Search Results for conjecture

Biographies

1. Freedman biography
   
   • Freedman was awarded a Fields Medal in 1986 for his work on the Poincare conjecture.
     
   • The Poincare conjecture, one of the famous problems of 20th-century mathematics, asserts that a simply connected closed 3-dimensional manifold is a 3-dimensional sphere.
     
   • The higher dimensional Poincare conjecture claims that any closed n-manifold which is homotopy equivalent to the n-sphere must be the n-sphere.
     
   • When n = 3 this is equivalent to the Poincare conjecture.
     
   • Smale proved the higher dimensional Poincare conjecture in 1961 for n at least 5.
     
   • Freedman proved the conjecture for n = 4 in 1982 but the original conjecture remained open until settled by G Perelman who was offered the 2006 Fields medal for his proof.
     
   • After the discovery in the early 60s of a proof for the Poincare conjecture and other properties of simply connected manifolds of dimension greater than four, one of the biggest open problems, besides the three dimensional Poincare conjecture, was the classification of closed simply connected four manifolds.
     
   • In his paper, The topology of four-dimensional manifolds, published in the Journal of Differential Geometry (1982), Freedman solved this problem, and in particular, the four-dimensional Poincare conjecture.
     

2. Goldbach biography
   
   • He is best remembered for his conjecture, made in 1742 in a letter to Euler (and still an open question), that every even integer greater than 2 can be represented as the sum of two primes.
     
   • Vinogradov made progress on this second conjecture in 1937.
     
   • The last conjecture was made by Goldbach in a letter written to Euler on 18 November 1752.
     
   • Euler replied on 16 December, saying he had checked Goldbach's conjecture up to 1000.
     
   • In fact the conjecture is false.
     
   • No other examples of numbers failing to satisfy this conjecture of Goldbach seem to be known.
     
   • It is interesting to ponder that Goldbach could, with some hard work, have tested this conjecture to 2500 as Euler did.
     
   • The Prime Pages (Goldbach's conjecture) .
     

3. Smale biography
   
   • One of Smale's impressive results was his work on the generalised Poincare conjecture.
     
   • The Poincare conjecture, one of the famous problems of 20th-century mathematics, asserts that a simply connected closed 3-dimensional manifold is a 3-dimensional sphere.
     
   • The higher dimensional Poincare conjecture claims that any closed n-dimensional manifold which is homotopy equivalent to the n-sphere must be the n-sphere.
     
   • When n = 3 this is equivalent to the Poincare conjecture.
     
   • Smale proved the higher dimensional Poincare conjecture in 1961 for n at least 5.
     
   • (Michael Freedman proved the conjecture for n = 4 in 1982 but the original conjecture remained open until settled by G Perelman who was offered the 2006 Fields medal for his proof.) .
     
   • In fact Smale attacked the generalised Poincare conjecture using Morse theory.
     

4. Yau biography
   
   • Yau was awarded a Fields Medal in 1982 for his contributions to partial differential equations, to the Calabi conjecture in algebraic geometry, to the positive mass conjecture of general relativity theory, and to real and complex Monge-Ampere equations.
     
   • On the Calabi conjecture, which was made in 1954, he writes that this:- .
     
   • Yau solved the Calabi conjecture in 1976.
     
   • Another conjecture solved by Yau was the positive mass conjecture, which comes from Riemannian geometry.
     
   • In joint work of Yau with Karen Uhlenbeck On the existence of Hermitian Yang-Mills connections in stable bundles (1986), they solved higher dimensional versions of the Hitchin-Kobayashi conjecture.
     
   • However, recently he has been involved in an unfortunate dispute regarding the proof of the Poincare conjecture.
     
   • The Russian mathematician Grigory Perelman sketched a proof of the conjecture in 2003 and several teams began work on giving a full comprehensive proof.
     

5. Artin biography
   
   • The new idea originated in work which Nikolai Chebotaryov published in 1924 where he had proved a conjecture made by Frobenius about the density of the set of prime ideals of a normal extension field.
     
   • First, the analogue of the Riemann conjecture for the zeta function of a curve over finite fields.
     
   • Thus, this conjecture of Artin was the origin of a wide range of activities in what is now called arithmetic geometry.
     
   • Second, there is Artin's conjecture on primitive roots.
     
   • Artin made this conjecture to Hasse on 27 September 1927 (according to an entry in Hasse's diary), and since then many mathematicians have tried to prove it.
     
   • Again, Artin's conjecture triggered a lot of interesting activities in number theory.
     

6. Iwasawa biography
   
   • In the late 1960s Iwasawa made a conjecture for algebraic number fields which, in some sense, was the analogue of the relationship which Weil had found between the zeta function and the divisor class group of an algebraic function field.
     
   • 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.
     
   • today it is no exaggeration to say that Iwasawa's ideas have played a pivotal role in many of the finest achievements of modern arithmetical algebraic geometry on such questions as the conjecture of B Birch and H Swinnerton-Dyer on elliptic curve; the conjecture of B Birch, J Tate, and S Lichtenbaum on the orders of the K-groups of the rings of integers of number fields; and the work of A Wiles on the modularity of elliptic curves and Fermat's Last Theorem.
     

7. Floer biography
   
   • Back in Germany he returned to the Ruhr University at Bochum where, supervised by Zehnder, he undertook research on V I Arnol'd's fixed-point conjecture for symplectic maps.
     
   • Together with Zehnder, he published Fixed point results for symplectic maps related to the Arnol'd conjecture in the Proceedings of the conference Dynamical systems and bifurcations held in Groningen in 1984.
     
   • Our aim is to present some recent results and open questions concerning the fixed point problem of symplectic maps related to the Arnol'd conjecture.
     
   • In this paper he proves a special case of Arnol'd's conjecture on the number of fixed points of an exact deformation of a compact symplectic manifold.
     
   • In particular he looks there at Floer's progress on the Arnol'd conjecture and instanton homology, and at Floer's instanton homology and 4-dimensional cobordisms.
     

8. Wiles biography
   
   • about ten years ago, G Frey suggested and K Ribet proved (building on ideas of B Mazur and J-P Serre) that Fermat's Last Theorem follows from the Shimura-Taniyama conjecture that every elliptic curve defined over the rational numbers is modular.
     
   • cannot be modular, thus violating the Shimura-Taniyama conjecture.
     
   • In fact Wiles abandoned all his other research when he heard what had been proved and, for seven years, he concentrated solely on attempting to prove the Shimura-Taniyama conjecture, knowing that a proof of Fermat's Last Theorem then followed.
     
   • Using Mazur's deformation theory of Galois representations, recent results on Serre's conjecture on the modularity of Galois representations, and deep arithmetical properties of Hecke algebras, Wiles (with one key step due jointly to Wiles and R Taylor) succeeded in proving that all semistable elliptic curves defined over the rational numbers are modular.
     
   • Although less than the full Shimura-Taniyama conjecture, this result does imply that the elliptic curve given above is modular, thereby proving Fermat's Last Theorem.
     

9. Mazur Barry biography
   
   • His achievement was already remarkable for by this time he had proved the Schoenflies Conjecture in geometric topology.
     
   • The proof of the Main Conjecture of Iwasawa theory by Mazur and Andrew Wiles, in "Class fields of abelian extensions of Q" (1984).
     
   • Ken Ribet's proof that the Taniyama-Shimura conjecture implies a proof of Fermat's Last Theorem, in "On modular representations of Gal(Q/Q) arising from modular forms" (1990).
     
   • Wiles's proof of the Taniyama-Shimura conjecture and of Fermat's Last Theorem, in "Modular elliptic curves and Fermat's last theorem" (1995), using results with R Taylor in "Ring-theoretic properties of certain Hecke algebras".
     
   • Loic Merel's proof of the uniform boundedness conjecture for the torsion of elliptic curves defined over number fields, in "Bornes pour la torsion des courbes elliptiques sur les corps de nombres" (1996).
     

10. Kuperberg biography
    
    • Kuperberg's most celebrated result, however, was discovered in 1993 and published in 1994 in A smooth counterexample to the Seifert conjecture in the Annals of Mathematics.
      
    • The Seifert conjecture (1950), is that all vector fields on the three-sphere have at least one closed orbit.
      
    • The paper is an important contribution to the theory of dynamical systems, and it solves in a simple but elegant way the long-standing Seifert conjecture.
      
    • In 1996 they published Generalized counterexamples to the Seifert conjecture which improved on Kuperberg's results from two years earlier.
      
    • They also solve the Seifert conjecture by giving counter-examples in dimension three or more.
      

11. Heilbronn biography
    
    • The problem which Heilbronn worked on for his doctorate was related to a conjecture made by Bertrand in 1845.
      
    • Chebyshev proved Bertrand's conjecture in 1850 and then in 1930 Hoheisel proved that there exists a t < 1 such that for all large x, there is a prime p between x and x + xt .
      
    • The first of the two papers proved a conjecture of Gauss on imaginary quadratic number fields using ideas of Hecke, Deuring and Mordell.
      
    • Heilbronn proved the conjecture which asserts that the class number of the quadratic number field Q(√-d) tends to infinity as d tends to infinity.
      

12. Wang Yuan biography
    
    • He looked at sieve methods and applied them to the Goldbach Conjecture.
      
    • He also applied circle methods to the Goldbach Conjecture.
      
    • Wang Yuan continued to improve his results on the Goldbach conjecture.
      
    • However he did write a number of books such as: (with Hua Loo Keng) Applications of number theory to numerical analysis (1978); Goldbach Conjecture (1984); (with Hua Loo Keng) Popularising mathematical methods in the People's Republic of China (1989); Diophantine equations and inequalities in algebraic number fields (1991); (with Fang Kai-Tai) Number theoretic methods in statistics (1994); Hua Loo Keng (1995); and (with Fong Yuen) Calculus (1997).
      

13. Haselgrove biography
    
    • In 1958 Haselgrove published his most famous number theory result in A disproof of a conjecture of Polya.
      
    • The conjecture of Polya claims that for every x > 1 there are at least as many numbers less than or equal to x having an odd number of prime factors as there are numbers with an even number of prime factors.
      
    • R S Lehman and W G Spohn had verified the conjecture for all numbers x up to 800,000 but Haselgrove found a counterexample using methods based on those developed by Ingham with the help of computations carried out on the EDSAC 1 computer at Cambridge.
      
    • In the same paper Haselgrove announced that he had also disproved a number theory conjecture of Turan.
      

14. Renyi biography
    
    • 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]:- .
      
    • The first result, regarding the representation of an even number, is an approximation to the unproved Goldbach conjecture and supersedes an earlier proof of the same proposition by Estermann (1932) which made use of an unproved generalized Riemann hypothesis for all Dirichlet L-series.
      
    • The second result is an approximation to the conjecture of the existence of infinitely many twin primes and is apparently a new result.
      
    • He published joint work with Erdos on random graphs, the most important being On the evolution of random graphs (1960), and also solved an outstanding conjecture concerning random space filling curves in On a one-dimensional problem concerning random space filling (1958).
      

15. Ribenboim biography
    
    • Ribenboim's big research breakthrough came when he found a counterexample to a conjecture of Krull.
      
    • He published his result in Sur une conjecture de Krull en theorie des valuations in Nagoya Math.
      
    • He also wnt to lectures by J Ax and these formed the basis of his next book La conjecture d'Artin sur les equations diophantiennes published in 1968.
      
    • 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).
      

16. Mordell biography
    
    • His work on modular functions and their applications to number theory led to his famous proof in 1917 of Ramanujan's conjecture on the tau-function.
      
    • During this time he discovered the result for which he is best known, namely the finite basis theorem, which proved a conjecture of Poincare.
      
    • This became known as the Mordell conjecture and it is discussed in detail in [Archimede 38 (1) (1986), 3-9',7)">7].
      
    • In 1983 Faltings proved the Mordell conjecture to be true.
      

17. Vandiver biography
    
    • A conjecture, now known as 'Vandiver's conjecture', concerning the class group of cyclotomic fields was so named since Vandiver frequently posed it.
      
    • He was not, however, the first to make the conjecture which should really be named 'Kummer's conjecture' since it first appears in 1849 in a letter which Kummer wrote to Kronecker.
      

18. Thompson John biography
    
    • Thompson's thesis, as is clear from its title, proved Frobenius's conjecture that a finite group with an automorphism which does not fix any group element is necessarily nilpotent.
      
    • The solution of Frobenius's conjecture was not done by simply pushing the existing techniques further than others had done; rather it was achieved by introducing many highly original ideas which were to lead to many developments in group theory.
      
    • Here, the authors proved a famous conjecture, to the effect that all non-cyclic finite simple groups have even order.
      
    • Fifty years ago [1920] this was already referred to as a very old conjecture.
      

19. Margulis biography
    
    • Margulis's most spectacular achievement has been the complete solution of that problem and, in particular, the proof of the conjecture in question.
      
    • The Oppenheim conjecture was made in 1929 and concerns values of indefinite irrational quadratic forms at integer points.
      
    • In the 1940s Davenport and Heilbronn contributed by proving special cases and in 1946 Watson extended their results showing the conjecture to be true for further special cases.
      
    • Margulis proved the full conjecture in 1986 and gives a beautiful survey of the work leading to this solution in [Fields Medallists Lectures (Singapore, 1997), 272-327.',3)">3].
      
    • At the centre of the work of Gregory Margulis lies his proof of the Selberg-Piatetskii-Shapiro Conjecture, affirming that lattices in higher rank Lie groups are arithmetic, a question whose origins date back to Poincare.
      
    • In a second tour de force, Margulis solved the 1929 Oppenheim Conjecture, stating that the set of values at integer points of an indefinite irrational nondegenerate quadratic form in more than three variables is dense in Rn.
      
    • This had been reduced (by Rhagunathan) to a conjecture about unipotent flows on homogeneous spaces, proved by Margulis.
      
    • Besides his celebrated results on super-rigidity and arithmeticity of irreducible lattices of higher rank semisimple Lie groups, and the solution of the Oppenheim conjecture on values of irrational indefinite quadratic forms at integral points, he has also initiated many other directions of research and solved a variety of famous open problems.
      

20. Takagi biography
    
    • On his return to Tokyo in 1903 Takagi proved a conjecture on abelian extensions of imaginary number fields made by Kronecker.
      
    • Kronecker described this conjecture as:- .
      
    • I tried my utmost to find a counterexample to the conjecture which seemed all too perfect.
      
    • 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.
      

21. Loewner biography
    
    • He wrote a series of papers on this topic, culminating in one where he proved a special case of the Bieberbach conjecture in 1923.
      
    • The Bieberbach conjecture states that if f is a complex function .
      
    • We should also note that Loewner's proof uses the Loewner differential equation which has been studied extensively since he introduced it, and was used by de Branges in his celebrated proof of the Bieberbach conjecture.
      

22. Adams Frank biography
    
    • The conjecture that Adams solved was the famous conjecture about the existence of H-structures on spheres.
      
    • Using this theory he solved another important conjecture, this one being about vector fields on spheres.
      

23. Roth Klaus biography
    
    • This was Roth's proof in 1952 of a conjecture made in 1935 by Erdos and Turan.
      
    • The conjecture concerned a sequence .
      
    • If N(x) denotes the number of terms of the sequence less than x, Roth proved the conjecture that N(x)/x → 0 as x → ∞.
      

24. Carleson biography
    
    • Kolmogorov proved results in 1928 which seemed to suggest that Luzin's conjecture must be false but Carleson amazed the world of mathematics when he proved Luzin's long-standing conjecture in 1966.
      
    • The citation emphasizes not only Carleson's fundamental scientific contributions, the best known of which perhaps are the proof of Luzin's conjecture on the convergence of Fourier series, the solutions of the corona problem and the interpolation problem for bounded analytic functions, the solution of the extension problem for quasiconformal mappings in higher dimensions, and the proof of the existence of 'strange attractors' in the Henon family of planar maps, but also his outstanding role as scientific leader and advisor.
      

25. Aryabhata I biography
    
    • Some conjecture that he was born in south India, perhaps Kerala, Tamil Nadu or Andhra Pradesh, while others conjecture that he was born in the north-east of India, perhaps in Bengal.
      
    • The conjecture that Aryabhata's value of π is of Greek origin is critically examined and is found to be without foundation.
      

26. Beurling biography
    
    • However, parts of the thesis were written in 1929, in particular his the proof of the Denjoy conjecture concerning asymptotic values of an entire function.
      
    • However, he was not the first to publish a proof of this conjecture since he took a vacation with his father (they went crocodile hunting!) and Lars Ahlfors published his proof of the conjecture in 1929.
      

27. Mertens biography
    
    • It was known that there is no x with M(x) > x but Mertens' conjecture was stronger, namely that there is no x with M(x) > √x.
      
    • The result is important since a proof of Mertens' conjecture would imply the truth of the Riemann hypothesis.
      
    • The conjecture stood for nearly 100 years before it was proved false in 1985 by A M Odlyzko and H J J te Riele.
      

28. Chebyshev biography
    
    • In it he generalised methods of Ostrogradski to show that a conjecture which Abel made in 1826 about the integral of f (x)/√R(x), where f (x) and R(x) are polynomials, was true.
      
    • Chebyshev's work on prime numbers included the determination of the number of primes not exceeding a given number, published in 1848, and a proof of Bertrand's conjecture.
      
    • Chebyshev proved Bertrand's conjecture in 1850.
      

29. Herstein biography
    
    • Before being appointed to Chicago he had published papers such as A proof of a conjecture of Vandiver (1950), On a conjecture on simple groups (1950), and Group-rings as *-algebras (1950).
      
    • The second paper proves a conjecture that the solubility of groups of odd order is equivalent to a condition on the group ring of a group, while the third paper takes methods from the study of Banach rings and topological groups to prove results about group rings over the complex numbers.
      

30. Novikov Sergi biography
    
    • In 1965 Novikov proved his famous theorem on the invariance of Pontryagin classes and stated the conjecture, now known as the Novikov conjecture, concerning the homotopy invariance of certain polynomials in the Pontryagin classes of a manifold, arising from the fundamental group.
      
    • Novikov discussed his conjecture in a lecture given at the 1970 International Congress of Mathematicians in Nice where he received a Fields Medal.
      
    • His early work in algebraic and differential topology includes such milestones as the calculation of cobordism rings and stable homotopy groups, proof of the topological invariance of rational Pontrjagin classes, formulation of the "Novikov Conjecture" on higher signature invariants, and proof of the existence of closed leaves in two-dimensional foliations of the 3-sphere.
      

31. Coates biography
    
    • He then proved certain special cases of Weil's conjecture on elliptic curves.
      
    • His 1977 article on the conjecture of Birch and Swinnerton-Dyer, written jointly with his research student Andrew Wiles, was a landmark contribution to number theory which introduced a panoply of new methods into the field of elliptic curves.
      
    • Coates's insights into the Iwasawa theory of the symmetric square of an elliptic curve were instrumental in the recent proof by Wiles of the Shimura-Taniyama conjecture for semistable elliptic curves over Q.
      

32. Szekeres biography
    
    • On the other hand he had outstanding mathematics publications such as the 1935 paper with Erdos; Ein Problem uber mehrere ebene Bereiche (1940) in which he gave an elementary proof a conjecture of G Grunwald; On an extremum problem in the plane (1941); On a certain class of metabelian groups (1948); and Countable Abelian groups without torsion (1948).
      
    • With this algorithm he then tests the conjecture that the best constant is 2/7, showing that the data supports the conjecture.
      

33. Shnirelman biography
    
    • Using these ideas of compactness of a sequence of natural numbers he was able to prove a weak form of the Goldbach conjecture showing that every number is the sum of ≤ 20 primes.
      
    • The Goldbach conjecture that every number is the sum of at most 3 primes still appears to be open.
      

34. Weil biography
    
    • However, Hadamard wanted his brilliant student to aim higher and try to prove the Mordell Conjecture.
      
    • My decision was a wise one: it was to take more than half a century to prove Mordell's Conjecture.
      

35. Springer biography
    
    • While there he proved a conjecture of Ernst Witt made in 1937, namely: .
      
    • Springer published a proof of the conjecture in Sur les formes quadratiques d'indice zero which appeared in Comtes Rendus of the Academy of Sciences in 1952.
      

36. Motzkin biography
    
    • For example, he once decided to present a seminar talk on Eberhard's conjecture that if every face of a trivalent convex polyhedron P has edge-number divisible by 3, then the number of edges of P is even.
      
    • To the astonishment of the audience, he proceeded in the talk to prove the conjecture, using properties of the group SL(2, 3) of order 24, which at first seemed to be completely unrelated to the problem.
      

37. Ore biography
    
    • 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.
      
    • 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.
      

38. Kurepa biography
    
    • There are many equivalent forms of the conjecture, but one of the most natural was given by Kurepa in the same 1971 paper, namely that !n is not divisible by n for any n > 2.
      
    • If the left factorial conjecture is false we certainly know that it will fail for n > 1000000.
      

39. Mayer Tobias biography
    
    • Eric Forbes makes a conjecture in [Ann.
      
    • Bearing in mind that the publication of the Gottingen Commentarii had been suspended owing to an unfortunate dispute with the printer, and that Mayer spoke Latin eloquently, it is not unreasonable to conjecture that he may have decided to base his talk upon [a] German tract preserved among his unpublished writings in the Gottingen University Library.
      

40. Zelmanov biography
    
    • The greatest early contribution to the Restricted Burnside problem was by Hall and Higman in 1956 where they showed that, if the Schreier conjecture holds, then the Restricted Burnside problem has a positive solution if it could be proved for all prime powers n.
      
    • The Schreier conjecture, that the outer automorphism groups of finite simple groups are soluble, was shown to be true as a consequence of the classification of finite simple groups.
      

41. Mori biography
    
    • He worked on algebraic manifolds with ample tangent bundles and was the first to prove the Hartshorne conjecture in 1978.
      
    • This conjecture, posed in 1970, claimed that projective spaces are the only smooth complete algebraic varieties with ample tangent bundles.
      

42. Cantor biography
    
    • Since a verification of Goldbach's conjecture up to 10000 had been done 40 years before, it is likely that this strange paper says more about Cantor's state of mind than it does about Goldbach's conjecture.
      

43. Poincare biography
    
    • Even today the Poincare conjecture remains as one of the most baffling and challenging unsolved problems in algebraic topology.
      
    • Surprisingly proofs are known for the equivalent of Poincare's conjecture for all dimensions strictly greater than three.
      

44. Bombieri biography
    
    • The award was made for his major contributions to the study of the prime numbers, to the study of univalent functions and the local Bieberbach conjecture, to the theory of functions of several complex variables, and to the theory of partial differential equations and minimal surfaces.
      
    • Chandrasekharan in [Proceedings of the International Congress of Mathematicians, Vancouver 1974 1 (Montreal, Que., 1975), 3-10.',3)">3] describes Bombieri's contributions to the distribution of primes, to univalent functions and the local Bieberbach conjecture and to functions of several complex variables.
      

45. Taniyama biography
    
    • These problems form the basis of a conjecture: every elliptic curve defined over the rational field is a factor of the Jacobian of a modular function field.
      
    • This conjecture proved to be a major factor in the proof of 'Fermat's Last Theorem' by Andrew Wiles.
      

46. Ahlfors biography
    
    • In Zurich Nevanlinna lectured on Denjoy's conjecture on the number of asymptotic values of an entire function.
      
    • I had the incredible luck of hitting upon a new approach, based on conformal mappings, which, with very considerable help from Nevanlinna and Polya, led to a proof of the full conjecture.
      

47. Mirsky biography
    
    • He obtained analogues of Vinogradov's result on the representation of an odd integer as the sum of three primes, the Goldbach conjecture on the representation of an even integer as the sum of two primes, and the twin primes conjecture.
      

48. Ingham biography
    
    • Polya, in 1919, made the following conjecture: .
      
    • The conjecture is that L(x) ≤ 0.
      

49. Gelfond biography
    
    • In 1934 he proved a special case of his conjecture namely that ax is transcendental if a is algebraic (a ≠ 0,1) and x is an irrational algebraic number.
      
    • (In 1966 Alan Baker proved Gelfond's Conjecture in general.) Gelfond's papers in 1933 and 1934, which include his remarkable achievement, are: Gram determinants for stationary series (written jointly with Khinchin) (1933); A necessary and sufficient criterion for the transcendence of a number (1933); Functions that take integer values at the points of a geometric progression (1933); On the seventh problem of D Hilbert (1934); and On the seventh problem of Hilbert (1934).
      

50. Kelly biography
    
    • In 1945 he published On isometries of square sets which proved a special case of a conjecture made by Ulam.
      
    • In this paper Kelly proved the conjecture for the case of certain isometries of a class of finite metric spaces.
      

51. Langlands biography
    
    • 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.
      
    • While he was in Ankara in 1967-68 he wrote to Serre with ideas which would eventually be formulated as the Deligne-Langlands conjecture; this was proved eventually by Kazhdan and Lusztig.
      

52. Pacioli biography
    
    • Although we know little of Pacioli's early life, the conjecture that he may have received at least a part of his education in the studio of della Francesca in Sansepolcro must at least have a strong chance of being correct.
      
    • During this time Pacioli worked with Scipione del Ferro and there has been much conjecture as to whether the two discussed the algebraic solution of cubic equations.
      

53. Cramer Harald biography
    
    • Note that if a = b = 1 then the question of whether this equation has a solution for all c is Goldbach's conjecture, while if a = 1, b = -1, c = 2, then the question about prime solutions to x = y + 2 is the twin prime conjecture.
      

54. Bieberbach biography
    
    • Bieberbach is best remembered (other than for his anti-Jewish views) for the Bieberbach Conjecture (1916).
      
    • Perhaps there is an irony in the fact that de Branges became the first winner of the Ostrowski Prize for solving the Bieberbach conjecture.
      

55. 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 .
      

56. Ricci Giovanni biography
    
    • During that time he published work on the Goldbach conjecture which concerns writing numbers as the sum of primes, and also on Hilbert's Seventh Problem which asks whether or not aõ was transcendental when a and b are algebraic.
      
    • Hilbert himself remarked that he expected this Seventh Problem to be harder than the solution of the Riemann conjecture.
      

57. Kostrikin biography
    
    • In the 1960's, Kostrikin studied infinite-dimensional Lie algebras of Cartan type for finite characteristic and, with Shafarevich, formulated a conjecture describing all simple Lie p-algebras for characteristic p > 5.
      
    • This conjecture motivated research in the field for many years and was finally proved by R E Black and R L Wilson.
      

58. Zeeman biography
    
    • in 1953 for his thesis Dihomology but he spent the first year as a research student trying unsuccessfully to solve the 3-dimensional Poincare Conjecture.
      
    • I suppose I am particularly fond of having unknotted spheres in 5-dimensions, of spinning lovely examples of knots in 4-dimensions, of proving Poincare's Conjecture in 5-dimensions, of showing that special relativity can be based solely on the notion of causality, and of classifying dynamical systems by using the Focke-Plank equation.
      

59. Drinfeld biography
    
    • Drinfeld's main achievements are his proof of the Langlands conjecture for GL(2) over a functional field; and his work in quantum group theory.
      
    • Although he only proved a special case of the Langlands conjecture, Drinfeld has introduced important new ideas in his solution and made a real breakthrough.
      

60. Lakatos biography
    
    • Mathematics develops, according to Lakatos, in a much more dramatic and exciting way - by a process of conjecture, followed by attempts to 'prove' the conjecture (i.e.
      

61. Catalan biography
    
    • The most famous of all, however, is the 'Catalan Conjecture' made in 1844 in a letter he sent to Crelle's Journal:- .
      
    • Progress on solving the conjecture was slow.
      

62. Polya biography
    
    • Szego at this time was a student at Budapest and Polya discussed a conjecture he had made on Fourier coefficients with Szego.
      
    • In fact Szego went on to prove Polya's conjecture and this became his first publication.
      

63. Stewart Dugald biography
    
    • Natural philosophy became science, [Stewart] held, when inquiry, freed from metaphysical conjecture, was directed towards discovering by observation and experiment the laws governing the connection of physical phenomena.
      
    • It requires that the phenomena of consciousness be approached without conjecture and that the laws of connection be inductively established.
      

64. Ollerenshaw biography
    
    • She emphasised the need for 'proof' and the difference between conjecture and logical mathematical proof.
      
    • In 1982, Dame Kathleen Ollerenshaw and Hermann Bondi published a paper, Magic squares of order four, in which they prove the conjecture of Frenicle de Bessy showing that there are 880 essentially different normal magic squares of order 4.
      

65. Olive biography
    
    • She has had a special interest in the polynomials which are generated by her generalised powers, and hopes that someone will prove or disprove her conjecture, now about 30 years old, that all their zeros lie on the unit circle.
      
    • This conjecture has now been verified for infinitely many special cases.
      

66. Schramm biography
    
    • His research in probability was sparked by his interest in the conjecture that the limit of two-dimensional critical percolation was conformally invariant.
      
    • In collaboration with G Lawler and W Werner, Schramm has used SLE to solve a number of open problems, in particular Mandelbrot's conjecture that the outer boundary of planar Brownian motion has dimension 4/3 and the determination of the scaling limit of loop-erased work.
      

67. Kaprekar biography
    
    • One would be tempted to conjecture that n! is a Harshad number for every n - this however would be incorrect.
      

68. Honda biography
    
    • An important consequence is that Honda was able to give a short proof of Manin's conjecture about formal groups.
      

69. Yates biography
    
    • Together they proved a longstanding conjecture on 6 × 6 Latin squares in 1934.
      

70. Tutte biography
    
    • In the same year he published a paper on perhaps the most famous of all graph theory problems On the four-colour conjecture.
      

71. Bing biography
    
    • He considered several different aspects of 3-manifolds including decompositions, maps, approximating surfaces, recognizing tameness, triangulation and the Poincare conjecture.
      

72. Witten biography
    
    • The first major contribution which led to Witten's Fields Medal was his simpler proof of the positive mass conjecture which had led to a Fields Medal for Yau in 1982.
      

73. Curry biography
    
    • There are always many parties and other, less formal gatherings, and we conjecture that Virginia's cooking has also played a role in the growth of interest in combinatory logic.
      

74. Harriot biography
    
    • This seems intuitively obvious, but resisted proof until 1998 when Thomas Hales of the University of Michigan (with the help of hours of computer generated data) finally proved the conjecture.
      

75. Bertrand biography
    
    • This conjecture, similar to one stated by Euler one hundred years earlier, was proved by Chebyshev in 1850.
      

76. Collatz biography
    
    • The problem asks if, for every starting value m, the sequence a(i) always reaches 1? The problem remains unsolved, but before you try a few small numbers yourself looking for a counterexample, let us say that the conjecture has been verified for all numbers m up to about 1014 .
      

77. Manin biography
    
    • He has written papers on: algebraic geometry including ones on the Mordell conjecture for function fields and a joint paper with V Iskovskikh on the counter-example to the Luroth problem; number theory including ones about torsion points on elliptic curves, p-adic modular forms, and on rational points on Fano varieties; and differential equations and mathematical physics including ones on string theory and quantum groups.
      

78. Thurston biography
    
    • Thurston's work on Kleinian groups yielded many new results and established a well known conjecture.
      

79. Theaetetus biography
    
    • Bulmer-Thomas prefers the conjecture that although Book X is based on Theaetetus's work there is much due to Euclid presented there too.
      

80. Straus biography
    
    • In On the maximal number of pairwise orthogonal Latin squares of a given order (1960) Straus, together with Erdos and Chowla, solved Euler's conjecture by showing that the number of pairwise orthogonal Latin squares of order n is greater than (1/3)n1/91.
      

81. Faltings biography
    
    • He received the medal primarily for his proof of the Mordell Conjecture which he achieved using methods of arithmetic algebraic geometry.
      

82. Koebe biography
    
    • He did make other important contributions, however, and his circle domain conjecture is still being attacked.
      

83. Theon biography
    
    • The Optics on the other hand is elementary and written in a totally different style and some historians conjecture that it is really a set of lecture notes by one of Theon's students.
      

84. Samuel biography
    
    • Another text is Lectures on old and new results on algebraic curves which was published in 1963 and explained the results of Grauert and Samuel on Mordell's conjecture over function fields.
      

85. Menaechmus biography
    
    • One conjecture is that it appeared in Menaechmus's commentaries on Plato's Republic referred to in the quote above from the Suda Lexicon.
      

86. Zeuthen biography
    
    • He suggested that the end of Theodorus's proof somehow involved the continued fractions for 17 and 19, a conjecture which is very much in line with modern ideas about Greek mathematics.
      

87. Suzuki Michio biography
    
    • Regarding the above conjecture of Burnside, Brauer said:- .
      

88. Linnik biography
    
    • From this he was able to produce a whole series of papers proving powerful arithmetical consequences, including a variant of the Goldbach Conjecture.
      

89. Watson biography
    
    • This layer in the atmosphere, now called the Heaviside layer, was only a conjecture in 1918 but it was suggested to Watson that, having shown the previous model to be wrong, he now look at the model resulting from the postulated Heaviside layer.
      

90. Faber biography
    
    • Two mathematicians independently verified Rayleigh's conjecture, Faber and Edgar Krahn.
      

91. Rudin biography
    
    • These confirm a conjecture by J Nikiel that they are precisely the compact Hausdorff monotonically normal spaces.
      

92. Householder biography
    
    • Hypothesis, conjecture and tentative theory flew in all directions and there was a period of great ferment.
      

93. Baker Alan biography
    
    • Hilbert himself remarked that he expected this problem to be harder than the solution of the Riemann conjecture.
      

94. Gromoll biography
    
    • provided one of the cornerstones of the Poincare Conjecture solution.
      

95. Robinson Raphael biography
    
    • If this conjecture were true (it has been shown to be false), then a general decision method would exist; namely, we systematically tile larger and larger square arrays of cells in every possible way with the given set of tiles.
      

96. Feit biography
    
    • This brought together many leading group theorists, and in particular it provided the opportunity for Feit and Thompson to enbark on the ambitious project of attempting to prove the conjecture that all groups of odd order are soluble.
      

97. Ramanujan biography
    
    • MacMahon had produced tables of the value of p(n) for small numbers n, and Ramanujan used this numerical data to conjecture some remarkable properties some of which he proved using elliptic functions.
      

98. Bass biography
    
    • 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.
      

99. Selberg biography
    
    • AMS (Selberg's eigenvalue conjecture) [registration required] .
      

100. Erdos biography
     
     • Chebyshev proved Bertrand's conjecture in 1850 but when Erdos was only an eighteen year old student in Budapest he found an elegant elementary proof of this result.
       

101. Thabit biography
     
     • 16 (4) (1989), 373-378.',9)">9] where the authors conjecture how Thabit might have discovered the rule.
       

102. Newman biography
     
     • He published a highly significant paper in 1966 which proved the Poincare Conjecture for topological manifolds of dimension greater than 4.
       

103. Euler biography
     
     • Goldbach asked Euler, in 1729, if he knew of Fermat's conjecture that the numbers 2n + 1 were always prime if n is a power of 2.
       

104. Diophantus biography
     
     • We conjecture the existence of a lost theoretical treatise of Diophantus, entitled "Teaching of the elements of arithmetic".
       

105. Fomin biography
     
     • 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.
       

106. Cohen biography
     
     • .for his paper, On a conjecture of Littlewood and idempotent measures, American Journal of Mathematics 82 (1960), 191-212.
       

107. Gordan biography
     
     • For the next twenty years Gordan tried to prove the finite basis theorem conjecture for n-ary forms.
       

108. Matsushima biography
     
     • The first paper which Matsushima published contained a proof that a conjecture of Zassenhaus was false.
       

109. Enriques biography
     
     • What leads a mathematician to make a conjecture? .
       

110. Korkin biography
     
     • He had given an upper bound for the minimum of an n-ary form of fixed given determinant and made a conjecture for a better upper bound.
       

111. Vandermonde biography
     
     • Lebesgue's conjecture in [Enseignement Math.
       

112. Friedrichs biography
     
     • Open problems and unresolved difficulties are carefully noted, and the reader is never left in doubt as to whether he is presented with a mathematical theorem or with a conjecture based on physical experience.
       

113. Metzler biography
     
     • William Henry Metzler, for instance, had found a proof of a conjecture about the relation between the coefficients of the characteristic polynomial of a square matrix and the determinants of some of its minors.
       

114. Goldstine biography
     
     • Goldstine and von Neumann also produced papers using computation to attack mathematical question, such as their application to number theory in A numerical study of a conjecture of Kummer (1953).
       

115. Zermelo biography
     
     • Hilbert saw this as one of the most fundamental questions which mathematicians should attack in the 1900s and he went further in proposing a method to attack the conjecture.
       

116. Droz-Farny biography
     
     • Looking at other work by Droz-Farny, one is led to conjecture that indeed he would have constructed a proof of the theorem.
       

117. Hilbert biography
     
     • Hilbert's problems included the continuum hypothesis, the well ordering of the reals, Goldbach's conjecture, the transcendence of powers of algebraic numbers, the Riemann hypothesis, the extension of Dirichlet's principle and many more.
       

118. Archimedes biography
     
     • Certainly he was completely familiar with the mathematics developed there, but what makes this conjecture much more certain, he knew personally the mathematicians working there and he sent his results to Alexandria with personal messages.
       

119. Ferro biography
     
     • There has been much conjecture as to whether del Ferro came to work on the solution to cubic equations as a result of a visit which Pacioli made to Bologna.
       

120. Thom biography
     
     • I am thinking too of Barry Mazur's demonstration of the Schonflies conjecture: Every sphere Sn-1 in Rn with regular boundary is the boundary of an n-ball.
       

121. Benjamin biography
     
     • In the Journal of Fluid Mechanics in 1995 he published the paper Verification of the Benjamin-Lighthill conjecture about steady water waves.
       

122. Waring biography
     
     • This result, now known as the Goldbach conjecture, is one of the most famous unsolved problems of mathematics.
       

123. Miller biography
     
     • He became a severe critic of historical methodology in mathematics and was zealous in rooting out error in conjecture or assumed fact.
       

124. Menelaus biography
     
     • It is an attractive conjecture but incapable of proof on present evidence.
       

125. De Groot biography
     
     • The main conjecture made by him has recently been solved.
       

126. Specker biography
     
     • Specker here proves the following important results which were long a subject for conjecture: .
       

127. Brauer Alfred biography
     
     • Brauer made major contributions to number theory, for example on the non-existence of odd perfect numbers of certain forms, and the Khinchin conjecture which was later proved and extended by Henry B Mann.
       

128. Thomason biography
     
     • For example in a 1983 paper he found a partial solution of Grothendieck's absolute cohomological purity conjecture.
       

129. Pontryagin biography
     
     • One of the 23 problems posed by Hilbert in 1900 was to prove his conjecture that any locally Euclidean topological group can be given the structure of an analytic manifold so as to become a Lie group.
       

130. Vinogradov biography
     
     • 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.
       

131. Askey biography
     
     • If one were to single out one paper by Askey of being of particular importance, it must be the one which contained a result which was used by Louis de Branges in giving his complete proof of the Bieberbach Conjecture in 1984.
       

132. Bernoulli Daniel biography
     
     • Based on experimental evidence he was able to conjecture certain laws which were not verified until many years later.
       

133. Killing biography
     
     • By the time he wrote to Engel on 23 May Killing had discovered that his conjecture about simple algebras was wrong, for he had discovered G, and by 18 October he had discovered the complete list of simple algebras.
       

134. Yamabe biography
     
     • It was during his two years at Princeton that he published the two papers On the conjecture of Iwasawa and Gleason and A generalization of a theorem of Gleason.
       

135. Picard Jean biography
     
     • It must remain a matter of conjecture as to how far Picard's studies were interrupted by the civil unrest which was so prevalent in that part of France in the 1630s and 1640s; and the Jesuit College itself was a highly volatile institution as the breakdown of social order affected the conduct of the students too.
       

136. Moser William biography
     
     • In fact Dirac was the right person to review this paper for he had made a famous conjecture that there are at least n/2 lines with the property that each of them passes through exactly two of the n points.
       

History Topics

1. The four colour theorem
   
   • The Four Colour Conjecture first seems to have been made by Francis Guthrie.
     Go directly to this paragraph
     
   • Before continuing with the history of the Four Colour Conjecture we will complete details of Francis Guthrie.
     
   • Charles Peirce in the USA attempted to prove the Conjecture in the 1860's and he was to retain a lifelong interest in the problem.
     Go directly to this paragraph
     
   • Cayley also learnt of the problem from De Morgan and on 13 June 1878 he posed a question to the London Mathematical Society asking if the Four Colour Conjecture had been solved.
     Go directly to this paragraph
     
   • The paper explains where the difficulties lie in attempting to prove the Conjecture.
     Go directly to this paragraph
     
   • On 17 July 1879 Alfred Bray Kempe announced in Nature that he had a proof of the Four Colour Conjecture.
     Go directly to this paragraph
     
   • Story reported the proof to the Scientific Association of Johns Hopkins University in November 1879 and Charles Peirce, who was at the November meeting, spoke at the December meeting of the Association of his own work on the Four Colour Conjecture.
     Go directly to this paragraph
     
   • The Four Colour Theorem returned to being the Four Colour Conjecture in 1890.
     Go directly to this paragraph
     
   • Heawood was to make further contributions to the Four Colour Conjecture.
     
   • The Four Colour Conjecture now asks if the vertices of the graph can be coloured with 4 colours so that no two adjacent vertices are the same colour.
     
   • Renewed interest in the USA was due to Veblen who published a paper in 1912 on the Four Colour Conjecture generalising Heawood's work.
     Go directly to this paragraph
     
   • However the final ideas necessary for the solution of the Four Colour Conjecture had been introduced before these last two results.
     Go directly to this paragraph
     
   • Heesch thought that the Four Colour Conjecture could be solved by considering a set of around 8900 configurations.
     
   • The year 1976 saw a complete solution to the Four Colour Conjecture when it was to become the Four Colour Theorem for the second, and last, time.
     Go directly to this paragraph
     

2. Fermat's last theorem
   
   • Further work by Weil and Shimura produced a conjecture, now known as the Shimura-Taniyama-Weil Conjecture.
     Go directly to this paragraph
     
   • In 1986 the connection was made between the Shimura-Taniyama- Weil Conjecture and Fermat's Last Theorem by Frey at Saarbrucken showing that Fermat's Last Theorem was far from being some unimportant curiosity in number theory but was in fact related to fundamental properties of space.
     Go directly to this paragraph
     
   • Further work by other mathematicians showed that a counter-example to Fermat's Last Theorem would provide a counter -example to the Shimura-Taniyama-Weil Conjecture.
     Go directly to this paragraph
     
   • In fact Wiles had proved the Shimura-Taniyama-Weil Conjecture for a class of examples, including those necessary to prove Fermat's Last Theorem.
     Go directly to this paragraph
     
   • The key reduction of (most cases of) the Taniyama-Shimura conjecture to the calculation of the Selmer group is correct.
     

3. Knots and physics
   
   • This became known as Tait's first conjecture.
     
   • If we interpret Tait in a form that he seems to have used the conjecture, namely that two alternating diagrams without nugatory crossings representing the same prime knot are related by a sequence of twists, then we get what has been called Tait's second conjecture.
     
   • This would lead him to Tait's first conjecture for alternating knots.
     
   • In fact it is not, but for alternating knots, it is an invariant and this fact is a consequence of Tait's second conjecture (a theorem since 1993).
     

4. Prime numbers
   
   • The Twin Primes Conjecture that there are infinitely many pairs of primes only 2 apart.
     
   • Goldbach's Conjecture (made in a letter by C Goldbach to Euler in 1742) that every even integer greater than 2 can be written as the sum of two primes.
     Go directly to this paragraph
     
   • (The fact that there is always a prime between n and 2n was called Bertrand's conjecture and was proved by Chebyshev.) .
     Go directly to this paragraph
     

5. The four colour theorem references
   
   • N L Biggs, E K Lloyd and R J Wilson, C S Peirce and De Morgan on the four-colour conjecture, Historia Mathematica 4 (1977), 215-216.
     
   • J Wilson, New light on the origin of the four-color conjecture, Historia Mathematica 3 (1976), 329-330.
     

6. The four colour theorem references
   
   • N L Biggs, E K Lloyd and R J Wilson, C S Peirce and De Morgan on the four-colour conjecture, Historia Mathematica 4 (1977), 215-216.
     
   • J Wilson, New light on the origin of the four-color conjecture, Historia Mathematica 3 (1976), 329-330.
     

7. Burnside problem
   
   • above is the so-called Schreier Conjecture) .
     
   • Even earlier it was known for n odd by Feit-Thompson (the "odd-order paper" of 1962), and at the time of publication must have been a reasonable conjecture.
     

8. Orbits
   
   • He conjectured that there are infinitely many periodic solutions of the restricted problem, the conjecture being later proved by Birkhoff.
     Go directly to this paragraph
     

9. Ledermann interview
   
   • He would be walking along dreaming that one day he would prove Herr Fermat's theorem, or this or that other conjecture.
     

10. Babylonian Pythagoras
    
    • J., 1999).',2)" onmouseover="window.status='Click to see reference';return true">2] and [The crest of the peacock (London, 1991).',4)" onmouseover="window.status='Click to see reference';return true">4], conjecture that the Babylonians used a method equivalent to Heron's method.
      

11. Perfect numbers
    
    • Of course if this conjecture were true it would solve the still open question of whether there are an infinite number of Mersenne primes (and also solve the still open question of whether there are infinitely many perfect numbers).
      

12. Babylonian numerals
    
    • One of the nicest things about these theories is that it may be possible to find written evidence of the two mixing systems and thereby give what would essentially amount to a proof of the conjecture.
      

13. Egyptian Papyri
    
    • The third conjecture in [Historia Math.
      

Famous Curves

Societies etc

1. European Mathematical Society Prizes
   
   • His results on the Mumford-Shah conjecture in the theory of computer vision meant a breakthrough.
     
   • Among others, his results include the resolution of a conjecture of Veys and the answer to a long-standing question of Mumford on moduli spaces.
     
   • He proved a conjecture of Gromov concerning an estimation of the product of weights, and the Cheeger-Gromov conjecture.
     
   • whose work is a major advance in the K-theory of operator algebras: the proof of the Baum-Connes conjecture for discrete co-compact subgroups of SL3(R), SL3(C), SL3(Qp) and some other locally compact group (and of more general objects).
     
   • The conjecture plays a central role in non-commutative geometry and has far-reaching connections to the Novikov conjecture on higher signatures in topology, to harmonic analysis on discrete groups and the theory of C*-algebras.
     
   • Lafforgue's result is the first passage of the barrier which property T of Kazhdan has posed for many years in the proof of the Baum-Connes conjecture.
     
   • The proof involves several remarkable technical and conceptual developments, like a bivariant K-theory for Banach algebras (versus Kasparov's by now classical one for C*-algebras) or establishing the conjecture for various completions of the L1 algebras of the groups.
     
   • Among these results one can mention a new proof of Bloch's conjecture on holomorphic curves in closed subvarieties of abelian varieties, the proof of the conjecture of Green and Griffiths that a holomorphic curve in a surface of general type cannot be Zariski-dense, and the hyperbolicity for generic hypersurfaces in a projective space P3 of high enough degree (Kobayashi conjecture).
     
   • Although he has worked widely in ergodic theory, his recent proof of the quantum unique ergodicity conjecture for arithmetic hyperbolic surfaces breaks fertile new ground, with great promise for future applications to number theory.
     
   • Already, in joint work with Katok and Einsiedler, he has used some of the ideas in this work to prove the celebrated conjecture of Littlewood on simultaneous diophantine approximation for all pairs of real numbers lying outside a set of Hausdorff dimension zero.
     
   • This goes far beyond what was known earlier about Littlewood's conjecture, and spectacularly confirms the high promise of the methods of ergodic theory in studying previously intractable problems of diophantine approximation.
     
   • His early results include a proof of a conjecture of Olshanski on the representations theory of groups with infinite-dimensional duals.
     
   • Okounkov gave the first proof of the celebrated Baik-Deift-Johansson conjecture, which states that the asymptotics of random partitions distributed according to the Plancherel measure coincides with that of the eigenvalues of large Hermitian matrices.
     
   • Answering affirmatively Melnikov's conjecture, Tolsa provides a solution of the Painleve problem in terms of the Menger curvature.
     
   • In arithmetic geometry, Iwasawa theory is the only general technique known for studying the mysterious relations between exact arithmetic formulae and special values of L-functions, as typified by the conjecture of Birch and Swinnerton-Dyer.
     

2. AMS Cole Prize in Algebra
   
   • for their solution of Abhyankar's conjecture.
     
   • This work appeared in the papers Revetements de la droite affine en characteristique p > 0" and "Abhyankar's conjecture on Galois groups over curves".
     
   • for his outstanding achievements in the theory of rationally connected varieties and for his illuminating work on a conjecture of Nash.
     

3. BMC 2006
   
   • Mihailescu, P Cyclotomic norm equations and Catalan's Conjecture .
     
   • Scholl, A J Recent progress on Serre's conjecture .
     

4. AMS Veblen Prize
   
   • for his paper "Stable homeomorphisms and the annulus conjecture".
     
   • for his work in differential geometry and, in particular, the solution of the four-dimensional Poincare conjecture.
     

5. AMS Steele Prize
   
   • for his expository work in his book "Algebraic theory of quadratic forms", and four of his papers "K0 and K1-an introduction to algebraic K-theory", "Ten lectures on quadratic forms over fields", "Serre's conjecture", and "The theory of ordered fields".
     
   • for his proof of the Bieberbach Conjecture.
     

6. BMC 1977
   
   • Brannan, D ASixty years of the Bieberbach conjecture for univalent functions .
     
   • Coates, JThe conjecture of Birch and Swinnerton-Dyer .
     

7. BMC 2000
   
   • Anantharaman-Delaroche, CAmenable dynamical systems and groupoids, and the Novikov conjecture .
     

8. BMC 2008
   
   • Weiss, MMumford conjecture, Pontryagin-Thom construction and sheaf theoretic methods.
     

9. NAS Award in Mathematics
   
   • for his proof of Fermat's Last Theorem by discovering a beautiful strategy to establish a major portion of the Shimura-Taniyama conjecture, and for his courage and technical power in bringing his idea to completion.
     

10. International Congress Speakers
    
    • Louis de Branges, Underlying Concepts in the Proof of the Bieberbach Conjecture.
      
    • Richard Hamilton, The Poincare Conjecture.
      

11. Fermat Prize
    
    • for his works on Shimura-Taniyama-Weil's conjecture which resulted in the demonstration of Fermat's Last Theorem.
      
    • for his proof, in collaboration with Jean-Pierre Wintenberger, of Serre's modularity conjecture in number theory.
      

12. MAA Chauvenet Prize
    
    • Ludwig Bieberbach's Conjecture and Its Proof by Louis de Branges, Amer.
      

13. AMS Cole Prize in Number Theory
    
    • for his work on the Shimura-Taniyama conjecture and Fermat's Last Theorem, published in "Modular elliptic curves and Fermat's Last Theorem".
      

14. BMC 1998
    
    • Bushnell, C J The local Langlands conjecture - some recent progress .
      

15. BMC 1992
    
    • Kronheimer, P B Milnor's conjecture and the unknotting number of algebraic knots .
      

16. BMC 1973
    
    • Laxton, RSolutions of a conjecture on linear recurrences .
      

17. BMC 1986
    
    • Rourke, C PThe resolution of the Poincare conjecture .
      

18. BMC 2003
    
    • Harris, M The local Langlands conjecture .
      

19. BMC 2002
    
    • Parnovski, LBethe-Sommerfeld conjecture and periodic differential operators .
      

20. AMS Bôcher Prize
    
    • for his paper "On a conjecture of Littlewood and idempotent measures".
      

References

1. References for Novikov Sergi
   
   • S C Ferry, A Ranickiand J Rosenberg, A history and survey of the Novikov conjecture, in Novikov conjectures, index theorems and rigidity 1 (Cambridge, 1995), 7-66.
     
   • S Weinberger, Aspects of the Novikov conjecture, in Geometric and topological invariants of elliptic operators (Providence, RI, 1990), 281-297.
     

2. References for Mordell
   
   • F Gherardelli, Two famous problems of number theory : the Fermat 'theorem' and the Mordell 'conjecture' (Italian), Archimede 38 (1) (1986), 3-9 .
     

3. References for De Morgan
   
   • N L Biggs, E K Lloyd and R J Wilson, C S Peirce and De Morgan on the four-colour conjecture, Historia Math.
     

4. References for Hadamard
   
   • P Gunther, Huygens' principle and Hadamard's conjecture, The Mathematical Intelligencer 13 (2) (1991), 56-63.
     

5. References for Peirce Charles
   
   • N L Biggs, E K Lloyd and R J Wilson, C S Peirce and De Morgan on the four-colour conjecture, Historia Math.
     

6. References for Huygens
   
   • P Gunther, Huygens' principle and Hadamard's conjecture, Math.
     

7. References for Mertens
   
   • H J J te Riele, Some historical and other notes about the Mertens conjecture and its recent disproof, Nieuw Arch.
     

8. References for Descartes
   
   • H J M Bos, Descartes, Pappus' problem and the Cartesian parabola : a conjecture, in The investigation of difficult things (Cambridge, 1992), 71-96.
     

9. References for MacMahon
   
   • E K Lloyd, Redfield's proofs of MacMahon's conjecture, Historia Mathematica 17 (1) (1990), 36-47.
     

10. References for Bieberbach
    
    • J Korevaar, Ludwig Bieberbach's conjecture and its proof by Louis de Branges, Amer.
      

11. References for Margulis
    
    • G A Margulis, Oppenheim conjecture, in M Atiyah and D Iagolnitzer (eds.), Fields Medallists Lectures (Singapore, 1997), 272-327.
      

12. References for Euler
    
    • H H Frisinger, The solution of a famous two-centuries-old problem : The Leonhard Euler Latin square conjecture, Historia Math.
      

13. References for Carlitz
    
    • G L Mullen, Permutation polynomials : A matrix analogue of Schur's conjecture and a survey of recent results, Finite Fields Appl.
      

Additional material

1. EMS obituary
   
   • He encountered, in the study of algebraic correspondences between two curves, precisely two such batches, on one of the curves, whose independence he strongly suspected but was only able to conjecture.
     
   • The conjecture was, he showed in 13, equivalent to a lemma of Severi (1905) which was well-known both because of its implications in the theory of Picard integrals on a surface and because there was some hesitation, and not only on Baker's part, in accepting Severi's proof as valid.
     
   • But just before Baker published 13 Hodge, who saw it in manuscript, succeeded in showing that Baker's conjecture was indeed true.
     
   • There is an incident relating to this conjecture that is not generally known.
     
   • In the summer of 1929 Baker was lecturing on these topics and put the conjecture to his class.
     

2. Charles Bossut on Leibniz and Newton Part 2
   
   • The reason for this is not difficult to conjecture: he had not resolved his own problem.
     
   • In this conjecture he was cruelly mistaken: and his challenge, which was something more than indiscreet, drew on him a reprimand from the Swiss geometrician that was so much the more poignant as the only mode of answering it satisfactorily was by a solution of the problem which he could neither effect by his own skill nor by the assistance of his friends.
     

3. Eulogy to Euler by Fuss
   
   • Euler of this conjecture that the trajectories that are described around one or more centers of force could be determined by the same method.
     
   • Dolland who had found two different types of glass in the meanwhile which were capable of examining in greater detail finally admitted to Euler's conjecture in 1757 by the invention of achromatic glasses which made considerable impression in both astronomy and dioptics.
     

4. Eulogy to Euler by Fuss
   
   • Euler of this conjecture that the trajectories that are described around one or more centers of force could be determined by the same method.
     
   • Dolland who had found two different types of glass in the meanwhile which were capable of examining in greater detail finally admitted to Euler's conjecture in 1757 by the invention of achromatic glasses which made considerable impression in both astronomy and dioptics.
     

5. Bronowski and retrodigitisation
   
   • Emil Artin (1898--1962) distilled the ensuing investigations of special cases in a general conjecture in 1927: any integer m, other than 0 or -1, and not divisible by a square, is the primitive root of infinitely many primes; and such primes have positive denisity in the set of primes independent of the choice of m.
     
   • Although much progress has been made on this conjecture, it has been of a conditional or non-constructive kind, and, as yet, no m is known which is a primitive root for infinitely many primes.
     

6. Mark Kac on education, physics and mathematics
   
   • Now I think you would agree with me because, especially with things like geometry, of which Stan's a past master, seeing things - not always leading neatly to a proof, but certainly leading to the understanding - ultimately results in the correct conjecture.
     

7. EMS obituary
   
   • A long-standing conjecture on the convergence properties of the former was settled in one of her papers.
     

8. James Clerk Maxwell on the nature of Saturn's rings
   
   • These appearances seem to indicate the same slow progress of the rings towards separation which we found to he the result of theory, and the remark, that the inner edge of the inner ring is most distinct, seems to indicate that the approach towards the planet is less rapid near the edge, as we had reason to conjecture.
     

9. Charles Bossut on Leibniz and Newton
   
   • What renders this conjecture very probable is that Mr Oldenburg, secretary to the Royal Society, sending a copy of Sluze's Method of Tangents, which had been printed at London, to the author on the 10th of July 1673, encloses him an extract from a letter of Newton's; in which, after having observed that this method justly belongs to Sluze, Newton goes on thus: 'as to the methods,' (he is speaking of that of Sluze and his own) 'they are the same, though I believe they are derived from different principles.
     

10. Twenty-Five Years of Groups St Andrews Conferences
    
    • In 1979 we held a small meeting in St Andrews at which Joachim Neubuser from RWTH Aachen spoke on Counterexamples to the class-breadth conjecture.
      

11. NAS Award in Mathematics
    
    • for his proof of Fermat's Last Theorem by discovering a beautiful strategy to establish a major portion of the Shimura-Taniyama conjecture, and for his courage and technical power in bringing his idea to completion.
      

12. James Jeans addresses the British Association in 1934
    
    • The earthquake shocks were, of course, new facts of observation, and the building fell because it was not built on the solid rock of ascertained fact, but on the ever-shifting sands of conjecture and speculation.
      

13. Julia Robinson: Hilbert's 10th Problem
    
    • If you really are 22 [he was], I am especially pleased to think that when I first made the conjecture you were a baby and I just had to wait for you to grow up! .
      

14. ELOGIUM OF EULER
    
    • The proof of this conjecture might have depended in the glass and whether the elimination of iridescence of colors which project through the glass lenses.
      

15. De Coste on Mersenne 1.html
    

16. Julia Robinson: Hilbert's 10th Problem
    
    • If you really are 22 [he was], I am especially pleased to think that when I first made the conjecture you were a baby and I just had to wait for you to grow up! .
      

17. Andrew Forsyth addresses the British Association in 1905, Part 2
    
    • No doubt our astronomers will be ready for it: and the added knowledge of electrical science, in connection particularly with the properties of matter, may enable them to review Bessel's often-discussed conjecture as to an explanation of the emission of a sunward tail.
      

18. H L F Helmholtz: 'Theory of Music' Introduction
    
    • 540-510) knew that when strings of different lengths but of the same make, and subjected to the same tension, were used to give the perfect consonances of the Octave, Fifth, or Fourth, their lengths must be in the ratios of 1 to 2, 2 to 3, or 3 to 4 respectively, and if, as is probable, his knowledge was partly derived from the Egyptian priests, it is impossible to conjecture in what remote antiquity this law was first known.
      

19. G H Hardy addresses the British Association in 1922, Part 2
    
    • It seems now that Mersenne's assertion, so far from hiding unplumbed depths of mathematical profundity, was a conjecture based on inadequate empirical evidence, and a rather unhappy one at that.
      

20. Christiaan Huygens' article on Saturn's Ring
    
    • Astonished by what he saw, he tried to reach by conjecture the cause of the appearance, and made a few predictions as to the time when the former phase was due to recur.
      

21. Christiaan Huygens' article on Saturn's Ring
    
    • Astonished by what he saw, he tried to reach by conjecture the cause of the appearance, and made a few predictions as to the time when the former phase was due to recur.
      

22. Gibson History 7 - Robert Simson
    
    • The text of Pappus' Collection was itself in a very unsatisfactory state so that the opportunities for conjecture were endless.
      

23. Julia Robinson: Hilbert's 10th Problem
    
    • If you really are 22 [he was], I am especially pleased to think that when I first made the conjecture you were a baby and I just had to wait for you to grow up! .
      

24. Kuratowski: 'Introduction to Set Theory
    
    • In connection with this theorem, there arises the fundamental conjecture: does there exist a relation for any set which establishes its well ordering? - We shall prove that this is in fact so, if we assume, the axiom of choice.
      

25. James Clerk Maxwell on the nature of Saturn's rings
    
    • These appearances seem to indicate the same slow progress of the rings towards separation which we found to he the result of theory, and the remark, that the inner edge of the inner ring is most distinct, seems to indicate that the approach towards the planet is less rapid near the edge, as we had reason to conjecture.
      

26. G H Hardy addresses the British Association in 1922
    
    • It seems now that Mersenne's assertion, so far from hiding unplumbed depths of mathematical profundity, was a conjecture based on inadequate empirical evidence, and a rather unhappy one at that.
      

27. De Coste on Mersenne
    
    • Just as fountain-makers take it to be a good omen when they see vapour coming out of the earth in the morning, because it is one of the signs which make them hope they will find a good spring; so in the same way, those who have the best knowledge of our souls, rejoice in noting at a tender age, a passionate desire to learn and a rapturous ardour for knowledge and for virtue, because from that they can conjecture almost certainly the good quality of our minds and the excellence which we must one day achieve.
      

28. Euler Elogium.html.html
    
    • The proof of this conjecture might have depended in the glass and whether the elimination of iridescence of colors which project through the glass lenses.
      

Quotations

1. Quotations by Leibniz
   
   • The art of discovering the causes of phenomena, or true hypothesis, is like the art of decyphering, in which an ingenious conjecture greatly shortens the road.
     
   • The art of discovering the causes of phenomena, or true hypothesis, is like the art of decyphering, in which an ingenious conjecture greatly shortens the road.
     

2. Quotations by Bernoulli Jacob
   
   • We define the art of conjecture, or stochastic art, as the art of evaluating as exactly as possible the probabilities of things, so that in our judgments and actions we can always base ourselves on what has been found to be the best, the most appropriate, the most certain, the best advised; this is the only object of the wisdom of the philosopher and the prudence of the statesman.
     
   • It seems that to make a correct conjecture about any event whatever, it is necessary to calculate exactly the number of possible cases and then to determine how much more likely it is that one case will occur than another.
     

3. Quotations by Wiles
   
   • Concluding the lecture in which he claimed to have proved the Taniyama-Weil Conjecture for a class of examples, including those necessary to prove Fermat's Last Theorem.
     

4. Quotations by Bernoulli Johann
   
   • We define the art of conjecture, or stochastic art, as the art of evaluating as exactly as possible the probabilities of things, so that in our judgements and actions we can always base ourselves on what has been found to be the best, the most appropriate, the most certain, the best advised: this is the only object of the wisdom of the philosopher and the prudence of the statesman.
     

Chronology

1. Mathematical Chronology
   
   • Jacob Bernoulli's book Ars conjectandi (The Art of Conjecture) is an important work on probability.
     
   • It is not yet known whether Goldbach's conjecture is true.
     
   • Euler makes Euler's Conjecture, namely that it is impossible to exhibit three fourth powers whose sum is a fourth power, four fifth powers whose sum is a fifth power, and similarly for higher powers.
     
   • Gauss proves Fermat's conjecture that every number can be written as the sum of three triangular numbers.
     
   • He proves Bertrand's conjecture there is always at least one prime between n and 2n for n > 1.
     
   • Francis Guthrie poses the Four Colour Conjecture to De Morgan.
     
   • Riemann makes a conjecture about the zeta function which involves prime numbers.
     
   • The problems include the continuum hypothesis, the well ordering of the real numbers, Goldbach's conjecture, the transcendence of powers of algebraic numbers, the Riemann hypothesis, the extension of "Dirichlet's principle" and many more.
     
   • Poincare proposes the Poincare Conjecture, namely that any closed 3-dimensional manifold which is homotopy equivalent to the 3-sphere must be the 3-sphere.
     
   • Bieberbach formulates the Bieberbach Conjecture.
     
   • Gelfond makes his Conjecture about the linear independence of algebraic numbers over the rational numbers.
     
   • This is a major contribution to the solution of the Goldbach conjecture.
     
   • Hodge puts forward the "Hodge Conjecture" on projective algebraic varieties.
     
   • Taniyama poses his conjecture on elliptic curves which will play a major role in the proof of Fermat's Last Theorem.
     
   • Smale proves the higher dimensional Poincare conjecture for n > 4, namely that any closed n-dimensional manifold which is homotopy equivalent to the n-sphere must be the n-sphere.
     
   • Sergi Novikov's work on differential topology, in particular in calculating stable homotopy groups and classifying smooth simply-connected manifolds, leads him to make the "Novikov Conjecture".
     
   • Lander and Parkin use a computer to find a counterexample to Euler's Conjecture.
     
   • Alan Baker proves "Gelfond's Conjecture" about the linear independence of algebraic numbers over the rational numbers.
     
   • It makes a major contribution to the Goldbach Conjecture.
     
   • Appel and Haken show that the Four Colour Conjecture is true using 1200 hours of computer time to examine around 1500 configurations.
     
   • Mori proves the "Hartshorne conjecture", that projective spaces are the only smooth complete algebraic varieties with ample tangent bundles.
     
   • This proves a further case of the higher dimensional Poincare conjecture following Smale's work in 1961.
     
   • Shing-Tung Yau is awarded a Fields Medal for his contributions to partial differential equations, to the "Calabi conjecture" in algebraic geometry, to the positive mass conjecture of general relativity theory, and to real and complex Monge-Ampere equations.
     
   • Faltings proves the "Mordell conjecture".
     
   • Louis de Brange solves the Bieberbach Conjecture.
     
   • Margulis proves the "Oppenheim conjecture" on the values of indefinite irrational quadratic forms at integer points.
     
   • Zelmanov proves an important conjecture about when an infinite dimensional Lie algebra is nilpotent.
     
   • Elkies finds a counterexample to Euler's Conjecture with n = 4, namely 958004 + 2175194 + 4145604 = 4224814.
     
   • Menasco and Thistlethwaite prove the knot theory conjecture known as "Tait's Second Conjecture", namely that any two reduced alternating diagrams of the same prime knot are related by a sequence of twists.
     
   • Krystyna Kuperberg solves the "Seifert Conjecture" about the topology of dynamical systems.
     
   • A large prize is offered by banker Andrew Beal for a solution to the Beal Conjecture: the equation xp + yq = zr has no solutions for p, q, r > 2 and coprime integers x, y, z.
     
   • Conrad and Taylor prove the "Taniyama-Shimura conjecture".
     

2. Chronology for 1980 to 1990
   
   • This proves a further case of the higher dimensional Poincare conjecture following Smale's work in 1961.
     
   • Shing-Tung Yau is awarded a Fields Medal for his contributions to partial differential equations, to the "Calabi conjecture" in algebraic geometry, to the positive mass conjecture of general relativity theory, and to real and complex Monge-Ampere equations.
     
   • Faltings proves the "Mordell conjecture".
     
   • Louis de Brange solves the Bieberbach Conjecture.
     
   • Margulis proves the "Oppenheim conjecture" on the values of indefinite irrational quadratic forms at integer points.
     
   • Zelmanov proves an important conjecture about when an infinite dimensional Lie algebra is nilpotent.
     
   • Elkies finds a counterexample to Euler's Conjecture with n = 4, namely 26824404 + 153656394 + 187967604 = 206156734.
     

3. Chronology for 1990 to 2000
   
   • Menasco and Thistlethwaite prove the knot theory conjecture known as "Tait's Second Conjecture", namely that any two reduced alternating diagrams of the same prime knot are related by a sequence of twists.
     
   • Krystyna Kuperberg solves the "Seifert Conjecture" about the topology of dynamical systems.
     
   • A large prize is offered by banker Andrew Beal for a solution to the Beal Conjecture: the equation xp + yq = zr has no solutions for p, q, r > 2 and coprime integers x, y, z.
     
   • Conrad and Taylor prove the "Taniyama-Shimura conjecture".
     
   • Called the Millennium Prize Problems they are: P versus NP; The "Hodge Conjecture"; The Poincare Conjecture; The Riemann Hypothesis; "Yang-Mills Existence and Mass Gap"; "Navier-Stokes Existence and Smoothness"; and The "Birch and Swinnerton-Dyer Conjecture".
     

4. Chronology for 1960 to 1970
   
   • Smale proves the higher dimensional Poincare conjecture for n > 4, namely that any closed n-dimensional manifold which is homotopy equivalent to the n-sphere must be the n-sphere.
     
   • Sergi Novikov's work on differential topology, in particular in calculating stable homotopy groups and classifying smooth simply-connected manifolds, leads him to make the "Novikov Conjecture".
     
   • Lander and Parkin use a computer to find a counterexample to Euler's Conjecture.
     
   • Alan Baker proves "Gelfond's Conjecture" about the linear independence of algebraic numbers over the rational numbers.
     

5. Chronology for 1850 to 1860
   
   • He proves Bertrand's conjecture there is always at least one prime between n and 2n for n > 1.
     
   • Francis Guthrie poses the Four Colour Conjecture to De Morgan.
     
   • Riemann makes a conjecture about the zeta function which involves prime numbers.
     

6. Chronology for 1970 to 1980
   
   • It makes a major contribution to the Goldbach Conjecture.
     
   • Appel and Haken show that the Four Colour Conjecture is true using 1200 hours of computer time to examine around 1500 configurations.
     
   • Mori proves the "Hartshorne conjecture", that projective spaces are the only smooth complete algebraic varieties with ample tangent bundles.
     

7. Chronology for 1950 to 1960
   
   • Hodge puts forward the "Hodge Conjecture" on projective algebraic varieties.
     
   • Taniyama poses his conjecture on elliptic curves which will play a major role in the proof of Fermat's Last Theorem.
     

8. Chronology for 1900 to 1910
   
   • The problems include the continuum hypothesis, the well ordering of the real numbers, Goldbach's conjecture, the transcendence of powers of algebraic numbers, the Riemann hypothesis, the extension of "Dirichlet's principle" and many more.
     
   • Poincare proposes the Poincare Conjecture, namely that any closed 3-dimensional manifold which is homotopy equivalent to the 3-sphere must be the 3-sphere.
     

9. Chronology for 1800 to 1810
   
   • Gauss proves Fermat's conjecture that every number can be written as the sum of three triangular numbers.
     

10. Chronology for 1910 to 1920
    
    • Bieberbach formulates the Bieberbach Conjecture.
      

11. Chronology for 1930 to 1940
    
    • This is a major contribution to the solution of the Goldbach conjecture.
      

12. Chronology for 1940 to 1950
    
    • Hodge puts forward the "Hodge Conjecture" on projective algebraic varieties.
      

13. Chronology for 1920 to 1930
    
    • Gelfond makes his Conjecture about the linear independence of algebraic numbers over the rational numbers.
      

14. Chronology for 1840 to 1850
    
    • He proves Bertrand's conjecture there is always at least one prime between n and 2n for n > 1.
      

15. Chronology for 1740 to 1760
    
    • It is not yet known whether Goldbach's conjecture is true.
      

16. Chronology for 1890 to 1900
    
    • The problems include the continuum hypothesis, the well ordering of the real numbers, Goldbach's conjecture, the transcendence of powers of algebraic numbers, the Riemann hypothesis, the extension of "Dirichlet's principle" and many more.
      

17. Chronology for 1760 to 1780
    
    • Euler makes Euler's Conjecture, namely that it is impossible to exhibit three fourth powers whose sum is a fourth power, four fifth powers whose sum is a fifth power, and similarly for higher powers.
      

18. Chronology for 1700 to 1720
    
    • Jacob Bernoulli's book Ars conjectandi (The Art of Conjecture) is an important work on probability.