.
       
     • Kingman gave a beautiful description of the development of the subject in his 1973 paper Subadditive ergodic theory published in the Annals of Probability.
       
     • I was appointed Professor of Mathematics to raise the profile of probability theory (but not statistics) in the Faculty of Mathematics.
       
     • This work was part of the theory of regenerative phenomena and, in addition to a number of other articles, Kingman published his classic text Regenerative phenomena in 1972.
       
     • He wrote on the theory of Markov processes and published an important series of articles on Markov transition probabilities which we gave details of above.
       
     • Among a large number of other topics to which he made major contributions were ergodic theorems, random walks and the theory of queues, in particular applying queuing theory to problems of traffic flow.
       
     • It is shown how the theory can be applied to interesting problems of astronomy, queuing and traffic etc., and these examples are studied very thoroughly and deeply, giving even the specialist new insights.
       

154. Poincare biography
     
     • He made contributions to numerous branches of mathematics, celestial mechanics, fluid mechanics, the special theory of relativity and the philosophy of science.
       
     • In a correspondence between Klein and Poincare many deep ideas were exchanged and the development of the theory of automorphic functions greatly benefited.
       
     • Homotopy theory reduces topological questions to algebra by associating with topological spaces various groups which are algebraic invariants.
       
     • Poincare is also considered the originator of the theory of analytic functions of several complex variables.
       
     • His first major contribution to number theory was made in 1901 with work on [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .
       
     • In applied mathematics he studied optics, electricity, telegraphy, capillarity, elasticity, thermodynamics, potential theory, quantum theory, theory of relativity and cosmology.
       
     • He is acknowledged as a co-discoverer, with Albert Einstein and Hendrik Lorentz, of the special theory of relativity.
       
     • It is interesting that this error is now regarded as marking the birth of chaos theory.
       

155. Menshov biography
     
     • Menshov's first degree was awarded in 1916 for the thesis which he wrote on The Riemann theory of trigonometric series which was examined by Egorov and Luzin.
       
     • he had already been acknowledged as one of the world's most outstanding specialists on the theory of functions of a real and a complex variable.
       
     • In 1933 a new chair of Analysis and Theory of Functions was created at Moscow University and Lavrentev appointed.
       
     • In 1938 the Faculty of Mechanics and Mathematics at Moscow University founded two chairs, the chair of the Theory of Functions and the chair of Functional Analysis.
       
     • Privalov held this first chair up to 1941 but then, on Privalov's early death in that year, Menshov was appointed to the chair of the Theory of Functions.
       
     • In 1943 these two chairs were combined and the Department of Theory of Functions and Functional Analysis was created with Menshov as its head.
       
     • His scientific interests relate principally to the theory of trigonometric series, the theory of orthogonal series and the problem of monogenity of functions of a complex variable.
       
     • He published more than eighty papers on these subjects, which have had an exceptionally great effect on the development of the whole theory of functions.
       
     • A characteristic feature of scientific activity is that in his work on the theory of functions he solved a number of extremely difficult key problems which had baffled many eminent mathematicians.
       

156. Waring biography
     
     • We shall comment further below on this important work, covering topics in the theory of equations, number theory and geometry.
       
     • Meditationes Algebraicae, covering the theory of equations and number theory, appeared in 1770 with an expanded version in 1782.
       
     • This work makes Waring one of the earliest contributers to Galois theory.
       
     • This is, in essence, the first result in the theory of symmetric functions (beyond the basic building blocks which appeared in Chapter 1), a theory whose systematic development was not to appear until the 19th century (Lagrange, Gauss, and others) and was ultimately followed by the theory of permutation groups (Galois, Jordan, ..
       
     • The rest of the book deals with number theory, a topic in which Waring made some interesting advances.
       
     • Hilbert's proof led to major new theorems in number theory.
       

157. Kuratowski biography
     
     • Soon I could answer some of the more difficult questions in the set theory course, and I began to pose other problems.
       
     • At Lvov, however, Kuratowski worked with Banach and they answered some fundamental problems on measure theory.
       
     • Kuratowski's main work was in the area of topology and set theory.
       
     • His 1930 work on non-planar graphs is of fundamental importance in graph theory, he showed that a necessary and sufficient condition for a graph G to be planar is that it does not contain a subgraph homeomorphic to either K5 or K3,3.
       
     • His work in set theory considered a function as a set of ordered pairs and this made the function notion as proposed by Frege, Charles Peirce and Schroder redundant.
       
     • He also considered the topology of the continuum, the theory of connectivity, dimension theory, and answered measure theory questions.
       
     • Kuratowski's Foreword to Set Theory and Topology .
       
     • Kuratowski's Introduction to Set Theory .
       

158. Cole biography
     
     • Cole returned to Harvard and wrote a thesis A Contribution to the Theory of the General Equation of the Sixth Degree which, as the title indicates, studied equations of degree 6.
       
     • His main research contributions are to number theory, in particular to prime numbers, and to group theory.
       
     • In number theory he achieved the distinction of being the first to factor 267 - 1 and he did this using quadratic remainders.
       
     • Another important contribution to the theory of groups made by Cole was his work in publishing his English translation of Netto's book on group theory.
       
     • This appeared as The theory of substitutions and its applications to algebra (A A Register Publishing Company, 1892) and was undertaken with Netto's agreement.
       
     • It was the first book on group theory in English and was important in stimulating interest in group theory in the English speaking world.
       
     • He established the Frank Nelson Cole Prizes in algebra and number theory and today these are highly prestigious awards.
       

159. Wintner biography
     
     • George W Hill had published an account of his lunar theory in 1878.
       
     • Wintner wrote a number of papers putting Hill's theory on a rigorous mathematical foundation.
       
     • In fact at this time the development of Hilbert spaces had become particularly important for the study of quantum theory since this mathematics underlay the theory.
       
     • Wintner published on analysis, number theory, differential equations and probability (with several joint papers with Norbert Wiener).
       
     • These innovators were more concerned with the underlying theory, less concerned with quantitatively accurate prediction of celestial body motion.
       
     • led Wintner to various interests: first, his interest in almost periodic functions as such; second, analytic number theory and summability; third, the asymptotic distributions of almost periodic functions; and finally, the theory of distribution functions as such.
       
     • In this connection, Wintner used to like to point out the debt of analytic number theory to dynamics, noting that in a certain sense the oldest Tauberian theorems date back to the dynamical work of Sundman and Hadamard.
       
     • These were Lectures on asymptotic distributions and infinite convolutions (1938), Analytical foundations of celestial mechanics (1941), Eratosthenian averages (1943), Theory of measure in arithmetical semigroups (1944), The Fourier transforms of probability distributions (1947), and An arithmetical approach to ordinary Fourier series (1945).
       

160. Pauli biography
     
     • Within two months of leaving school he had submitted his first paper on the theory of relativity.
       
     • While still an undergraduate at Munich he wrote two further articles on the theory of relativity.
       
     • I was not spared the shock which every physicist accustomed to the classical way of thinking experienced when he came to know Bohr's basic postulate of quantum theory for the first time.
       
     • Pauli received his doctorate, which had been supervised by Sommerfeld, in July 1921 for a thesis on the quantum theory of ionised molecular hydrogen.
       
     • Looking at it now one can see that it showed that quantum theory, as then formulated, was not in itself going to provide the necessary structure on which to build a logical theory of atomic structure which agreed with experimental evidence.
       
     • Two months after the award of his doctorate Pauli's survey of the theory of relativity appeared, by this time having grown into a work of 237 pages.
       
     • Pauli, who before that had begun to feel that further advances could not be made with the theory as it then existed, quickly made progress using Heisenberg's new ideas and before the end of 1925 he had derived the hydrogen spectrum from the new theory.
       
     • In 1925 and 1926 essential progress of another kind was made in the quantum theory, which is the foundation of atomic physics.
       

161. Springer biography
     
     • As research interests, Springer gives algebra and the theory of linear algebraic groups.
       
     • For the associatively-inclined this book expunges the dread word "nonassociative" from Jordan theory, since there is nothing nonassociative about inversion; most importantly, it makes Jordan structure theory accessible to the growing audience of persons familiar with root systems.
       
     • Springer's next book was Invariant theory (1977).
       
     • These notes had their origin in a course in invariant theory, given at the University of Utrecht in the autumn of 1975.
       
     • The purpose of the course was to give an introduction to invariant theory on an elementary level, illustrated by some examples from 19th century invariant theory.
       
     • The notes are an enjoyable, readable account of the invariant theory of reductive algebraic groups, concentrating on delicate finiteness theorems.
       
     • The general theory is illustrated by a detailed analysis of SL(2, K) and finite groups.
       
     • These notes contain an introduction to the theory of linear algebraic groups over an algebraically closed ground field.
       

162. Brown biography
     
     • Before this, in 1889, he had joined the Royal Astronomical Society and in the year he left England his first paper was published by that Society on lunar theory.
       
     • While he had worked in Cambridge, before going to the United States, Brown had read Hill's Researches in the lunar theory (1878) and published his own ideas on that theory.
       
     • He published An Introductory Treatise on the Lunar Theory in 1896, then embarked on a new theory of the orbit of the Moon based on Hill's ideas.
       
     • However the Moon stubbornly refused to follow the path that mathematicians computed for it and, despite attempts to take all reasonable effects into account, there were still fluctuations in the motion not predicted by theory [The Times [available on the Web]',2)">2]:- .
       
     • Brown proposed an attack from the observational side and proposed utilising the observation of occultations of stars to a larger extent than hitherto to map the moon's path and improve the theory.
       
     • Not only did Brown contribute greatly to the theory of the Moon's orbit but he also worked on planetary motion.
       
     • His book on this topic, Planetary Theory, written jointly with C A Shook, appeared in 1933.
       
     • He received the Gold Medal of the Royal Astronomical Society in 1907 for his Researches in the lunar theory, and the Royal Medal of the Royal Society in 1914.
       

163. Birkhoff biography
     
     • In 1901 he began a correspondence with Harry Vandiver on a problem in number theory.
       
     • While at Harvard he submitted the results he had obtained with Vandiver to the Annals of Mathematics in 1904 and this joint number theory paper became his first publication.
       
     • However we should note that his main work was on dynamics and ergodic theory.
       
     • His ergodic theorem transformed the Maxwell-Boltzmann kinetic theory of gases into a rigorous principle through the use of Lebesgue measure.
       
     • This theory, which resolved in principle one of the fundamental problems arising in the theory of gases and statistical mechanics, has been influential not only in dynamics itself but also in probability theory, group theory, and functional analysis.
       
     • He developed a mathematical theory of aesthetics which he applied to art, music and poetry.
       
     • He has told us that the formal structure of western music, the riddle of melody, began to interest him in undergraduate days; somewhat intense consideration of the mathematical elements here involved led him to apply his theory also to aesthetic objects such as polygons, tilings, vases, and even poetry.
       

164. FitzGerald biography
     
     • George FitzGerald was a brilliant mathematical physicist who today is known by most scientists as one of the proposers of the FitzGerald-Lorentz contraction in the theory of relativity.
       
     • FitzGerald immediately saw Maxwell's work as providing the framework for further development and he began to work on pushing forward the theory.
       
     • Very few seemed to see the theory as a starting point, rather most saw it only as a means to produce Maxwell's own results.
       
     • Maxwell's theory was for many years, in the words of Heaviside, "considerably underdeveloped and little understood" but a few others were to see it in the same light as FitzGerald including Heaviside, Hertz and Lorentz.
       
     • He had already begun to contribute to Maxwell's theory and, as well as theoretical contributions, he was conducting experiments in electromagnetic theory.
       
     • His first major theoretical contribution was On the electromagnetic theory of the reflection and refraction of light which he sent to the Royal Society in October 1878.
       
     • At a meeting of the British Association in Southport in 1883, FitzGerald gave a lecture discussing electromagnetic theory.
       
     • My scientific sympathy and alliance with him have greatly ripened during the last six or seven years over the undulatory theory of light and the aether theory of electricity and magnetism.
       

165. Haselgrove biography
     
     • His dissertation was Some Theorems in the Analytic Theory of Numbers.
       
     • Haselgrove's first published research paper was on number theory.
       
     • This was A connection between the zeros and the mean values of z(s) (1949) followed by Some theorems in the analytic theory of numbers (1951), On Ingham's Tauberian theorem for partitions (1952), and (with H N V Temperley) Asymptotic formulae in the theory of partitions (1954).
       
     • However Haselgrove was involved in much more than number theory.
       
     • An implementation of this procedure, designed by Todd and Coxeter for hand calculation, remains today as one of the main tools in computational group theory.
       
     • The state of the art implementation of coset enumeration today is the package ACE (available both as a stand-alone program or within all major group theory systems).
       
     • In 1958 Haselgrove published his most famous number theory result in A disproof of a conjecture of Polya.
       
     • In the same paper Haselgrove announced that he had also disproved a number theory conjecture of Turan.
       
     • These methods are based on the theory of Diophantine approximation.
       

166. Schur biography
     
     • Frobenius and Burnside had been the two main founders of the theory of representations of groups as groups of matrices.
       
     • This theory proved a very powerful tool in the study of groups and Schur was to learn the foundations of this subject from Frobenius.
       
     • Schur is mainly known for his fundamental work on the representation theory of groups but he also worked in number theory, analysis and other topics described below.
       
     • Schur returned to work on representation theory with renewed vigour and he was able to complete the programme of research begun in his doctoral dissertation and give a complete description of the rational representations of the general linear group.
       
     • First there was pure group theory, in which Schur adopted the surprising approach of proving without the aid of characters, theorems that had previously been demonstrated only by that means.
       
     • Fourth, he worked in number theory; .
       
     • and lastly in function theory.
       
     • The school which Schur built at Berlin was of major importance not only for the representation theory of groups but, as indicated above, for other areas of mathematics.
       
     • They worked on topics such as soluble groups, combinatorics, and matrix theory.
       

167. Planck biography
     
     • Planck returned to Munich and received his doctorate in July 1879 at the age of 21 with a thesis on the second law of thermodynamics entitled On the Second Law of Mechanical Theory of Heat.
       
     • Following this Planck continued to work for his habilitation which was awarded on 14 June 1880, after he had submitted his thesis on entropy and the mechanical theory of heat, and he became a Privatdozent at Munich University.
       
     • At first the theory met resistance but, due to the successful work of Niels Bohr in 1913 calculating positions of spectral lines using the theory, it became generally accepted.
       
     • Planck himself in [Scientific Autobiography, and Other Papers (1949).',7)">7] explains how despite having invented quantum theory he did not understand it himself at first:- .
       
     • I tried immediately to weld the elementary quantum of action somehow in the framework of classical theory.
       
     • My futile attempts to put the elementary quantum of action into the classical theory continued for a number of years and they cost me a great deal of effort.
       
     • Planck who was 42 years old when he made his historic quantum announcement, took only a minor part in the further development of quantum theory.
       
     • This was left to Einstein with theories of light quanta, Poincare who proved mathematically that the quanta was a necessary consequence of Planck's radiation law, Niels Bohr with his theory of the atom, Paul Dirac and others.
       
     • Max Planck: Quantum Theory .
       

168. Ostrowski biography
     
     • He read through mathematical journals, in his own words, from cover to cover, occupied himself with the study of foreign languages, music and valuation theory ..
       
     • These are determinants, linear algebra, algebraic equations, multivariate algebra, formal algebra, number theory, geometry, topology, convergence, theory of real functions, differential equations, differential transformations, theory of complex functions, conformal mappings, numerical analysis and miscellany.
       
     • To illustrate some of the areas which Ostrowski wrote on we mention his results on the theory of norms of matrices and the applications to finding inequalities, to studying methods to solve linear systems, and to the methods for both finding and approximating eigenvalues.
       
     • His work on aglebraic equations involved a study of the fundamental theorem of algebra, Galois theory, and estimating the roots of algebraic equations.
       
     • He gave a definitive version of the theory of valuations in 1934 after publishing initial results in 1917.
       
     • On number theory he studied a variety of topics including Waring's problem, Diophantine approximation and some problems in algebraic number theory.
       
     • you are able, on the one hand, to emphasise the abstract and axiomatic side of mathematics, as for example in your theory of general norms, or, on the other hand, to concentrate on the concrete and constructive aspects of mathematics, as in your study of numerical methods, and to do both with equal ease.
       

169. Schwartz biography
     
     • It was during this period of his career that he produced his famous work on the theory of distributions described below.
       
     • The outstanding contribution to mathematics which Schwartz made in the late 1940s was his work in the theory of distributions.
       
     • The theory of distribution is a considerable broadening of the differential and integral calculus.
       
     • Schwartz's development of the theory of distributions put methods of this type onto a sound basis, and greatly extended their range of application, providing powerful tools for applications in numerous areas.
       
     • Because of the demands of differentiability in distribution theory, the spaces of test-functions and their duals are somewhat more complicated.
       
     • Harald Bohr presented a Fields Medal to Schwartz at the International Congress in Harvard on 30 August 1950 for his work on the theory of distributions.
       
     • I think every reader of his cited paper, like myself, will have left a considerable amount of pleasant excitement, on seeing the wonderful harmony of the whole structure of the calculus to which the theory leads and on understanding how essential an advance its application may mean to many parts of higher analysis, such as spectral theory, potential theory, and indeed the whole theory of linear partial differential equations ..
       

170. Hurwitz biography
     
     • In particular he attended the one semester course by Weierstrass Introduction to the theory of analytic functions and the notes taken by Hurwitz at this time are reproduced as the book [Einleitung in die Theorie der analytischen Funktionen (Braunschweig, 1988).',2)">2].
       
     • The lectures contained Weierstrass's version of the arithmetisation of analysis including his "construction" of the real numbers, the ε, δ approach to analysis and his theory of complex functions based on power series.
       
     • Klein's new view on modular functions, uniting geometrical aspects such as the fundamental domain with group theory tools such as the congruence subgroups and with topological notions such as the genus of the Riemann surface, was fully exploited by Hurwitz.
       
     • 101 (3) (1999), 97-115.',11)">11] how this work, together with Schur's work on orthogonality relations and the character formula for the orthogonal groups, led to Weyl's papers on the representation theory of semisimple Lie groups.
       
     • Further topics studied by Hurwitz include complex function theory, the roots of Bessel functions, and difference equations.
       
     • This remarkably influential paper was reprinted 100 years later in the proceedings of the Hurwitz Symposium on Stability theory in Ascona in 1995.
       
     • The excellent review [Stability theory, Ascona, 1995, Internat.
       
     • Hurwitz did excellent work in algebraic number theory.
       
     • For example he published a paper on a factorisation theory for integer quaternions in 1896 and applied it to the problem of representing an integer as the sum of four squares.
       
     • History Topics: The beginnings of set theory .
       

171. Dinghas biography
     
     • Right from the time he began his studies in 1931, Dinghas became interested in Nevanlinna theory.
       
     • His work is in many areas of mathematics including differential equations, functions of a complex variable, functions of several complex variables, measure theory and differential geometry.
       
     • His most important work was in function theory, in particular Nevanlinna theory and the growth of subharmonic functions.
       
     • However, the gaps have largely been filled in and the vision of the basic ideas will secure a permanent niche for their author in the theory of functions.
       
     • This treatise presents an amazing amount of function theory in its modest 400 pages.
       
     • Examples are the formula of Plana-Abel-Cauchy, the theorem of Julia-Wolff-Caratheodory, and the theory of Nevanlinna and of Hallstrom.
       
     • This book will clearly prove valuable as a reference or as a text for any student who already knows a modest amount of elementary function theory.
       
     • The final part containing chapters on the maximum principle and the distribution of values, geometric function theory and conformal mapping, and Nevanlinna theory.
       

172. Church biography
     
     • His work is of major importance in mathematical logic, recursion theory, and in theoretical computer science.
       
     • Church's Thesis appears in An unsolvable problem in elementary number theory published in the American Journal of Mathematics 58 (1936), 345-363.
       
     • Another area of interest to Church was axiomatic set theory.
       
     • He published A formulation of the simple theory of types in 1940 in which he attempted to give a system related to that of Whitehead and Russell's Principia Mathematica which was designed to avoid the paradoxes of naive set theory.
       
     • Church bases his form of the theory of types on his λ-calculus.
       
     • Other work by Church in this area includes Set theory with a universal set published in 1971 which examines a variant of ZF-type axiomatic set theory and Comparison of Russell's resolution of the semantical antinomies with that of Tarski published in 1976.
       
     • For example he published Remarks on the elementary theory of differential equations as area of research in 1965 and A generalization of Laplace's transformation in 1966.
       
     • The first examines ideas and results in the elementary theory of ordinary and partial differential equations which Church feels may encourage further investigation of the topic.
       

173. Kepler biography
     
     • Kepler did not express himself in terms of percentage errors, and his is in fact the first mathematical cosmological model, but it is easy to see why he believed that the observational evidence supported his theory.
       
     • Kepler saw his cosmological theory as providing evidence for the Copernican theory.
       
     • Before presenting his own theory he gave arguments to establish the plausibility of the Copernican theory itself.
       
     • Kepler asserts that its advantages over the geocentric theory are in its greater explanatory power.
       
     • For instance, the Copernican theory can explain why Venus and Mercury are never seen very far from the Sun (they lie between Earth and the Sun) whereas in the geocentric theory there is no explanation of this fact.
       
     • Meanwhile, in response to concerns about the different apparent diameter of the Moon when observed directly and when observed using a camera obscura, Kepler did some work on optics, and came up with the first correct mathematical theory of the camera obscura and the first correct explanation of the working of the human eye, with an upside-down picture formed on the retina.
       
     • In the coach, on his journey to Wurttemberg to defend his mother, he read a work on music theory by Vincenzo Galilei (c.1520 - 1591, Galileo's father), to which there are numerous references in The Harmony of the World.
       

174. Robinson Raphael biography
     
     • in December 1934 for his thesis Some results in the theory of Schlicht functions.
       
     • In 1939 Robinson taught a course in number theory and one of his students was Julia Bowman.
       
     • His doctoral dissertation was on complex analysis, but he also worked on logic, set theory, geometry, number theory, and combinatorics.
       
     • He also examined the concept of 'essentially undecidable' introduced by Tarski, and answered an important open question by constructing a theory with a finite number of axioms that is essentially undecidable.
       
     • The book gives an introductory account of the methods introduced by Tarski for establishing the undecidability of several fairly simple branches of mathematics (group theory, lattices, abstract projective geometry, closure algebras and others).
       
     • As we mentioned above, Robinson worked in number theory and he used the earliest computers to obtain results.
       
     • A number theory colleague wrote the following about Robinson's number theory papers:- .
       
     • This paper not only makes a considerable contribution in simplifying a tangled body of theory; it is wonderfully clear in exposition.
       

175. Alling biography
     
     • He had, by this time, begun working in new areas of algebra publishing papers such as On exponentially closed fields (1962), An application of valuation theory to rings of continuous real and complex-valued functions (1963), and The valuation theory of meromorphic function fields over open Riemann surfaces (1963).
       
     • Both Norman and I shared a strong interest in the theory of valuations and this topic continues to link us.
       
     • cover a wide range of mathematics, from the theory of ordered groups to Riemann surfaces.
       
     • One such book, written jointly with N Greenleaf, is 'Foundations of the theory of Klein surfaces'.
       
     • The third book 'Foundations of analysis over surreal number fields' appeared in 1987, and includes an account of Conway's theory of surreal numbers.
       
     • C Earle, reviewing Foundations of the theory of Klein surfaces, writes:- .
       
     • the authors develop basic function theory on Klein surfaces and explore the relation between compact Klein surfaces and real algebraic function fields.
       
     • The first two parts offer a detailed and scrupulous presentation of the historical development of the theory of elliptic integrals and functions in the 18th and 19th centuries, from Giulio Fagnano and Euler through Legendre, Gauss, Abel and Jacobi to Riemann and Weierstrass.
       
     • Although these two parts amount to roughly two-thirds of the book and form an effective self-contained whole, they only supply the background for the last part, in which the author deals with the actual theme of the book and also with his own contribution to the theory of real elliptic curves.
       

176. Robinson biography
     
     • At Cranfield his interests in the aerodynamic theory of wings, both in subsonic and supersonic flow, broadened and became increasingly comprehensive.
       
     • from London in 1949 for pioneering work in model theory and the metamathematics of algebraic systems.
       
     • A collection of papers Model theory and algebra was published in 1975 as a memorial tribute to Robinson.
       
     • The one applied mathematics book is Wing theory written jointly with J A Laurmann and published in 1956.
       
     • Robinson's contributions to model theory were developed during his time at the University of Toronto.
       
     • He weaved his many contributions and papers into a treatise Introduction to model theory and to the metamathematics of algebra published in 1963.
       
     • the first attempt to write a connected exposition of the new subject of model theory.
       
     • Fenyo has explained the ideas behind the theory:- .
       
     • [Robinson's] theory is based on the metamathematical fact that the system of real numbers is incomplete.
       
     • This book, which appeared just 250 years after Leibniz's death, presents a rigorous and efficient theory of infinitesimals obeying, as Leibniz wanted, the same laws as the ordinary numbers.
       

177. Stoilow biography
     
     • After two years in Bucharest, he was named Professor of Function Theory and Higher Algebra at Cernauti University.
       
     • Stoilow returned to Bucharest in 1939 when he was appointed Head of the Department of the Theory of Functions at the Polytechnic Institute, succeeding Dimitrie Pompeiu.
       
     • His scientific production was influential in the development of the modern theory of analytic functions, and spanned the period from 1914 through 1972, with 77 titles listed in the paper.
       
     • Before he took up his first university appointment in 1919, Stoilow concentrated on the theory of partial differential equations in the complex domain.
       
     • After this he changed somewhat the direction of his work and began to undertake research on the theory of functions of a real variable and on topology.
       
     • From around 1927 he began to work on the topological theory of analytic functions.
       
     • Three theorems of Stoilow, published in 1928, 1932 and 1935, constitute his main contribution to the topological theory of analytic functions, a field of which he must be considered one of the founders.
       
     • Those who study this deeply original book, epoch-making for topology as well as for function theory, are struck by the exceptional variety and richness of the results and the mastery with which the author passes from the concrete intuition of geometrical facts to the most abstract generalisations.
       
     • In the two volumes of Theory of functions of a complex variable (Romanian) (1954, 1958), we see lecture courses which Stoilow gave at the University of Bucharest.
       
     • After a fairly standard introduction to the general theory, beginning with power series, he goes on, in volume 1, to look at topics such as entire and meromorphic functions, doubly periodic functions, conformal mapping on the boundary of a Jordan region, multiple-valued functions, and applications of modular functions to the Picard circle of ideas.
       

178. Atiyah biography
     
     • Atiyah showed how the study of vector bundles on spaces could be regarded as the study of cohomology theory, called K-theory.
       
     • Grothendieck also contributed substantially to the development of K-theory.
       
     • His first major contribution (in collaboration with F Hirzebruch) was the development of a new and powerful technique in topology (K-theory) which led to the solution of many outstanding difficult problems.
       
     • The K-theory and the index theorem are studied in Atiyah's book K-theory (1967, reprinted 1989) and his joint work with G B Segal The Index of Elliptic Operators I-V in the Annals of Mathematics, volumes 88 and 93 (1968, 1971).
       
     • The index theorem could be interpreted in terms of quantum theory and has proved a useful tool for theoretical physicists.
       
     • More recently Atiyah has been influential in stressing the role of topology in quantum field theory and in bringing the work of theoretical physicists, notably E Witten, to the attention of the mathematical community.
       
     • The theories of superspace and supergravity and the string theory of fundamental particles, which involves the theory of Riemann surfaces in novel and unexpected ways, were all areas of theoretical physics which developed using the ideas which Atiyah was introducing.
       

179. Lemaitre biography
     
     • However in 1929 Hubble published work presenting considerably more evidence of an expanding universe, contradicting the then accepted theory of a static universe.
       
     • Eddington and other members of the Royal Astronomical Society began to undertake work to try to solve the problem brought about by the discrepancy between theory and observation.
       
     • There was still a part of Lemaitre's theory that scientists, including Eddington, found impossible to accept, namely the implication that the universe had a beginning at a finite time in the past.
       
     • Lemaitre responded to the objections against his theory in a paper published in Nature in May 1931.
       
     • This was the first explicit formulation of the currently accepted 'big bang' theory.
       
     • We should note that, although accepted by most scientists, Fred Hoyle did not accept this theory and the term 'big bang' was Hoyle's scornful description of Lemaitre's theory in a 1950 radio broadcast.
       
     • After listening to Lemaitre explain his theory in one of these seminars, Einstein stood up and said:- .
       
     • Lemaitre published a more detailed version of his theory in L'univers en expansion in 1933.
       
     • The paper opens with a rapid expository review of the general relativity theory of gravitation, including discussion of kinematics, conservation laws, spherical symmetry, and the solutions of Schwarzschild and de Sitter in terms of comoving coordinates.
       

180. Fermat biography
     
     • However it was during this time that Fermat worked on number theory.
       
     • Fermat is best remembered for this work in number theory, in particular for Fermat's Last Theorem.
       
     • Unsuccessful attempts to prove the theorem over a 300 year period led to the discovery of commutative ring theory and a wealth of other mathematical discoveries.
       
     • Their short correspondence set up the theory of probability and from this they are now regarded as joint founders of the subject.
       
     • Fermat however, feeling his isolation and still wanting to adopt his old style of challenging mathematicians, tried to change the topic from probability to number theory.
       
     • His problems did not prompt too much interest as most mathematicians seemed to think that number theory was not an important topic.
       
     • Frenicle de Bessy was perhaps the only mathematician at that time who was really interested in number theory but he did not have sufficient mathematical talents to allow him to make a significant contribution.
       
     • This grew out of Huygens interest in probability and the correspondence was soon manipulated by Fermat onto topics of number theory.
       
     • The handicap imposed by the awkward notations operated less severely in Fermat's favourite field of study, the theory of numbers, but here, unfortunately, he found no correspondent to share his enthusiasm.
       
     • History Topics: The development of Ring Theory .
       

181. Nielsen Jakob biography
     
     • Dehn greatly influenced Nielsen and introduced him to the newest ideas in topology and group theory.
       
     • His work on group theory is important as he was one of the founders of combinatorial group theory.
       
     • He also produced work on fixed point theory related to that of Dehn and the theory of discontinuous groups of isometries of the hyperbolic plane.
       
     • Jakob Nielsen initiated much of the topology of surfaces and of combinatorial group theory, and for this reason alone he occupies an important place in the history of 20th century mathematics.
       
     • Tidskr 8 (1960), 5-10.',3)">3], and was based on a course of lectures Nielsen gave on the theory of discontinuous groups of isometries of the hyperbolic plane in 1938-39.
       
     • This piece of mathematics, now known as Fenchel-Nielsen theory, had a rather strange history.
       
     • Fenchel and Nielsen decided to write a monograph on the theory but neither were happy with the first draft of the manuscript which they produced.
       
     • Eventually Fenchel wrote a book Elementary geometry in hyperbolic space which was intended to put give an approach with would make presentation of Fenchel-Nielsen theory much clearer.
       

182. Schoenberg biography
     
     • During this time he was engaged in research on a topic in analytic number theory suggested by Schur.
       
     • Theory 63 (1) (1990), 1-2.',1)">1]:- .
       
     • Theory 63 (1) (1990), 1-2.',1)">1]:- .
       
     • It was during this war work that he initiated the work for which he is most famous, the theory of splines.
       
     • Approximation Theory 8 (1973), vi-ix.',4)">4]:- .
       
     • Schoenberg is noted worldwide for his realisation of the importance of spline functions for general mathematical analysis and in approximation theory, their key relevance in numerical procedures for solving differential equations with initial and/or boundary conditions, and their role in the solution of a whole host of variational problems.
       
     • Theory 63 (1) (1990), 1-2.',1)">1] state:- .
       
     • Schoenberg made further outstanding contributions in a series of papers between 1950 and 1959 on the theory of Polya frequency functions.
       
     • He investigated their wide applications in approximation theory in a series of three papers between 1969 and 1973.
       
     • Approximation Theory 8 (1973), vi-ix.',4)">4], written at the time he retired in 1973, his interests were described:- .
       

183. James biography
     
     • James has done wide ranging work in topology, particularly in homotopy theory.
       
     • These papers were on fibre spaces and the homotopy theory of sphere bundles over spheres.
       
     • These volumes covered Henry Whitehead's work in differential geometry, complexes and manifolds, homotopy theory, and algebraic and classical topology.
       
     • There are a number of fascinating problems concerning the Stiefel manifolds which are of importance in the geometric applications of homotopy theory, and in homotopy theory itself.
       
     • K-theory characteristic classes, J-theory, Samelson products) as needed.
       
     • general topology and homotopy theory (1984), another book by James which was based on his lectures, is described as follows:- .
       
     • In this monograph, based on a set of sixteen lectures to students, the author expounds certain parts of general topology which are particulary relevant to homotopy theory.
       
     • Fibrewise topology (1988) is a treatise on general topology, uniform spaces, and homotopy theory from the point of view of fibres.
       

184. Kochina biography
     
     • In 1996 she was awarded the M V Keldysh Gold Medal for a series of studies into hydrodynamics and the theory of filtration.
       
     • An application of the theory of linear differential equations to some problems of ground-water motion published in 1940 is quite typical of many of her papers.
       
     • In 1947 she wrote a major survey The theory of seepage of a fluid in porous media examining many contributions to the theory of seepage of an incompressible fluid through a porous medium.
       
     • Other major texts included Theory of motion of ground water in 1952 with a second completely revised and enlarged edition appearing in 1977.
       
     • This book is a widely quoted source for researchers in filtration theory.
       
     • For example in 1948 she studied numerical solutions of a partial differential equation in On a nonlinear partial differential equation arising in the theory of filtration.
       
     • In 1991 a volume of selected works by Kochina on hydrodynamics and filtration theory was published.
       
     • The papers in this book are divided into eight sections: Kinematics of atmospheric motions; Hydrodynamics; Applications of the analytical theory of linear differential equations in filtration theory; Steady flow in the presence of porous media; Unsteady motion of groundwater; Problems on oil filtration; Gas filtration through coal layers; and Filtration of liquids through porous media.
       

185. Rota biography
     
     • He then undertook doctoral studies, supervised by Jacob T Schwartz, and he was awarded a PhD from Yale in 1956 for his thesis Extension theory of differential operators.
       
     • The topics were wide-ranging: differential equations, ergodic theory, nonstandard analysis, probability, and of course, combinatorics.
       
     • As we have indicated above, Rota worked on functional analysis for his doctorate and, up to about 1960, he wrote a series of papers on operator theory.
       
     • Two papers in 1959-60, although still in the area of operator theory, looked at ergodic theory which is an area which requires considerable combinatorial skills.
       
     • These papers seem to have led Rota away from operator theory and into the area of combinatorics.
       
     • His first major work on combinatorics, which was to change the direction of the whole subject, was On the Foundations of Combinatorial Theory which Rota published in 1964.
       
     • The Prize citation singles out the 1964 paper On the Foundations of Combinatorial Theory as:- .
       
     • This paper was the first of a series of ten papers with this main title, all ten have subtitles (for example this first one was subtitled Theory of Mobius functions ) and all the remaining nine have between one and three additional co-authors.
       
     • Rota observes that combinatorics is providing the essential continuing link between mathematics and the sciences: biology (structure of large molecules), linguistics (context-free languages, automata theory), physics (statistical mechanics, phase transition problems, elementary particles).
       

186. Gnedenko biography
     
     • He became deeply interested in probability theory after attending seminars by Kolmogorov and Khinchin.
       
     • In June 1937 Gnedenko was examined on his doctoral dissertation on the theory of infinitely divisible distributions.
       
     • He held these posts until 1960 when he returned to Moscow University, becoming Head of the Department of Probability Theory in 1966.
       
     • One of Gnedenko's most famous books is Course in the Theory of Probability which first appeared in 1950.
       
     • In 1966, along with I V Kovalenko, he published Introduction to queuing theory.
       
     • This is an attractively written systematic exposition of the basic probabilistic methods of congestion theory.
       
     • The book is written on a mature level, but little probability theory beyond very elementary concepts or very intuitive ones is presupposed.
       
     • For example, with Solov'yev, he wrote the elementary work Mathematics and reliability theory in 1982 which aimed to give a popular account of the mathematical theory of reliability.
       
     • In an elementary way the book describes the basic notions of reliability, lifetime distributions, redundant systems, renewal and maintenance theory, inclusion-exclusion principle, and the estimation of reliability.
       

187. Stueckelberg biography
     
     • The Michigan summer physics programme was held every year and, in 1928, Kramers lectured on quantum theory and Ehrenfest on statistical physics.
       
     • In September 1934 Stueckelberg submitted the paper Relativistisch invariante Storungstheorie des Diracschen Elektrons (The relativistically invariant perturbation theory of the direct electron) to Annalen der Physik.
       
     • Concerning the formalisation of scattering theory, I wish to draw your attention to a paper of Stueckelberg (1934).
       
     • This paper is not written very well, but the basic idea (which goes back to Wentzel) seems to me reasonable; it consists of establishing relativistic invariance by the fact that one removes space and time totally from the theory, and directly examines the coefficients of the four-dimensional Fourier expansion of the wave function.
       
     • A big advance in theoretical physics was the renormalization programme in quantum field theory.
       
     • He then quoted Stueckelberg's 1934 paper as giving an example of such a covariant theory.
       
     • for his theory for critical phenomena in connection with phase transitions.
       
     • Wilson built his theory on an essential modification of a method in theoretical physics called renormalization group theory, which was developed already during the fifties and was applied with varying success to different problems.
       
     • Stueckelberg, having such a problem with his student T A Green, in the work they were doing in S-matrix theory, asked me if I would be interested in working at the Swiss Atomic Energy Commission, in order to deal with this problem in a mathematical way, according to Schwartz, Sobolev and others.
       

188. Kahler biography
     
     • He entered the University of Leipzig in 1924 and took a course on Galois theory in his first semester.
       
     • This paper was the starting point for Kahler geometry with its notions of Kahler manifolds and Kahler groups which are today fundamental ideas in string theory and the study of space-time.
       
     • He had just published his booklet entitled Einfuhrung in die Theorie der Systeme von Differentialgleichungen, which gives a treatment of the theory developed by Elie Cartan.
       
     • [For my thesis I] received much advice from Kahler, from whom I learned the subject of exterior differential calculus and what is now known as the Cartan-Kahler theory.
       
     • [He] gave a five semester course with sometimes more than 10 hours of classes per week, in which he expostulated on algebra, algebraic geometry, function theory and arithmetic.
       
     • He especially dwells on the theory of monades of Leibniz and on the work "Also sprach Zarathustra" of F Nietzsche.
       
     • His speculative considerations are illustrated by suggestive examples from set theory, mathematical logic, abstract algebra and differential, algebraic and analytic geometry.
       
     • The main thesis of the paper is that algebraic geometry is a prolegomenon to a mathematical theory of monades.
       
     • In The Poincare group (1986) Kahler defines a new Poincare group using the theory of quaternions and puts forward the hypothesis that a general theory of matter must be founded on this new Poincare group.
       

189. Grauert biography
     
     • The papers are arranged under three headings which indicate the main areas of his research: General theory of complex spaces, Levi problem and pseudoconvexity, Fibre bundles, Direct images, q-convexity and cohomology, Deformation of complex objects, Decomposition of complex spaces, and Special results (which contains three papers on complex manifolds).
       
     • Other areas on which Grauert wrote papers, but are not included in [Hans Grauert : Selected papers (Springer-Verlag, Berlin, 1994).',2)">2], are hyperbolicity, non-Archimedean function theory and quantum physics.
       
     • Hans Grauert has been the leading mathematician in the theory of several complex variables in his generation.
       
     • This text is an excellent introduction to the classical themes of modern several complex variables theory: domains of holomorphy, holomorphic complexity, pseudoconvexity, the ring of convergent power series, analytic subvarieties and the several variables version of the Mittag-Leffler and Weierstrass problems ..
       
     • The development of the basic theory is done very well, with pedagogically careful definitions, motivating examples and precise (sometimes even a little pedantic) proofs of various important results.
       
     • In 1977 Grauert and Remmert published Theorie der Steinschen Raume (an English translation Theory of Stein spaces appeared in 1979 and was reprinted in 2004) and the same two authors published Coherent analytic sheaves in 1984.
       
     • Note that on one hand we place great importance on geometry, particularly on the interface between geometric visualization and mathematical-logical formulation; on the other hand, we also treat in this book a large part of the basic theory that is needed, say, by students of mathematical economics or physics, even though it only reflects the contents of the first part of the two-semester course 'Analytical geometry and linear algebra'.
       
     • Much space is occupied by the treatment of systems of linear equations (the Gaussian algorithm), the theory of determinants and the theory of eigenvalues of linear mappings in 'Euclidean' vector spaces (transformation of principal axes).
       
     • to give an understandable introduction to the theory of complex manifolds.
       

190. Fock biography
     
     • Schrodinger published his two fundamental papers on quantum theory in the spring of 1926 and Fock immediately started to develop the ideas and by the end of the year two of his own important papers on the Schrodinger equation had been published.
       
     • the fundamental paper of 1935 in which the full symmetry structure of the hydrogen atom energy levels was shown to be given by the full Lorentz group; and the 1937 paper on the proper time parametrization of the Dirac equation, seminal for the later development of Schwinger's theory of field propagators and for the whole subject of parametrised field theories.
       
     • He set out these views clearly in his textbook Theory of space, time and gravitation published in 1955.
       
     • Second, it is an eloquent and at times somewhat polemical plea for an unorthodox interpretation of Einstein's theory of gravitation.
       
     • [Fock] violently objects to the "principle of equivalence" which Einstein regarded as the corner-stone of the theory.
       
     • He objects to the use of the name "general relativity" for Einstein's theory of gravitation.
       
     • In this sense the gravitation theory contains less relativity than what is usually called "special relativity", and not more.
       
     • So he vehemently insists that we call "special relativity" "relativity", and "general relativity" "gravitation theory." ..
       
     • The reviewer feels that the author has made a major contribution to the understanding of gravitation theory, especially by his insistence on studying the solutions of the field equations and not merely the formal properties of the equations.
       
     • The articles presented also possess a great historical value, most of them representing important steps in the development of quantum mechanics and quantum field theory during the first half of last century, and should be subject to careful and detailed analysis from historians of science.
       

191. Gentzen biography
     
     • The paper studies the theory of 'sentence systems' and answers a major open problem in the subject by constructing a counterexample to show that not all sentence systems have independent axiom systems.
       
     • The idea of levels, probably first introduced by Weyl, considers number theory as the first level since it deals with the natural numbers, analysis as the second level since it deals with the real numbers, and set theory as the third level where the full extent of Cantor's cardinal and ordinal numbers would be studied.
       
     • Gentzen wrote several papers on these concepts, particularly examining the occurrence of set theory paradoxes.
       
     • We can paraphrase it by saying that for number theory no once-and-for-all sufficient system of forms of inference can be specified, but that on the contrary, new theorems can always be found whose proof requires a new form of inference.
       
     • The aim of the present paper is to prove the consistency of elementary number theory or, rather, to reduce the question of consistency to certain fundamental principles.
       
     • This faith in the indubitable certainty of mathematical proofs was sadly shaken around 1900 by the discovery of the antinomies or paradoxes of set theory.
       
     • I shall carry out such a consistency proof for elementary number theory.
       
     • to what extent the Gentzen proof can be accepted as securing classical number theory in the sense of that problem formulation is in the present state of affairs a matter of individual judgement.
       
     • In the summer of 1942 he submitted his Habilitation thesis Provability and nonprovability of restricted transfinite induction in elementary number theory to Gottingen and, on the award of the degree, he became entitled to teach in universities.
       

192. Parry biography
     
     • There followed a number of papers on ergodic theory.
       
     • In 1963 he published An ergodic theorem of information theory without invariant measure generalising the individual version of McMillan's ergodic theorem of information theory without the hypothesis of an invariant probability function.
       
     • Particularly important in Parry's development as a research mathematician was the year 1962-63 during which he worked at Yale University in the United States with a group of other young mathematicians interested in ergodic theory.
       
     • There he worked on entropy theory showing, amongst other things, that each aperiodic measure-preserving transformation could be viewed as the shift on the realisation space of a stationary, countable state, stochastic process indexed by the integers or the natural numbers.
       
     • These books included Entropy and generators in ergodic theory (1969), Topics in ergodic theory (1981), and (with Selim Tuncel) Classification problems in ergodic theory (1982).
       
     • This book is an attractive and clear introduction to entropy and generators in ergodic theory which allows the reader who is not an expert in ergodic theory to gain an appreciation of the flavour of the subject and an understanding of the important theorems.
       

193. Livsic biography
     
     • However, as he wrote, see [Topics in operator theory : Essays dedicated to M S Livsic on the occasion of his 70th birthday (Basel-Boston, 1988).',1)">1]:- .
       
     • In fact [Topics in operator theory : Essays dedicated to M S Livsic on the occasion of his 70th birthday (Basel-Boston, 1988).',1)">1]:- .
       
     • He used the ideas, techniques and methods of analytic function theory throughout all his research.
       
     • His Master's Degree was achieved with a thesis in quasianalytic functions, then he became interested in operator theory which came out of earlier work he had done on the moment problem.
       
     • It was 1942 before he was able to defend his doctoral thesis on Hermitian operator theory and the generalised moment problem.
       
     • His habilitation thesis on generalisations of von Neumann's extension theory was examined in 1945 by a powerful groups of mathematicians, namely Banach, Gelfand, Naimark and Plessner at the Steklov Institute.
       
     • He remained there until 1957, publishing results on applications of his functional analysis results to quantum theory.
       
     • Again he began building a research school in operator theory.
       
     • He is described in [Topics in operator theory : Essays dedicated to M S Livsic on the occasion of his 70th birthday (Basel-Boston, 1988), 6-15.',2)">2] as follows:- .
       
     • His recent breakthroughs in the theory of characteristic functions for several commuting operators indicate that in spite of his seventy years, mathematically Moshe is still a young man.
       

194. Pontryagin biography
     
     • lies not merely in its effect on the further development of topology; of equal significance is the fact that his theorem enabled him to construct a general theory of characters for commutative topological groups.
       
     • This theory, historically the first really exceptional achievement in a new branch of mathematics, that of topological algebra, was one of the most fundamental advances in the whole of mathematics during the present century..
       
     • In 1934 Pontryagin was able to prove Hilbert's Fifth Problem for abelian groups using the theory of characters on locally compact abelian groups which he had introduced.
       
     • The essential tool of cobordism theory is the Pontryagin-Thom construction.
       
     • He began to study applied mathematics problems, in particular studying differential equations and control theory.
       
     • From the 1930s Pontryagin had been friendly with the physicist A A Andronov and had regularly discussed with him problems in the theory of oscillations and the theory of automatic control on which Andronov was working.
       
     • In 1961 he published The Mathematical Theory of Optimal Processes with his students V G Boltyanskii, R V Gamrelidze and E F Mishchenko.
       
     • He then produced a series of papers on differential games which extends his work on control theory.
       
     • Pontryagin's work in control theory is discussed in the historical survey [SIAM Journal on Control and Optimization 27 (5) (1989), 916-939.',3)">3].
       

195. Suetuna biography
     
     • This was an exciting period to study at Tokyo University for Takagi published his famous paper on class field theory in 1920.
       
     • When Suetuna was in his final undergraduate year his studies were supervised by Takagi and this inspired Suetuna to work on number theory.
       
     • In particular he read Hardy and Littlewood's paper The approximate functional equation in the theory of the zeta function with applications to the divisor problems of Dirichlet and Piltz which appeared in the Proceedings of the London Mathematical Society.
       
     • However he did read books on probability and statistics while in Germany and by the time he returned to Japan he had gained considerable expertise in probability and statistics despite concentrating on his research topics in algebra and number theory.
       
     • He continued to publish research papers on topics related to Artin's 1927 paper and he also wrote several books: one on algebra and number theory, one on analytic number theory, and one on probability.
       
     • The Analytical theory of numbers was originally published in the form of lecture notes but in 1950 a revised edition was published which incorporated recent developments of the theory.
       
     • This book, based mainly on the Riemann zeta-functions and L-functions, is a unique exposition of the analytical theory of numbers in a modern sense as can be seen from the chapter headings: I) Riemann's zeta-functions; II) Hecke's L-functions; III) Dirichlet's L-functions; and IV) Artin's L-series.
       
     • The paper [Sugaku 23 (1) (1971), 49-53.',2)">2] details his publications consisting of thirty works on number theory, twelve on the foundations of mathematics and eight on Buddhist philosophy.
       

196. Penrose biography
     
     • of the mathematical apparatus of gravitation theory, with emphasis on the geometrical theory of the Riemann tensor.
       
     • Penrose looked for a unified theory combining relativity and quantum theory since quantum effects become dominant at the singularity.
       
     • One of Penrose's major breakthroughs was his introduction of twistor theory in an attempt to unite relativity and quantum theory.
       
     • This is a remarkable mathematical theory combining powerful algebraic and geometric methods.
       
     • taken as reducing mental activity to the carrying out of an algorithmic process, and to propose that a more adequate theory of mind will have to be founded on an as yet not existing physical theory adequate to the known nature of the material world.
       
     • In the process of the argument elegant expositions, at a level suitable for the unlearned but reasonably sophisticated reader, are given of a wide variety of topics ranging from the nature of algorithms and abstract computability, through results on undecidability and incompleteness, the basic structures of classical physics, the basic structures and philosophical puzzles in quantum mechanics, the basic features of entropic asymmetry and its relation to cosmological structure, the search for an adequate quantum theory of gravity, to some of the results of neuro-anatomy and research into the functioning of the brain.
       
     • His development of Twistor Theory has produced a beautiful and productive approach to the classical equations of mathematical physics.
       
     • Sir Roger, Emeritus Rouse Ball Professor of Mathematics at the University of Oxford, has made outstanding contributions to general relativity theory and cosmology, most notably for his work on black holes and the Big Bang.
       
     • He applied new mathematical techniques to Einstein's theory, and led the renaissance in gravitation theory in the 1960s.
       

197. Hua biography
     
     • By now he had published widely on questions within the orbit of Waring's problem (also on other topics in diophantine analysis and function theory) and he was well prepared to take advantage of the stimulating environment of the Hardy-Littlewood school, then at the zenith of its fame.
       
     • In the years ahead, even though Hua's scientific activities branched out in other directions, Hua was always ready to return to Waring's problem, to number theory in general and especially to questions involving exponential sums; thus as late as 1959 he published an important monograph on Exponential Sums and Their Applications in Number Theory for the Enzyklopadie der Matematischen Wissenschaften.
       
     • His instinct for what was important and his marvellous command of technique make his papers on number theory even now virtually an index to the major activities in that subject during the first half of the twentieth century.
       
     • In September 1946, shortly after returning from Russia, Hua did depart for Princeton, bringing with him projects not only in matrix theory but also in functions of several complex variables and in group theory.
       
     • In 1956 his voluminous text, Introduction to Number Theory, appeared.
       
     • In connection with the last of these, the study of the Monte Carlo method and the role of uniform distribution led them to invent an alternative deterministic method based on ideas from algebraic number theory.
       
     • Their theory was set out in Applications of Number Theory to Numerical Analysis, which was published much later, in 1978, and by Springer in English translation in 1981.
       

198. Hutton James biography
     
     • First his remarkable theory of the age of the Earth was inspired by Newton's world view as presented in the teaching of Colin Maclaurin, and second that one of his main collaborators in his geological research was John Playfair.
       
     • It was around this time that Hutton began to read books on the topic; reading the works Discourse on earthquakes (1705) by Hooke, New theory of the Earth (1696) by Whiston, Protogaea (1749) by Leibniz and Histoire naturelle by Buffon, as well as Steno's treatise Dissertation concerning a solid body enclosed by the process of nature within a solid.
       
     • Some time during 1784 the Royal Society of Edinburgh invited Hutton to give two lectures on his theory.
       
     • Hutton's reaction was what one would expect of an outstanding scientist - he undertook trips to view rock formations to try to gain further evidence to prove that his theory was correct.
       
     • He then worked on a book which would explain his theory in more detail.
       
     • This treatise The theory of the earth appeared in 1795 but sadly it was not nearly such a good book as it might have been since Hutton wrote it during a time of deteriorating health.
       
     • Hutton had a rather peculiar written style of presentation which made his theory less intelligible and, as a result, he received less acclaim than he deserved.
       
     • Playfair's published Illustrations of the Huttonian Theory of the Earth in 1802.
       
     • He presented Hutton's theories in a different style from that of The theory of the earth.
       
     • His simple and eloquent style consisted of a series of chapters clearly stating the Huttonian theory, giving the facts to support it, and the arguments given against it.
       

199. Mises biography
     
     • Ludwig, who was about eighteen months older than Richard, went on to become an economist who contributed to liberalism in economic theory and made his belief in consumer power an important part of that theory.
       
     • Von Mises worked on fluid mechanics, aerodynamics, aeronautics, statistics and probability theory.
       
     • His studies of wing theory for aircraft led him to investigate turbulence.
       
     • His most famous, and at the same time most controversial, work was in probability theory.
       
     • One has even forgiven him his theory of probability.
       
     • After the measure theory approach by Kolmogorov had become favoured by almost all statisticians over von Mises' limiting frequency theory approach, there was a return to von Mises ideas and there was an attempt to incorporate them into the measure theoretic approach of Kolmogorov who wrote himself in 1963:- .
       
     • that the basis for the applicability of the results of the mathematical theory of probability to real 'random phenomena' must depend on some form of the frequency concept of probability, the unavoidable nature of which has been established by von Mises in a spirited manner.
       
     • von Mises' notion of a random sequence in the context of his approach to probability theory.
       

200. Bondi biography
     
     • Bondi is perhaps best known as a creator of the steady-state theory of the universe which goes further than the accepted theory that the universe looks essentially the same from every place, and proposes in addition that the universe essentially remains 'the same' for all time.
       
     • They first put the theory forward in 1946.
       
     • For the extrapolative theories, such as the theory of relativity, this "principle" is merely an aid to the formulation of a problem which is to be attacked within the framework of the extrapolated physical laws; for the deductive theories, such as Milne's kinematical relativity, it is an a priori requirement, a sort of categorical imperative, to which physical experience must conform.
       
     • Of special note is the case of the "Perfect Cosmological Principle", which would require that the world-views obtained by equivalent observers are in addition stationary (but not necessarily static); upon it are based the Bondi-Gold steady-state theory (deductive) and the Hoyle theory of continuous creation (mainly extrapolative).
       
     • It would certainly be a mistake to think that this represents the most important part of Bondi's scientific work, however, for he was a leading expert on many topics in applied mathematics, in particular in relativity theory.
       
     • When evidence began to accumulate showing that the steady-state theory did not hold, Bondi's reputation was not seriously affected.
       
     • With R A Littleton, he wrote two papers On the dynamical theory of the rotation of the earth (1948 and 1953).
       
     • With W H McCrea he published Energy transfer by gravitation in Newtonian theory (1960) and then went further with his single authored paper On the physical characteristics of gravitational waves (1962).
       

201. Yano biography
     
     • Everyone in Japan was saying that the theory of relativity was so difficult that only twelve people in the world could understand it.
       
     • I do not know how difficult the theory of relativity is to understand, but it was not created by God, it was created by a human being called Albert Einstein.
       
     • So Kentaro! I am sure that if you study hard you may someday understand what the theory of relativity is.
       
     • This stuck with Kentaro so when he was in secondary school and discovered that his physics textbook had an appendix on the theory of relativity, he tried to read it.
       
     • Not finding it too difficult he spoke to his physics teacher who explained to him that what he had read was the special theory of relativity.
       
     • To understand the general theory of relativity one had to study differential geometry.
       
     • In the following year his book The theory of Lie derivatives and its applications was published.
       
     • This is a comprehensive treatise of the theory of Lie derivatives.
       
     • Although the book is self-contained it continues to develop the material treated in Yano's earlier books Curvature and Betti numbers and Theory of Lie derivatives and its applications.
       
     • The purpose of this book is to provide an introduction to the theory of various differential-geometric structures on manifolds and to gather and organize the results on submanifolds of Riemannian manifolds and of Kahlerian manifolds with such structures.
       

202. Solitar biography
     
     • in group theory with Emil Artin as his advisor.
       
     • This did not work out as he had hoped since Artin was working in class field theory and Solitar was very certain of the area in which he wanted to undertake research.
       
     • Abe had visited Donald while he had been studying at Princeton and the two had begun undertaking research into group theory which Karrass was studying with Magnus.
       
     • In 1957 they published Note on a theorem of Schreier then in the following year the two papers On free products and Subgroup theorems in the theory of groups given by defining relations.
       
     • Also during this time the classic text Combinatorial group theory: Presentations of groups in terms of generators and relations by Wilhelm Magnus, Abe Karrass, and Donald Solitar, was published.
       
     • Here is the Preface of Combinatorial group theory.
       
     • This book is an excellent and detailed account, with many examples, of some aspects of group theory closely connected with generators and relations.
       
     • this book covers a section of group theory not adequately treated elsewhere in the literature.
       
     • One of the subtleties which the beginning student of group theory has to master is this: When a group is given by generators and defining relations, there is no obvious way of telling when different expressions represent the same element (even the elements represented by different generators need not be distinct).
       
     • "Combinatorial group theory" .
       

203. Hoehnke biography
     
     • Around this time, almost certainly motivated by his investigation of groupoids, he embarked on an ambitious programme to examine the structure theory of semigroups.
       
     • The bulk of Hoehnke's work on semigroups relies on ring theory and is based on the observation that, in most aspects, congruences of semigroups play the role of ideals of rings.
       
     • The natural starting point for the structure theory of semigroups is the theory of transformation semigroups, in parallel to Jacobson's structure theory of rings which uses linear transformations of vector spaces.
       
     • The main directions of this theory should be: the theory of (0)-primative semigroups, the 0-radical, simplifications of the theory under finiteness conditions, a study of the congruence lattice, Kronecker products (coming from matrix theory) for semigroups and acts, topological methods, comparison of the lift- and right-sided behaviour of a semigroup and finally, finding the place of all former results ..
       
     • in this theory.
       

204. Remez biography
     
     • He continued to undertake research in approximation theory, particularly in the constructive theory of functions, and in numerical analysis for the rest of his career.
       
     • His main work was on the constructive theory of functions and approximation theory.
       
     • Remez generalised Chebyshev-Markov characterisation theory and used it to obtain approximate solutions of differential equations.
       
     • This is the first volume to include between its two covers a fairly complete development of Chebyshev approximation, its theory and its practice.
       
     • An appendix sets forth the theory of convex bodies as far as it is required in the main text.
       
     • Remez lectured at this meeting on Chebyshev's work on approximation theory.
       
     • In 1974 Remez published Some of the principal divisions of P L Chebyshev's scientific activity (Ukrainian) which discussed Chebyshev's contributions to number theory, the theory of mechanisms, approximation of functions, minimax problems, and cartography.
       

205. Riemann biography
     
     • On one occasion he lent Bernhard Legendre's book on the theory of numbers and Bernhard read the 900 page book in six days.
       
     • He learnt much from Eisenstein and discussed using complex variables in elliptic function theory.
       
     • It was during his time at the University of Berlin that Riemann worked out his general theory of complex variables that formed the basis of some of his most important work.
       
     • Riemann's thesis studied the theory of complex variables and, in particular, what we now call Riemann surfaces.
       
     • It therefore introduced topological methods into complex function theory.
       
     • The work builds on Cauchy's foundations of the theory of complex variables built up over many years and also on Puiseux's ideas of branch points.
       
     • The general theory of relativity splendidly justified his work.
       
     • The paper Theory of abelian functions was the result of work carried out over several years and contained in a lecture course he gave to three people in 1855-56.
       
     • Prior to the appearance of his most recent work [Theory of abelian functions], Riemann was almost unknown to mathematicians.
       

206. Lorentz biography
     
     • Lorentz refined Maxwell's electromagnetic theory in his doctoral thesis The theory of the reflection and refraction of light presented in 1875.
       
     • Lorentz developed his mathematical theory of the electron for which he received the Nobel Prize in 1902.
       
     • Lorentz transformations, which he introduced in 1904, form the basis of Einstein's special theory of relativity.
       
     • This conference looked at the problems of having two approaches, namely that of classical physics and of quantum theory.
       
     • However Lorentz never fully accepted quantum theory and always hoped that it would be possible to incorporate it back into the classical approach.
       
     • They embodied the first systematic appearance of the electrodynamic principle of relativity, and in 1920 he brought out "The Einstein Theory of Relativity: A Concise Statement".
       
     • In 1909 he published his "Theory of Electrons", based on a series of lectures at Columbia University, and in 1916 he published in French at Leipzig an account of statistical thermodynamic theories, based on lectures delivered at the College de France in 1912.
       
     • He was also the author of a textbook of the differential and integral calculus; "Visible and Invisible Movements", 1901; and "Clerk Maxwell's Electromagnetic Theory", 1924.
       

207. Dirac biography
     
     • Dirac had been hoping to have his research supervised by Ebenezer Cunningham, for by this time Dirac had become fascinated in the general theory of relativity and wanted to undertake research on this topic.
       
     • Fowler was then the leading theoretician in Cambridge, well versed in the quantum theory of atoms; his own research was mostly on statistical mechanics.
       
     • No doubt Fowler aroused his interest in the quantum theory, and in May 1924 Dirac completed his first paper dealing with quantum problems.
       
     • This similarity provided the clue which led him to formulate for the first time a mathematically consistent general theory of quantum mechanics in correspondence with Hamiltonian mechanics.
       
     • His lectures at Cambridge were closely modelled on [The principles of Quantum Mechanics], and they conveyed to generations of students a powerful impression of the coherence and elegance of quantum theory.
       
     • Both children adopted the name Dirac and Gabriel Andrew Dirac went on the became a famous pure mathematician, particularly contributing to graph theory, becoming professor of pure mathematics at the University of Aarhus in Denmark.
       
     • He published his famous paper on classical electron theory, which included mass renormalisation and radiative reaction in 1938.
       
     • Dirac unified the theories of quantum mechanics and relativity theory, but he also is remembered for his outstanding work on the magnetic monopole, fundamental length, antimatter, the d-function, bra-kets, etc.
       
     • we vividly see everywhere the brilliant imprints of Dirac, unifier of quantum mechanics and relativity theory.
       

208. Tits biography
     
     • In 1973 Tits accepted the Chair of Group Theory at the College de France.
       
     • This paper is an essay on how the development of group theory led to the discovery of various families of simple groups, and how these in turn led to the theory of buildings.
       
     • Jordan, in his famous Traite des substitutions et des equations algebriques, published in 1870, promoted Galois' work and put the theory of groups on a firm foundation.
       
     • At this time groups were treated as groups of permutations, but other aspects of group theory were soon on the way.
       
     • Lie visited Paris in 1870 as a graduate student, and went on to create the theory of continuous transformation groups.
       
     • During this time Tits was gradually developing the theory of buildings, and his book "Buildings of spherical type and finite BN-pairs" in 1974 produced a fully-fledged theory that has since found many uses.
       
     • a later approach to buildings, also due to Tits, is mentioned, and we return at the end to the construction of the exceptional groups of Lie type using building theory.
       
     • for their profound achievements in algebra and in particular for shaping modern group theory.
       
     • The theory of buildings is a central unifying principle with an amazing range of applications, for example to the classification of algebraic and Lie groups as well as finite simple groups, to Kac-Moody groups (used by theoretical physicists), to combinatorial geometry (used in computer science), and to the study of rigidity phenomena in negatively curved spaces.
       
     • They complement each other and together form the backbone of modern group theory.
       

209. Bugaev biography
     
     • His research was mainly on analysis and number theory.
       
     • His work in Moscow was to lead to the creation of the Moscow school of the theory of functions of a real variable in 1911, eight years after his death by Egorov, one of his students.
       
     • Bugaev's most important work in number theory was based on an analogy between some operations in number theory and the operations such as differentiation and integration in analysis.
       
     • Bugaev built a systematic theory of discontinuous functions which he called arithmology.
       
     • In this work Bugaev describes mathematics as based on the theory of functions, with geometry and the theory of probability having a minor role.
       
     • established an important component of the philosophic context in which the formation of the Moscow school of the theory of functions of one real variable unfolded at the beginning of the twentieth century.
       
     • The theorem relating convergence almost everywhere and uniform convergence by D F Egorov, one of Bugaev's pupils, in 1911 is seen as marking the beginning of the Moscow school of the theory of functions of a real variable.
       

210. Boersma biography
     
     • Christoffel J Bouwkamp was a mathematician, an expert on diffraction theory, who was working at the Laboratories and he became Boersma's advisor.
       
     • Also in 1960 he published Mathematical theory of the two-body problem with one of the masses decreasing with time.
       
     • In 1964 he was awarded his doctorate by the University of Groningen for his thesis Boundary value problems in diffraction theory and lifting surface theory.
       
     • The discussion centres around the use of integral representation theory to reduce such problems to Fredholm integral equations which are suitable for the study of low frequency oscillations.
       
     • Various problems in lifting surface theory are also discussed in the paper.
       
     • Complex Function Theory, Applied Analysis and Partial Differential Equations which provided the interesting combination of mathematical theory applied to physics problems.
       
     • An author's theory had to be absolutely meticulous to earn John's approval.
       

211. Landau biography
     
     • His doctoral work there was supervised by Frobenius, and Landau received his doctorate in 1899 for a thesis on number theory.
       
     • He submitted this habilitation thesis in 1901, only two years after his doctorate, consisted of his work on Dirichlet series, a topic in analytic number theory.
       
     • He taught courses for beginners, which he did not have to do, and also lectured on his own speciality of number theory.
       
     • In addition he gave lecture courses on the foundations of mathematics, irrational numbers, and set theory.
       
     • Landau's main work was in analytic number theory and the distribution of primes.
       
     • His masterpiece of 1909 was a treatise Handbuch der Lehre von der Verteilung der Primzahlen a two volume work giving the first systematic presentation of analytic number theory.
       
     • He also wrote important works on the theory of analytic functions of a single variable.
       
     • contains a collection of interesting and elegant theorems of the theory of analytic functions of a single variable.
       
     • Landau wrote over 250 papers on number theory which had a major influence on the development of the subject.
       

212. Privat de Molieres biography
     
     • Many around this time voiced such an opinion (Newton himself realised this was a weakness in his theories) but where Privat de Molieres differed from other critics of Newton's theory of gravitation is that he attempted to make a mathematically sound theory based on the idea of vortices.
       
     • Understanding the accuracy of the theory of gravitation, Privat attempted to bring Newton's calculations into the vortex theory of matter of Malebranche.
       
     • The problem was Kepler's laws, easily explained by Newton, but the cause of real problems for Descartes' vortex theory of planetary motion.
       
     • Nakata gives details of Privat's theory in [Historia Sci.
       
     • Although his arguments were very effective particularly as he was able to incorporate a theory of electrical and chemical phenomena within his little vortices theory, eventually Newtonian physics came to the fore in France.
       
     • His Lecons de physique, contenant les elements de la physique determines par les seules lois des mechaniques (1734-1739), was a four volume work based on his lectures at the College Royal and contains details of the mathematical theory of his elastic "little vortices".
       

213. Campbell biography
     
     • Campbell's book Lie's Theory of Finite Continuous Groups (1903) introduced Lie's ideas to British mathematicians.
       
     • Mr Campbell's interest in the theory of continuous groups was first shown in two papers on "A Law of Combination of Operators" in volumes 28 and 29 of the Proceedings of the London Mathematical Society.
       
     • In these papers he deals, from a point of view which is essentially his own, with the formal results which are at the base of Lie's theory.
       
     • In a paper published two years later "On the Theory of Simultaneous Partial Differential Equations" he develops a system of formulas by which it may be determined whether such a system is or is not integrable.
       
     • His next contribution to the subject was a "Proof of the Third Fundamental Theorem in Lie's Theory".
       
     • In 1903, not very long after the date of the last paper, Mr Campbell's "Introductory Treatise on Lie's Theory of Finite Continuous transformation Groups" was published.
       
     • It gives a wonderfully clear and complete account of a modern theory which, although it already had a literature of its own on the Continent, had, at the time when the book was published, attracted little or no attention in this country.
       
     • The theory in which Mr Campbell was so specially interested underlies most of his more recent work on differential geometry in general, and on that particular branch of it connected with Einstein's gravitational theory.
       

214. Wiener Norbert biography
     
     • In 1914 he went to Gottingen to study differential equations under Hilbert, and also attended a group theory course by Edmund Landau.
       
     • this study introduced me to the theory of probability.
       
     • All these concepts have combined with the engineering preoccupations of a professor of the Mathematical Institute of Technology to lead me to make both theoretical and practical advances in the theory of communication, and ultimately to found the discipline of cybernetics, which is in essence a statistical approach to the theory of communication.
       
     • From 1923 he investigated Dirichlet's problem, producing work which had a major influence on potential theory.
       
     • Wiener had an extraordinarily wide range of interests and contributed to many areas in addition to those we have mentioned above including communication theory, cybernetics (a term he coined), quantum theory and during World War II he worked on gunfire control.
       
     • [Cybernetics] has contributed to popularising a way of thinking in communication theory terms, such as feedback, information, control, input, output, stability, homeostasis, prediction, and filtering.
       
     • Some of Wiener's publications which we have not mentioned include Nonlinear Problems in Random Theory (1958), and God and Golem, Inc.: A Comment on Certain Points Where Cybernetics Impinges on Religion (1964).
       

215. Gelfand biography
     
     • While he did this evening teaching he also attended lectures at Moscow University, the first course he attended being the theory of functions of a complex variable by Lavrentev.
       
     • Gelfand's next major achievement was the theory of commutative normed rings which he created and studied in his D.Sc.
       
     • it was [Gelfand] who brought to light the fundamental concept of a maximal ideal which made it possible to unite previously uncoordinated facts and to create an interesting new theory.
       
     • Gelfand's theory of normed rings revealed close connections between Banach's general functional analysis and classical analysis.
       
     • One important area which he started work on in the early 1940s was the theory of representations of non-compact groups.
       
     • This work followed on from the representation theory of finite groups by Frobenius and Schur and the representation theory of compact groups by Weyl.
       
     • He saw the importance of the work of Sobolev and Schwartz on the theory of generalised functions and distributions, and he developed this theory in a series of monographs.
       

216. Lefschetz biography
     
     • Poincare had studied curves on a surface but Lefschetz pushed the ideas into much more general settings by building a theory of subvarieties of an algebraic variety.
       
     • far reaching theory of the intersection of complexes on a manifold.
       
     • Lefschetz worked on results which provided a deep generalisation of Emile Picard's theorems in function theory to several complex variables.
       
     • In doing this he developed a theory of algebraic topology of algebraic varieties of higher dimension.
       
     • He made extensive use of product spaces; he developed intersection theory, including the theory of the intersection ring of a manifold; and he made essential contributions to various kinds of homology theory, notably relative homology, singular homology, and cohomology.
       
     • He tackled problems related to dissipative nonlinear ordinary differential equations but did not take the usual approach of using linear theory to tackle nonlinear differential equations.
       
     • Lefschetz had many students working in this area and, between 1950 and 1960, a series of important publications Contributions to the theory of nonlinear oscillations appeared in the Annals of Mathematics Studies, published by Princeton University Press.
       

217. Yule biography
     
     • Yule's work entitled On the Theory of Correlation was first published in 1897.
       
     • In the ordinary theory of statistical correlation, normal or otherwise, we are always supposed to be dealing with material susceptible of continuous variation, or at least of variation by a considerable number of discontinuous steps.
       
     • These lectures became the basis for Yule's famous text Introduction to the Theory of Statistics which he first published in 1911.
       
     • He wrote papers on time-correlation in which he introduced the correlogram and he did fundamental work on the theory of autoregressive series.
       
     • In 1937 Yule produced a thorough revision of the text of Introduction to the Theory of Statistics for the eleventh edition published in that year.
       
     • The fourteenth and last edition of Introduction to the Theory of Statistics was written jointly with Maurice Kendall and published in 1950, shortly before Yule's death.
       
     • The first half of the book deals with descriptive statistics: the theory of attributes, frequency distributions and their characteristics, correlation and regression, and curve fitting).
       
     • The second half of the book deals with sampling theory: large and small samples, chi-square, analysis of variance.
       
     • Yule did not develop any completely new branches of statistical theory but he took the first steps in many areas which proved important in their further development by later statisticians.
       

218. Petersen biography
     
     • His research was on a wide variety of topics from algebra and number theory to geometry, analysis, differential equations and mechanics.
       
     • He published The theory of algebraic equations in 1877 which was written in a concise style, treating as many topics as possible without using Galois theory.
       
     • A paper which he wrote in 1891 Die Theorie der regularen Graphs marks the birth of graph theory.
       
     • Watkins writes about Petersen's paper in [36th Southeastern International Conference on Combinatorics, Graph Theory, and Computing, Congr.
       
     • It is reasonable to ask why Petersen began working in graph theory.
       
     • In fact his aim was to discover results in invariant theory and he corresponded with Sylvester on this topic.
       
     • In the last part of his career, from 1888 to 1909, Petersen worked on function theory, latin squares, and number theory.
       

219. Kerekjarto biography
     
     • After Kerekjarto's visit to Gottingen in 1922, in the following year he gave courses at the University of Barcelona entitled Geometry and The theory of functions.
       
     • Examples of paper Kerekjarto published during this time are On a geometrical theory of continuous groups (1925) and On a geometrical theory of continuous groups.
       
     • He was succeeded in Szeged by Gyula Sz›kefalvi-Nagy, a distinguished geometer, founder of the theory of curves with maximal index, formerly professor of the Teacher's Training College in Szeged.
       
     • His most important scientific results were in the area of "classical" topology founded by Poincare and Brouwer and in the theory of continuous groups.
       
     • In fact, while the first theorem played an important role in Poincare's research in dynamics, the second was applied by Brouwer in the theory of continuous groups.
       
     • Also in his later work he preferred to work in topological problems which were closely connected with problems of classical geometry, theory of functions, etc.
       
     • It had also given impetus to the development of the theory of continuous groups.
       
     • The most beautiful results of this theory, in the case of dimension 2, are due to Kerekjarto.
       

220. Meyer biography
     
     • This impressive work extended apolarity theory as introduced by Reye to projective geometry in several dimensions using the theory of rational curves.
       
     • As indicated by the thesis which he submitted for his habilitation, the title of which we quoted above, Meyer's interests lie in the study of algebraic geometry, algebraic curves and projective invariant theory.
       
     • His expertise in invariant theory led to an invitation from the recently founded German Mathematical Society (Deutsche Mathematiker-Vereinigung) to write a report on the subject which he did, publishing his article of over 200 pages Bericht uber den gegenwartigen Stand der Invariantentheorie in Jahresberichte der Deutsche Mathematiker-Vereinigung in 1892.
       
     • In particular he wrote articles on invariant theory, third order surfaces, and surfaces of order higher than three.
       
     • He also co-authored an article on potential theory with H Burkhardt, and an article on the geometry of the triangle with G Berkhan.
       
     • gave lectures discussing the essential aspects of mathematical research in the spirit of Klein's Erlangen programme, and gave lectures discussing the essential aspects of mathematical research in the spirit of the time and emphasizing the importance of simple algebraic identities, the symmetries of group theory, and transformation principles as a source of geometric theorems.
       
     • In particular Hilbert's results on invariant theory were so powerful that they ended a chapter in the development of the subject and so Meyer's many contributions in this area are little remembered today.
       
     • Likewise the work in his habilitation thesis went into the geometrical theory of Segre manifolds set up by Corrado Segre.
       

221. Chudakov biography
     
     • In that year he returned once again to Saratov University, this time to become Head of the Department of Algebra and Number Theory which had just been set up.
       
     • Chudakov established a number of important results in number theory.
       
     • This problem has, from that time on, been one of the major unsolved problems of number theory.
       
     • Chudakov is also famed as the author of the classic monograph Introduction to the theory of Dirichlet L-functions (1947) which was widely used by number theory experts.
       
     • The text assumes only that the reader is familiar with the elements of number theory and complex variable theory, and goes on to develop the theory of characters and Dirichlet L-functions.
       
     • During his many years of academic and pedagogical work at Saratov University, Chudakov introduced more than one generation of mathematicians to research work in number theory and algebra.
       

222. Polya biography
     
     • In the following year he returned to Budapest where he was awarded a doctorate in mathematics having studied, essentially without supervision, a problem in the theory of geometric probability.
       
     • His basic research contributions span complex analysis, mathematical physics, probability theory, geometry, and combinatorics.
       
     • For example, in 1918 he published papers on series, number theory, combinatorics and voting systems.
       
     • It has applications such as the enumeration of chemical compounds and the enumeration of rooted trees in graph theory.
       
     • In fact a whole new area of graph theory called enumerative graph theory grew up based on Polya's ideas.
       
     • He also worked on conformal mappings and potential theory, and he was led to study boundary value problems for partial differential equations and the theory of various functionals connected with them.
       
     • Theory 1 (4) (1977), 289-290.',11)">11]:- .
       

223. Frohlich biography
     
     • The necessary extension of representation theory was published by the author in a previous paper [The representation of a finite group as a group of automorphisms on a finite Abelian group (1950)].
       
     • He and Cassels gave preliminary courses, respectively on local and global algebraic number theory and; the main courses, on local class field theory and on global class field theory, were given by J P Serre and J Tate.
       
     • Before Brighton, class field theory was a recondite mystery known only to a few (in Britain, only a very few indeed); after Brighton, it was a standard tool of mathematics, available to any professional.
       
     • The theory of Galois module structure of rings of algebraic integers has been developed by the author and others during the last twenty years, and the book under review contains a detailed survey of it.
       
     • Section 1 of Chapter I contains a very clearly written history of this theory and an outline of main problems and results.
       
     • In 1986 he published the book Tame representations of local Weil groups and of chain groups of local principal orders had his introductory textbook on algebraic number theory Algebraic number theory appeared in 1993 written jointly with M J Taylor.
       

224. Cohn biography
     
     • In research interests Cohn has worked widely in many areas of algebra but, in particular he has made outstanding contributions to non-commutative ring theory.
       
     • Over the next few years his work ranged across group theory, field theory, Lie rings, semigroups, abelian groups and ring theory.
       
     • From the mid 1960s his work concentrates on non-commutative ring theory and the theory of algebras.
       
     • It completes the formation of the theory of free associative algebras and related classes of rings as an independent domain of ring theory.
       
     • The theory of skew fields is still not so familiar as the commutative analogue.
       

225. Aleksandrov biography
     
     • It was not only the result which was important for set theory, but also the methods which Aleksandrov used which turned out to be one of the most useful methods in descriptive set theory.
       
     • After Aleksandrov's great successes Luzin did what many a supervisor might do, he realised that he had one of the greatest mathematical talents in Aleksandrov so he thought that it was worth asking him to try to solve the biggest open problem in set theory, namely the continuum hypothesis.
       
     • After taking his examinations in 1921, Aleksandrov was appointed as a lecturer at Moscow university and lectured on a variety of topics including functions of a real variable, topology and Galois theory.
       
     • Hausdorff, building on work by Frechet and others, had created a theory of topological and metric spaces in his famous book Grundzuge der Mengenlehre published in 1914.
       
     • Aleksandrov and Urysohn now began to push the theory forward with work on countably compact spaces producing results of fundamental importance.
       
     • He laid the foundations of homology theory in a series of fundamental papers between 1925 and 1929.
       
     • Aleksandrov's work on homology moved forward with his homological theory of dimension around 1928-30 .
       
     • He worked on the theory of continuous mappings of topological spaces.
       

226. Morse biography
     
     • Morse developed variational theory in the large with applications to equilibrium problems in mathematical physics.
       
     • This is now called Morse theory and it grew out of a major discovery which Morse made not long after returning to mathematics after the war and published in his important paper Relations between the critical points of a real function of n independent variables in 1925.
       
     • Morse theory is important in the field of global analysis which is the study of ordinary and partial differential equations from a global or topological point of view.
       
     • Morse's major works include Calculus of variations in the large (1934), Functional topology and abstract variational theory (1938), Topological methods in the theory of functions of a complex variable (1947) and Lectures on analysis in the large (1947).
       
     • However this theme, Morse theory, is perhaps the single greatest contribution of American mathematics.
       
     • These include papers on minimal surfaces, some on the theory of functions of a complex variable where he was particularly interested in applying topological methods, papers on differential topology and on dynamics.
       
     • He also wanted to produce a topological version of quantum theory, but this largely remained a dream which he never achieved.
       
     • In 1933 the American Mathematical Society awarded him the Bocher Prize for his memoir The foundations of a theory of the calculus of variations in the large in m-space published in Transactions of the American Mathematical Society in 1929 (which he shared with Norbert Wiener).
       

227. Pless biography
     
     • Both Emmy Noether and Irving Kaplansky were algebraists, particularly studying ring theory, so it is not surprising that their inspiration led her to algebra.
       
     • During this time she became one of the leading world experts on coding theory.
       
     • Further early papers on coding theory included On the uniqueness of the Golay codes published in the Journal of Combinatorial Theory in 1968 and, in the following year On a new family of symmetry codes and related new five-designs in the Bulletin of the American Mathematical Society.
       
     • In 1982 she published an important text on coding theory Introduction to the theory of error-correcting codes.
       
     • It was designed as a one semester course for undergraduates on algebraic coding theory and has:- .
       
     • proven its usefulness for professors teaching courses in coding theory.
       
     • a woman who is as friendly and engaging as she is unaffected by the dramatic impact she has had in the field of coding theory.
       

228. Rychlik biography
     
     • He also attended Georges Humbert's lectures on number theory at the College de France.
       
     • The first of these was A Contribution to the theory of forms (submitted November 1910), followed by A Contribution to the theory of forms II (submitted June 1911).
       
     • In this role Rychlik gave lectures on the Theory of Algebraic Fields and on the Theory of Algebraic Functions.
       
     • He did excellent work on algebra and number theory, for example he generalised Hensel's ideas on g-adic numbers in 1914, later approaching them via sequences and limits unlike the 'generalised decimal expansion' approach of Hensel.
       
     • He introduced the theory of pseudo-valuation in a paper in 1916, twenty years before the corresponding definition by Mahler, who is usually considered the founder of the theory.
       
     • Other works by Rychlik on Bolzano from this later period of his research career include Theory of real numbers in the manuscripts left by Bolzano (Czech) (1956), Theorie der reellen Zahlen im Bolzano's handschriftlichen Nachlasse (1957), Betrachtungen aus der Logik im Bolzano's handschriftlichen Nachlasse (Czech) (1958), Betrachtungen aus der Logik in Bolzanos handschriftlichem Nachlasse (1958), La theorie des nombres reels de Bolzano d'apres ses manuscrits inedits (Russian) (1958), and Theorie der reellen Zahlen in Bolzanos handschriftlichem Nachlasse (1962).
       

229. Miller biography
     
     • He lived there for two years and this was perhaps the most significant event for his mathematical development for Cole was interested in the theory of groups and he soon had Miller totally fascinated by this topic.
       
     • Miller spent the years from 1895 to 1897 in Europe attending lectures on group theory by Lie in Leipzig and Jordan in Paris.
       
     • Miller worked mostly on group theory but he was also interested in the history of mathematics.
       
     • Many of Miller's group theory papers enumerate the possible finite groups which satisfy given conditions such as: the prime factors which divide the order, the orders of two generating permutations and their product; the types of subgroups; or the degree of a representation as a permutation group.
       
     • Miller did not introduce new techniques to attack these group theory questions and one is tempted to say that he should have applied his undoubted skills to produce fewer yet more significant results.
       
     • His best historical papers are those which look at the history of group theory.
       
     • He also wrote papers such as The founder of group theory, Primary facts in the history of mathematics, and Some thoughts on modern mathematical research.
       
     • In addition Miller wrote a number of books: Determinants (1892); Historical introduction to mathematical literature (1916) and he co-authored Theory and application of finite groups (1916) with Blichfeldt and Dickson.
       
     • His historical writings outside the theory of groups often depended on secondary sources and reflected an attitude overly concerned with pointing out error in published accounts.
       

230. Steinhaus biography
     
     • by Steinhaus and Banach, concentrated mainly on functional analysis and its diverse applications, the general theory of orthogonal series, and probability theory.
       
     • In 1923 he published in Fundamenta Mathematicae the first rigorous account of the theory of tossing coins based on measure theory.
       
     • In 1925 he was the first to define and discuss the concept of strategy in game theory.
       
     • An important contribution to the series was a volume written by Steinhaus jointly with Kaczmarz in 1937, The theory of orthogonal series.
       
     • As we have noted above, other contributions by Steinhaus were on orthogonal series, probability theory, real functions and their applications.
       
     • In particular he is associated with the theory of independent functions, arising from his work in probability theory, and he was the first to make precise the concepts of "independent" and "uniformly distributed".
       

231. Dynkin biography
     
     • For ten years he worked both on the theory of Lie algebras and on probability theory although his main work during this period was in algebra.
       
     • From the time he was appointed to the chair, Dynkin's work became more and more devoted to probability theory.
       
     • His work from this period is contained in two major books Foundations of the Theory of Markov Processes (1959) and Markov Processes (1963) which have become classics of probability theory.
       
     • Following Kolmogorov, Feller, Doob, and Ito, Dynkin opened a new chapter in the theory of Markov processes.
       
     • During his short spell of work there he organized a group of young workers together with whom he obtained important results in the theory of economic growth and economic equilibrium that culminated in the first Soviet report on this topic at the International Mathematics Congress in Vancouver (to which, incidentally, in the usual way, he was not allowed to go).
       
     • Around 1980 Dynkin interpreted and vastly generalized an identity which had first come up in the context of quantum field theory.
       
     • In the last few years Dynkin has obtained exciting results in the theory of "superprocesses" ..
       
     • Eugene B Dynkin has made major contributions to the theory of Lie algebras and to probability theory.
       
     • Dynkin's most famous contribution to the theory of Lie algebras was his use of the "Coxeter-Dynkin diagrams" to describe and classify the Cartan matrices of semisimple Lie algebras.
       
     • Dynkin has laid much of the foundations of the general theory of Markov processes as we know it today.
       
     • He developed the semigroup theory of Markov processes and characterized Markov processes by the generator of their semigroup.
       
     • Dynkin further studied such topics as excessive functions, Martin boundary, additive functionals, entrance and exit laws, random time change, control theory, and mathematical economics.
       
     • Around 1980 Dynkin interpreted and vastly generalized an identity which had first come up in the context of quantum field theory.
       
     • In the last few years Dynkin has obtained exciting results in the theory of "superprocesses".
       

232. Oleinik biography
     
     • In 1990, together with G A Iosifyan and A S Shamaev, she published Mathematical problems in the theory of strongly inhomogeneous elastic media.
       
     • It is a work in three chapters which studies the mathematical theory of perforated elastic bodies and builds into one volume many papers that Oleinik and her two coauthors had produced in the preceding 10 years.
       
     • The three chapters are: Basic mathematical aspects of the theory of elasticity; Homogenization of the equations of linear elasticity; Composites and perforated materials and Spectral problems in homogenization theory.
       
     • Without a doubt, this book promotes an important new stage in homogenization theory.
       
     • In 1996 Oleinik published Some asymptotic problems in the theory of partial differential equations.
       
     • Oleinik published Mathematical methods in boundary-layer theory in 1997, jointly with V N Samokhin.
       
     • Oleinik and her many collaborators are responsible for developing most of this theory over the last four decades.
       
     • These replacement equations have a quite different mathematical character than the Navier-Stokes equations, and as one can easily see from this book, the theory goes along very different lines.
       

233. Lang biography
     
     • Your primary love has always been number theory and you have written, by one colleague's estimate, over 50 books and monographs [actually over 60], many of them concerned with this topic.
       
     • Lang's mathematical research ranged over a wide range of topics such as algebraic geometry, Diophantine geometry (a term Lang invented), transcendental number theory, Diophantine approximation, analytic number theory and its connections to representation theory, modular curves and their applications in number theory, L-series, hyperbolic geometry, Arakelov theory, and differential geometry.
       
     • Other books by Lang include Introduction to algebraic geometry (1958), Abelian varieties (1959), Diophantine geometry (1962), Introduction to differentiable manifolds (1962), Algebraic numbers (1964), Linear algebra (1966), Introduction to diophantine approximations (1966), Introduction to transcendental numbers (1966), Algebraic structures (1967), Algebraic number theory (1970), Introduction to algebraic and abelian functions (1972), Differential manifolds (1972), Elliptic functions (1973), SL(R) (1975), Introduction to modular forms (1976), Complex analysis (1977), Cyclotomic fields (1978), Elliptic curves: Diophantine analysis (1978), Undergraduate analysis (1983), Complex multiplication (1983), Riemann-Roch algebra (1985), The beauty of doing mathematics.
       
     • Three public dialogues (1985), Introduction to complex hyperbolic spaces (1987), Introduction to Arakelov theory (1988), Topics in Nevanlinna theory (1990), Basic analysis of regularized series and products (1993), Fundamentals of differential geometry (1999), and Math talks for undergraduates (1999).
       

234. Cartan Henri biography
     
     • His doctoral studies were supervised by Paul Montel, whose research interests were the theory of analytic functions of a complex variable, and Cartan received his Docteur es Sciences mathematiques in 1928.
       
     • Cartan worked on analytic functions, the theory of sheaves, homological theory, algebraic topology and potential theory, producing significant developments in all these areas.
       
     • The theory of functions of several complex variables has gone from its infancy with the work of Hartogs, Levi and Poincare shortly after the turn of the century to its current role as a central field of modern mathematics, much as its predecessor, function theory in one complex variable, did in the 19th century.
       
     • A central figure in this development has been Henri Cartan, whose series of papers in this field starting in the 1920's dealt with fundamental questions relating to Nevanlinna theory, generalizations of the Mittag-Leffler and Weierstrass theorems to functions of several variables, problems concerned with biholomorphic mappings and the biholomorphic equivalence problem, domains of holomorphy and holomorphic convexity, etc.
       
     • The major developments in the theory from 1930 to 1950 came from Cartan and his school in France, Behnke's school in Munster, and Oka in Japan.
       
     • In particular one cannot appreciate the importance of Cartan's contributions to sheaf theory, Stein manifolds and analytic spaces without studying his 1950/51, 1951/52 and 1953/54 Seminars.
       

235. Iyanaga biography
     
     • In his second year of study Iyanaga took further courses by Takagi which developed group theory, representation theory, Galois theory, and algebraic number theory.
       
     • In his third undergraduate year Iyanaga was allowed to take part in Takagi's seminar on class field theory.
       
     • A question by Takagi led Iyanaga to prove a result which further led to his first paper on class field theory (and his third published paper while he was an undergraduate).
       
     • I was very lucky to follow Artin's course on class field theory together with Chevalley.
       
     • I continued to give courses and organise seminars, in which I used to discuss class field theory with my younger friends.
       
     • As a member of the Science Council of Japan he was the main organiser of an International Symposium on Algebraic Number Theory held in Japan in 1955.
       

236. Christoffel biography
     
     • Christoffel published papers on function theory including conformal mappings, geometry and tensor analysis, Riemann's o-function, the theory of invariants, orthogonal polynomials and continued fractions, differential equations and potential theory, light, and shock waves.
       
     • Between 1865 and 1871 Christoffel published four important papers on potential theory, three of them dealing with the Dirichlet problem.
       
     • This was an early contribution to the theory of shock waves and followed earlier work on one dimensional gas flows by Riemann.
       
     • Christoffel was interested in the theory of invariants.
       
     • The core of the lectures was the course on complex function theory, distinguished by the inspirational name of Riemann.
       
     • Christoffel had developed Riemann's function theory independently, particularly in the area of ultraelliptic functions, but did not publish his research, presenting them only in his lectures, after the model of Weierstrass.
       
     • Christoffel not only contributed to all these fields, but his interests extended to orthogonal polynomials and continued fractions, and the applications of his work to the foundations of tensor analysis, to geodetical science, to the theory of shock waves, to the dispersion of light.
       

237. Tichy biography
     
     • He was awarded his doctorate in 1959 for his thesis An Exposition of Godel's Incompleteness Theorem in the Simple Theory of Types (Czech).
       
     • His first important paper, based on his doctoral thesis, An Exposition of Godel's Incompletness Proof in the Single Type Theory was published in 1962.
       
     • by the University of Exeter in 1971 for his thesis Contributions to the Theory of Postulate Systems.
       
     • Synthetic components of infinite classes of postulates (1971) extends, in certain cases, the fact that a theory with finitely many postulates can be broken into a Ramsey (synthetic) part and an analytic part to theories with infinitely many postulates.
       
     • Other papers include What do we talk about? (1975), Verisimilitude redefined (1976), A new theory of subjunctive conditionals (1978), The transiency of truth (1980), and Constructions (1986).
       
     • Perhaps his most enduring claim to fame lies in his theory called Transparent Intensional Logic, the culmination of his extensive work on semantics and logic.
       
     • He developed what he called Transparent Intensional Logic, a semantic theory within which to analyse both natural and artificial languages.
       
     • This theory is devoted to the central problem of saying exactly what it is that we learn, know and can communicate when we come to understand what a sentence means.
       
     • The theory remains one of the most inspiring and controversial doctrines of contemporary philosophical logic, attracting passionate defenders and equally fierce opponents.
       

238. Erdos biography
     
     • The problems which attracted him most were problems in combinatorics, graph theory, and number theory.
       
     • Erdos did receive the Cole Prize of the American Mathematical Society in 1951 for his many papers on the theory of numbers, and in particular for the paper On a new method in elementary number theory which leads to an elementary proof of the prime number theorem published in the Proceedings of the National Academy of Sciences in 1949.
       
     • Erdos and Graham met at a number theory conference in 1963 and soon began a mathematical collaboration.
       
     • He was one of the century's greatest mathematicians, who posed and solved thorny problems in number theory and other areas and founded the field of discrete mathematics, which is the foundation of computer science.
       
     • He had been a true child prodigy, publishing his first results at the age of eighteen in number theory and in combinatorial analysis.
       
     • AMS Cole Prize in Number Theory1951 .
       
     • AMS Cole Prize for number theory1951 .
       

239. Sneddon biography
     
     • Also considered are interatomic forces and valence, the theory of solids, collision problems, radiation theory and relativistic quantum theory.
       
     • More unusual applications are to topics such as the theory of cosmic ray showers.
       
     • The aim of this book is to present the elements of the theory of partial differential equations in a form suitable for the use of students and research workers whose main interest in the subject lies in finding solutions of particular equations rather than in the general theory.
       
     • Another major text which he published was Mixed boundary value problems in potential theory in 1966.
       
     • This was again a work at the cutting edge of research containing a very complete description of the classical problem of potential theory, namely to determine the electrostatic potential due to a thin circular disk raised to a prescribed potential.
       
     • In 1969 Sneddon published Crack problems in the classical theory of elasticity with M Lowengrub.
       

240. Knapowski biography
     
     • We spoke much on possible applications of the method in analytic number theory.
       
     • Indeed this is precisely what Knapowski began to work on and he obtained his doctorate in 1957 from the Adam Mickiewicz University in Poznan for his thesis Zastosowanie metod Turana w analitycznej teorii liczb (Applications of Turan's methods in analytic number theory).
       
     • Back in the Adam Mickiewicz University in Poznan Knapowski submitted his habilitation thesis On new "explicit formulae" in prime number theory which was published in Acta Arithmetica in 1960.
       
     • Despite his short mathematical career, Knapowski published 53 paper, mostly in algebra and number theory.
       
     • Two areas of number theory which received particular attention from Knapowski were the distribution of primes in different residue classes modulo k, and the sign changes in the remainder term in the prime number formula.
       
     • Among Knapowski's other number theory papers we mention: On prime numbers in arithmetical progression (1958), On the Mobius function (1958), Contributions to the theory of the distribution of prime numbers in arithmetical progressions (1961, 1962), On Linnik's theorem concerning exceptional L-zeros (1961), and Further developments in the comparative prime number theory (8 papers).
       
     • He also wrote on other mathematical topics such as On some criteria for indecomposability of polynomials (1955) and A theorem from finite group theory (1956).
       

241. Zorn biography
     
     • Zorn's aim in this paper was to study field theory and in particular to improve on the method used for obtaining results in the subject.
       
     • What Zorn proposed in the 1935 paper was to develop field theory from the standard axioms of set theory, together with his maximum principle rather than Zermelo's well-ordering principle.
       
     • The paper then indicated how the maximum principle could be used to prove the standard field theory results.
       
     • Zorn made other contributions to set theory, such as his 1944 paper Idempotency of infinite cardinals in which he proved that an infinite cardinal number is equal to its square.
       
     • In addition to his well known work in infinite set theory, Zorn worked on topology and algebra.
       
     • In Alternative rings and related questions I: existence of the radical published in 1941 Zorn considered the theory of the radical of an alternative ring.
       
     • We have looked briefly at Zorn's contributions to algebra and to set theory.
       
     • We may ask that the function be differentiable on one-dimensional (complex) subspaces; here one is led to the theory of the Gateaux differential.
       

242. Hahn biography
     
     • As Menger explains, Hahn was a pioneer in set theory and functional analysis.
       
     • These include a report on integral equation he wrote in 1911, his modification of Hellinger's theory of invariants of quadratic forms, in which he dispensed with the use of the Hellinger integral, and his work on duality in Banach spaces, culminating with his proof of the Hahn-Banach theorem in 1927.
       
     • He wrote papers on the theory of curves including one which gave a rigorous proof of the Jordan's theorem for simple closed polygons which he based on Veblen's geometrical axioms.
       
     • He also studied the theory of ordered abelian groups and ordered fields, initiating the theory in 1907 (see for example [From Dedekind to Godel, Boston, MA, 1992 (Kluwer Acad.
       
     • Another area on which Hahn did research was measure theory.
       
     • In this area he studied a construction of the Lebesgue integral as a limit of Riemann sums, an integral proposed by Borel around 1910, and worked on the theory of abstract measures, in particular product measures.
       
     • The book was co-authored by Arthur Rosenthal and continued to develop material which first appeared in Saks's Theory of the Integral (1937).
       
     • The book is divided into an introduction (containing an exposition of the relevant parts of set theory and point set topology) and five chapters, entitled (I) Additive and totally additive set functions, (II) Measure, (III) Measurable functions, (IV) Integration and (V) Differentiation.
       

243. Neumann Bernhard biography
     
     • In fact it was Remak, more than any of the others, who influenced Neumann to turn towards group theory for at first he intended to become a topologist.
       
     • Schur advised him to apply the same methods to prove results on wreath products of groups and his doctoral dissertation followed on naturally from this first excursion into group theory.
       
     • Returning to questions in group theory which he had studied while in Berlin, he made rapid progress and was awarded his second doctorate in 1935.
       
     • He remained at Cambridge for a year teaching a preparatory course to give students the right background to take Olga Taussky-Todd's algebraic number theory course.
       
     • Neumann is one of the leading figures in group theory who has influenced the direction of the subject in many different ways.
       
     • While still in Berlin he published his first group theory paper on the automorphism group of a free group.
       
     • However his doctoral thesis at Cambridge introduced a new major area into group theory research.
       
     • An indication of some of the topics which interested Neumann can be seen from looking at the material covered in Lectures on topics in the theory of infinite groups (1960).
       
     • Not only did he form a department of very able mathematicians at the ANU specialising in group theory and functional analysis, he also took a deep interest in the Australian Mathematical Society.
       

244. Redei biography
     
     • By 1940, the year he moved from secondary school teaching to become a lecturer at Szeged University, Redei had published over 35 papers on algebraic number theory, particularly on class groups of quadratic number fields.
       
     • In 1941 Redei was appointed to the Chair of Geometry in Szeged but later he was appointed to the Chair of Algebra and Number Theory.
       
     • Let us look now at Redei's work on algebraic number theory.
       
     • He gave a unified theory for the structure of class groups of real quadratic number fields and conditions for solvability of Pell's equation and other indeterminate equations.
       
     • The other two main areas to which Redei contributed are group theory and semigroup theory.
       
     • In group theory he worked for many years of factorisations of finite abelian groups, looking at properties of abelian groups in every element had a unique factorisation as a product of elements one from each of a number of specified subsets of the group.
       
     • One of Redei's most important contributions to semigroup theory is his proof that every finitely generated commutative semigroup is finitely presented.
       
     • This result appears in his book Theorie der endlich erzeugbaren kommutativen Halbgruppen (1963) which was translated into English as The theory of finitely generated commutative semigroups (1965).
       

245. Fogels biography
     
     • He showed considerable mathematical talents at this school and, when he won a mathematics competition and was awarded a book on number theory as the prize, his interest grew still further.
       
     • He would have liked to write a Master's Thesis on number theory but nobody at the University of Latvia could supervise him on that topic.
       
     • He became a dozent in 1937 and gave lectures on algebra and number theory up until the end of 1938 when he travelled to England to undertake research at Cambridge.
       
     • This was a very productive time for Fogels who published twelve papers on number theory over the three years 1947-1950.
       
     • He also became interested in finite methods in number theory.
       
     • Using such finite methods Fogels proved a number of the classical theorems of number theory.
       
     • This was a fruitful period for his research and among his papers were three On the Abstract Theory of Primes published in Acta Arithmetica in 1964, 1965 and 1966.
       
     • The abstract theory of primes had been introduced by Beurling and studied by many authors.
       
     • However, it presented some new interesting connections of the Riemann hypothesis with the theory of prime numbers.
       

246. Temple biography
     
     • Chapman obtained a scholarship for Temple to undertake further research and he spent a year at Imperial working on quantum theory before going to Cambridge where he worked with Eddington.
       
     • Relativity theory, aerodynamics and quantum mechanics have been mentioned above but he also worked on analysis contributing to the study of the Lebesgue integral.
       
     • He wrote seven books, two on quantum theory An introduction to quantum theory (1931) and The general principles of quantum theory (1934).
       
     • His other books included An introduction to fluid dynamics (1958) and The structure of the Lebesgue integration theory (1971).
       
     • to establish the consistency of set theory, abstract arithmetic and propositional logic and the method used is to construct a new and fundamental theory from which these theories can be deduced.
       
     • in recognition of his many distinguished contributions to applied mathematics, especially in his work on distribution theory.
       

247. Rankine biography
     
     • At first he was strongly attracted to number theory but when he was 14 years old one of his uncles gave him a Latin edition of Newton's Principia which he read eagerly.
       
     • He won a Gold Medal for an essay on The wave theory of light in 1836 and another Gold Medal for an essay on Methods in physical investigation two years later.
       
     • inaugural address espoused the harmony of theory with practice in mechanics, and outlined a tripartite theory of knowledge - theory, practice, and the application of theory to practice - which left room for a new breed of engineering scientists to bridge theoretical and practical domains.
       
     • Rankine apparently regarded energy, as we do today, as being classified into two kinds, viz., kinetic and potential, and his thermodynamic theory was developed by considering the transformation of one into the other.
       
     • 31 (106) (1981), 72-134.',10)">10], looks at the entropy function which Rankine defined and its implications for the theory of thermodynamics which he developed.
       
     • Among his most important works are Manual of Applied Mechanics (1858), Manual of the Steam Engine and Other Prime Movers (1859), Civil Engineering (1862), Machinery and Millwork (1869), Useful Rules and Tables (1866), Mechanical Textbook (1873), and On the Thermodynamic Theory of Waves of Finite Longitudinal Disturbance.
       

248. Banach biography
     
     • In 1922 the Jan Kazimierz University in Lvov awarded Banach his habilitation for a thesis on measure theory.
       
     • Banach founded modern functional analysis and made major contributions to the theory of topological vector spaces.
       
     • In addition, he contributed to measure theory, integration, the theory of sets, and orthogonal series.
       
     • The idea was introduced by others at about the same time, for example Wiener introduced the notion but did not develop the theory.
       
     • The importance of Banach's contribution is that he developed a systematic theory of functional analysis, where before there had only been isolated results which were later seen to fit into the new theory.
       
     • The theory generalised the contributions made by Volterra, Fredholm and Hilbert on integral equations.
       
     • The Banach-Tarski paradox was a major contribution to the work being done on axiomatic set theory around this period.
       

249. Eisenhart biography
     
     • From the first, methods of the theory of functions of a real variable are employed.
       
     • The second stage started after 1921 when Eisenhart, prompted by Einstein's general theory of relativity and the related geometries, studied generalisations of Riemannian geometry.
       
     • Riemann proposed the generalisation of the theory of surfaces as developed by Gauss, to spaces of any order, and introduced certain fundamental ideas in this general theory.
       
     • Important contributions to it were made by Bianchi, Beltrami, Christoffel, Schur, Voss, and others, and Ricci-Curbastro coordinated and extended the theory with the use of tensor analysis and his absolute calculus.
       
     • The book gave a presentation of the existing theory of Riemannian geometry after a period of considerable study and development of the subject by Levi-Civita, Eisenhart, and many others.
       
     • In 1933 Eisenhart published Continuous Groups of Transformations which continues the work of his earlier books looking at Lie's theory using the methods of the tensor calculus and differential geometry.
       
     • The new chapter began about 1920 with the extended studies of tensor analysis, Riemannian geometry and its generalizations, and the application of the theory of continuous groups to the new physical theories.
       
     • In fact he published 21 papers between 1951 and 1963, for example: Generalized Riemann spaces and general relativity (1953); A unified theory of general relativity of gravitation and electromagnetism (1956); The cosmology problem in general relativity (1960); and The Einstein generalized Riemannian geometry (1963).
       

250. Janovskaja biography
     
     • She was educated in classics and mathematics at the Gymnasium in Odessa where she was fortunate enough to be taught by Ivan Jure'vich Timchenko, a major figure in the study of the history of mathematics especially that of the theory of analytic functions.
       
     • There she studied mathematics under Timchenko, who we mentioned above, and also Samuil Osipovich Shatunovsky who was interested in a wide variety of mathematical topics including group theory, the theory of numbers, and geometry.
       
     • He used the axiomatic method to lay the logical foundations of geometry, algebraic fields, Galois theory and analysis and his areas of interest had a large influence on his student Neimark.
       
     • The modern crisis of capitalism robs mathematics of materialistic tools and methods (intuitionism), widens the gap between theory and practice, and aggravates its spontaneous and unplanned character.
       
     • The history of mathematics was another topic which attracted Janovskaja and she published work on Egyptian mathematics On the theory of Egyptian fractions (1947), Zeno of Elea's paradoxes, Rolle's criticisms of the calculus in Michel Rolle as a critic of the infinitesimal analysis (1947), Descartes's geometry (see below), and Lobachevsky's work on non-euclidean geometry in papers such as The leading ideas of N I Lobachevsky - a combat weapon against idealism in mathematics (1950), On the philosophy of N I Lobachevsky (1950), and On the Weltanschauung of N I Lobachevsky (1951).
       
     • There then follows general comments concerning the theory of algorithms, and the mathematical concepts of proof, construction and solution.
       
     • The article contains an extended general discussion of the mathematical method, and of such concepts of mathematical logic as the theory of plurality, the theory of algorithms, the law of full mathematical induction, recursive functions and Turing machines.
       

251. Kato biography
     
     • However he had published many papers by the time the doctorate was awarded including work on pair creation by gamma rays, the motion of an object through a fluid and results on the spectral theory of operators arising in quantum mechanics.
       
     • The course covered, thoroughly but efficiently, most of the standard material from the theory of functions through partial differential equations.
       
     • Frantisek Wolf at Berkeley had become interested in perturbation theory through Kato's work and played a major role in an effort which brought Kato to Berkeley in 1962 to become Professor of Mathematics.
       
     • In 1966 Kato published his classic text Perturbation theory for linear operators which was [Notices Amer.
       
     • This is a treatise on linear transformations in Hilbert space as seen from the point of view of perturbation theory, as opposed to commutator theory or invariant subspaces.
       
     • The tone of the book is set in the first two chapters, which are concerned with transformations in finite-dimensional spaces and can be read with no prior knowledge of operator theory.
       
     • In 1982 Kato published A short introduction to perturbation theory for linear operators which, in his own words from the Introduction:- .
       
     • is a slightly expanded reproduction of the first two chapters (plus Introduction) of my book Perturbation theory for linear operators.
       

252. Whitney biography
     
     • As impressive as each individual paper is, it is even more impressive to see them grouped together in two volumes, the first including papers in graphs and combinatorics, differentiable functions and singularities, and analytic spaces, and the second containing contributions to manifolds, bundles and characteristic classes, topology and algebraic topology, and geometric integration theory.
       
     • Whitney's doctoral thesis was on graph theory, in particular making a major contribution to the four colour problem.
       
     • Following this he published a number of papers on graph theory such as A theorem on graphs (1931), Non-separable and planar graphs (1932), Congruent graphs and the connectivity of graphs (1932), The coloring of graphs (1932), A numerical equivalent of the four color map problem (1937).
       
     • His main work, however, was in topology, particularly in the theory of manifolds.
       
     • Other work on algebraic varieties and integration theory was important.
       
     • He published the book Geometric integration theory In 1957 which describes his work on the interactions between algebraic topology and the theory of integration.
       
     • Grassmann algebra; Differential forms; Riemann integration theory; Smooth manifolds; Abstract integration theory; Some relations between chains and functions; General properties of chains and cochains; Chains and cochains in open sets; Flat cochains and differential forms; Lipschitz mappings; Chains and additive set functions.
       

253. Szekeres biography
     
     • Besides combinatorial geometry, he has also made contributions in the theory of partitions, graph theory, and other areas of combinatorics.
       
     • Another prominent topic in George's career is general relativity; George is perhaps best known for his role in developing the mathematical theory underlying the study of black holes.
       
     • He embraced the computer age with enthusiasm, making early contributions to techniques of numerical analysis, especially in the theory of computing high dimensional integrals.
       
     • More recently, his research interests include combinatorial geometry, Hadamard determinants, and chaos theory.
       
     • The reference to chaos theory here refers in particular to his interest in Feigenbaum's functional equation.
       
     • There was the 1957 paper Spinor geometry and general field theory which Szekeres described in the introduction as follows:- .
       
     • In 1958 Szekeres published the group theory paper On a problem of D R Hughes written jointly with E G Straus, then two years later he published On the singularities of a Riemannian manifold in which he discussed the problem of determining when an apparent singularity in a Riemann manifold is real, and when it may be eliminated by an extension of the space.
       
     • Throughout his career he loved combinatorics, graph theory in particular, and he published papers on this topic such as Polyhedral decompositions of cubic graphs (1973) and Non-colourable trivalent graphs (1975) in the few years before he retired.
       

254. Dyson biography
     
     • Foreign languages came easily to Dyson and when he became interested in number theory in 1938 he decided to read An introduction to the theory of numbers by Vinogradov.
       
     • In the following year he read Eddington's The mathematical theory of relativity.
       
     • Something happened at this time that greatly pleased Dyson; Tomanaga in Japan had developed significant work in relativistic quantum field theory.
       
     • The Tomonaga-Schwinger quantum electrodynamics is discussed with due emphasis on the physical ideas involved and the equivalence with a mainly unpublished theory by Feynman is established.
       
     • After finishing the S-matrix paper, Dyson turned to meson theory where he devised a method of separating the calculation of high and low frequency interactions.
       
     • He has written a number of expository articles such as Mathematics in the physical sciences (1964) in Scientific American, on the role of mathematics, in particular group theory, in the physical sciences.
       
     • It has a wealth of topics and of seminal contributions: the famous QED [quantum electrodynamics] papers, the stability of matter, the invention of the hierarchical Ising models, the disordered linear chain, random matrices, spin wave theory, etc.; Dyson has made his mark in all these varied subjects.
       

255. Steinfeld biography
     
     • Basically he was seeking appropriate analogues in ring theory for certain concepts used in the theory of groups and he did this by looking at the corresponding notions in semigroups.
       
     • It is interesting that most of Steinfeld's research went in the other direction for he often took ideal theoretic properties of rings and looked for analogues in semigroup theory, semiring theory, or the theory of partially ordered algebraic structures.
       
     • The paper also considers applications of ideal theory in rings to Schreier extension theory in groups in the same spirit as Redei's paper Holomorphentheorie fur Gruppen und Ringe (1954) which we mentioned above.
       
     • Two new sets of conditions are obtained for unique prime factorisation in a partially ordered semigroup (not necessarily commutative), generalising the fundamental theorem or commutative ideal theory.
       

256. Conway biography
     
     • He was awarded his BA in 1959 and began to undertake research in number theory supervised by Harold Davenport.
       
     • Knowing that he did not have the group theory skills necessary to prove his conjectures he tried to interest others, see [From error-correcting codes through sphere packings to simple groups (Washington, 1983).
       
     • Let us mention that, in addition to his innovative contributions to group theory and his creation of surreal numbers mentioned above, he has done leading research in knot theory, number theory, game theory, quadratic forms, coding theory, and tilings.
       
     • this book is a momentous addition to the mathematical literature: a new, exciting, and highly original theory is expounded by its creator in a style that is at once concise, literate, and delightfully whimsical.
       

257. Schwinger biography
     
     • While at the Radiation Laboratory Schwinger invented important methods in electromagnetic field theory, which were extensively employed in the development of the theory of wave guides.
       
     • Schwinger was one of the inventors in the 1940s of the theory of renormalization, mentioned above.
       
     • This theory allows individual particles to be considered from a distant viewpoint.
       
     • He invented source theory, which deals uniformly with strongly interacting particles, photons, and gravitons.
       
     • Schwinger was joint winner of the Nobel Prize for Physics (1965) for his work in formulating quantum electrodynamics and thus reconciling quantum mechanics with Einstein's special theory of relativity.
       
     • For the hydrogen atom, which has only one electron and consequently is the simplest atom to investigate theoretically, the calculation of the motion of the electron in the electric field of the nucleus led to results of such accuracy that 20 years elapsed until any error of the theory could be found experimentally.
       
     • The list of his contributions is staggering, from his early work leading to the Schwinger action principle, Euclidean quantum field theory, and the genesis of the standard model, to later valuable work on magnetic charge and the Casimir effect.
       

258. Kendall Maurice biography
     
     • Their brief conversation would prove significant for when Yule discussed a revision of his text An Introduction to the Theory of Statistics (first published in 1911) with his publisher the suggestion was made that a second author might be brought in to help Yule.
       
     • Despite his heavy workload by day and air-raid warden duties by night, he somehow contrived to find time to work on the project [the advanced treatise on mathematical statistics] single-handedly Volume One of the Advanced Theory of Statistics was published in 1943, and Volume Two appeared in 1946.
       
     • In 1994, more than ten years after Kendall's death, the sixth edition which is now called Kendall's advanced theory of statistics Vol.
       
     • It is fifty years since the first edition of Maurice Kendall's Volume 1 appeared, so it is fitting that a new edition sees a major restructuring of The advanced theory that, we hope, remains true to his two goals of presenting a 'systematic treatment of (statistical) theory as it exists at the present time' and keeping the volumes first and foremost as a treatment of 'statistics, not statistical mathematics'.
       
     • Kendall continued a remarkable stream of research papers on topics such as the theory of k-statistics, time series, and rank correlation methods and a monograph Rank Correlation in 1948.
       
     • Other monographs are A course in the geometry of n dimensions (1961) which aims to present that part of the theory of n-dimensional geometry which has statistical applications, and to sketch very briefly what those applications are.
       
     • In 1963 he published (jointly with P A P Moran) Geometrical probability followed by Time series (1973) in which Kendall states his objectives to bridge the gap between "sophisticated theory and practical applications" in the field of time series and to "treat the subject in its entirety for the benefit of the practising statistician".
       

259. Newton biography
     
     • However, perhaps because of Newton's already high reputation, his corpuscular theory reigned until the wave theory was revived in the 19th century.
       
     • It dealt with the theory of light and colour and with .
       
     • To explain some of his observations he had to use a wave theory of light in conjunction with his corpuscular theory.
       
     • Another argument, this time with the English Jesuits in Liege over his theory of colour, led to a violent exchange of letters, then in 1678 Newton appears to have suffered a nervous breakdown.
       
     • Newton's greatest achievement was his work in physics and celestial mechanics, which culminated in the theory of universal gravitation.
       
     • The Continental scientists certainly did not accept the idea of action at a distance and continued to believe in Descartes' vortex theory where forces work through contact.
       

260. Schramm biography
     
     • Schramm worked at the Institute for seven years until, in 1999, he returned to the United States to take up a position in the Theory Group at Microsoft Research in Redmond, Washington.
       
     • His work in a spectacular series of papers has led to major progress in probability theory, in the theory of percolation and of random walks, as well as in related topics of conformal field theory.
       
     • work in combining analytic power with geometric insight in the field of random walks, percolation, and probability theory in general, especially for formulating stochastic Loewner evolution.
       
     • In trying to understand this limit as well as limits of other models such as the loop-erased