Patrick Du Val

Born: 26 March 1903 in Cheadle Hulme (near Manchester), Cheshire, England
Died: 22 January 1987 in Cambridge, Cambridgeshire, England

Patrick Du Val's parents were Bartram Du Val, who was a cabinet maker, and his wife Margaret. Partick's parents had an unfortunately short-lived marriage, and many years after the marriage effectively ended, Margaret petitioned for a divorce in 1930. In fact Patrick was brought up by his mother in a single-parent family. He suffered from bad health as a child, due to an asthmatic condition, and was not well enough to attend school. He was educated at home by his mother, who was a truly remarkable woman capable of teaching her son to a high level. However, as he progressed to advanced mathematics and science around the age of eighteen she struggled to keep one step ahead in order to give him proper tuition. He took a university level correspondence course, receiving some excellent tuition from a private tutor by the name of Miss B H Poole, and was awarded a B.Sc. with First Class Honours in Mathematics from the University of London in 1926, having been examined as an external student. What is quite remarkable is that he began publishing mathematical articles while studying at home. These were all applied mathematical in nature and were published by the Philosophical Magazine: Geometrical note on de Sitter's world (1924), The relations between antisymmetric tensors and tensor-densities (1924), On discriminations between past and future (1925), and The derivation of energy from action (1927).

Talented in many different subjects, such as languages and history, in addition to mathematics, he still had not made up his mind as to the career he should pursue. Wishing to study Henrik Ibsen's Peer Gynt, a drama written in Norwegian in rhymed couplets, he felt that the only way he could fully appreciate the work was to learn Norwegian; he therefore bought himself books and taught himself Norwegian. Despite being pulled in different directions by his wide range of talents, it was mathematics which came to dominate more and more. Perhaps the most significant step in his life came when he and his mother moved to a village near Cambridge. They got to know Henry Baker and he persuaded Patrick to undertake research into algebraic geometry at Cambridge; Du Val matriculated as a research student at Cambridge in 1927. Baker's research group contained a remarkable collection of geometers: Donald Coxeter, Leonard Edge, William Hodge and John Semple were all close to Du Val in age (only one year separated them all), and there also was the younger John Todd. During his period as a research student these geometers, all later making outstanding mathematical contributions, became good friends and Du Val formed a particularly strong friendship with Coxeter and Semple. With Baker as his thesis advisor, Du Val wrote the thesis On certain configurations of algebraic geometry, having groups of self-transformations representable by the symmetry groups of certain polytopes for which he received a Ph.D. in 1930. In the same year he attended the British Association meeting in Bristol and wrote a report for the Mathematical Gazette. He begins his report showing where his own interests lie [2]:-

The 100th meeting of the British Association was unquestionably one of the very greatest interest to mathematicians of almost all kinds; though by a conjunction of accidents straightforward algebraic geometry found itself almost completely left out of the programme. Interesting as were Mr Richmond's investigations as to the possibility of expressing a given number as the sum of three cubes, many of his hearers must have regretted the loss of his very valuable researches on the Canonical Curve of genus 5, which he apparently decided at the last moment were not at a sufficiently advanced stage to form conveniently the subject for his paper. Other branches of geometry were represented by Mr Coxeter, who gave an account of the modern analytical methods of discovering and completely enumerating the regular polytopes (a subject not as much studied in this country as one might wish), and by Mr Hodge, who explained the work which has been done towards the use of topology for the discussion of algebraic surfaces, a surface being represented by a "Riemann fourfold" in the same way as a curve may be by a Riemann surface.

His final remarks about Paul Dirac's lecture show his humour and also his confidence, for although Dirac was only one year older than Du Val, he had already attained a high international reputation:-

In spite of some quantitative difficulties, Dr Dirac succeeded in arousing the enthusiasm of a large audience, though many of them seemed to feel that the chief beauty of his theory was its appearance of being utter nonsense.

In the same year of 1930, Du Val achieved the unusual feat of being awarded a Fellowship of Trinity College after only three years of research. In addition to the work of his Ph.D., which he did not publish until 1933, Du Val published two papers during his three undergraduate years: On questions of reality for twisted quartics of the first kind (1928) and On the Hesse-Cayley algorithm for a plane quartic whose bitangents are all real (1929).

Du Val held the Trinity Fellowship for four years during which time he visited Rome working with Federigo Enriques. This was an exciting time for Du Val who became deeply interested in the theory of algebraic surfaces. He studied classification problems for such surfaces publishing his first two papers on this topic in Italian: Superficie di genere uno che non sono base per un sistema di quadriche (1932) and Osservazioni sulle superficie di genere uno che non sono base per un sistema di quadriche (1932). Then in 1934 he visited Princeton and attended lectures by Alexander, Luther Eisenhart, Solomon Lefschetz, Oswald Veblen, Joseph Wedderburn and Hermann Weyl. He spent two years at Princeton, supported by a Rockefeller Foundation Fellowship, where he continued his friendship and discussions with Donald Coxeter, who also had a fellowship allowing him to visit. In July 1935 Du Val and Coxeter gave a joint presentation to the Pittsburgh meeting of the American Association for the Advancement of Science. They also attended meetings of the American Mathematical Society in New York and after one such meeting Coxeter records in his diary that "Pat Du Val took me to see some burlesque."

Du Val returned to England in early 1936 and back at Trinity College Coxeter notes that when dining in College "Pat Du Val got drunk and tried to show how he could sing." In March 1936 Coxeter arranged his first date with the girl he was to marry, suggesting they "have lunch in my rooms [and] meet my best friend Pat Du Val for tea ..." In the same year Du Val submitted an essay on the resolution of singularities of an algebraic surface for the Adams Prize at Cambridge. He was unlucky in that Hodge submitted a truly remarkable essay on harmonic integrals for the 1936 Adams Prize which won, beating not only Du Val but also Coxeter and others. He was appointed as an Assistant Lecturer at Manchester University in 1936 where, after being reappointed, he stayed for five years. During this time (in 1938) he published a pamphlet of 26 pages written jointly with H S M Coxeter, H T Flather, and J F Petrie entitled The Fifty-nine Icosahedra. The idea for the pamphlet was due to Coxeter who wanted to construct models of 59 polyhedral stellations and write a paper to explain them but, realising the immense amount of work involved, enlisted the help of three friends, one of whom was Du Val. Here is Miller's description of what was involved [3]:-

This pamphlet describes the application of the process of stellation to the regular solids. By stellation is meant the derivation of one polyhedron from another by extension of the faces of the original until they meet other such extensions, to form new faces, edges, and vertices. The definition is modified so as to include cases in which a "face" of the derived solid consists of several isolated portions all, of course, in the same plane. This process evidently leaves the number of such faces unaltered. In the paper reviewed here, restrictive rules have been formulated which mean, in effect, that the final polyhedron must retain the full symmetry of the parent regular solid, except possibly for reflexive symmetry.

The pamphlet was reprinted in 1951, then 44 years after it was first published, Du Val brought out a second edition in 1982. This second edition was republished in 2002.

After five years in Manchester, Du Val went to Istanbul in 1941 to take up the chair of Pure Mathematics. Of course this was in the middle of World War II and, indeed, the post was related to the War being part of a war-time scheme devised by the British Council [4];-

Despite the uncertain world conditions of the time, he very much enjoyed this period of his life. He developed a strong interest in Byzantine culture and quickly mastered the Turkish language, in which some of his work is written, including an elementary textbook on coordinate geometry of which he was rather proud.

In 1945 he made a short return visit to Cambridge, during which time he married Isobel Shimwell. The newly married couple went to Istanbul where they started a family and Du Val continued to undertake his duties at the University. He publication record during this period was no longer the remarkable one of the years 1932-36 when he had published 19 papers. In 1949 he moved to the United States where he spent three years as Professor at Georgia State University at Athens, Georgia. Rather unhappy in the United States, he returned to England, first taking up a post at Bristol University in 1952, then at University College, London in 1954 where he remained until he retired in 1970. Together with Semple he led the London Geometry Seminar during the time he spent in London. After he retired Du Val returned to Istanbul and for three years where he held the same post as he had held 30 years before. Then he returned to England and lived in Cambridge in his retirement.

As we noted above, Du Val's early work before he became a research student at Cambridge was on relativity. He published on the de Sitter model of the universe and Grassmann's tensor calculus. His doctorate was on algebraic geometry and in his thesis he generalised a result of Pieter Schoute. He worked on algebraic surfaces, especially during his time in Rome, and he published the important monograph Homographies, quaternions and rotations in 1964. Bernard d'Orgeval writes in a review:-

The final chapter deals with groups of involutions and is the most original, bringing new views on the theory of singularities of algebraic surfaces to which the author has devoted considerable work.

Later in his career Du Val became interested in elliptic functions publishing the book Elliptic functions and elliptic curves in 1973. Herschel Farkas writes in a review:-

In these notes the author treats the reader to a magnificent tour through the fantasyland of elliptic function theory.

We note that Du Val was an invited speaker at the British Mathematical Colloquium held in Nottingham in 1957 when he gave the lecture Quaternions and polytopes in four dimensions.

Du Val was always interested in teaching as well as research. His style as a lecturer is described in these terms in [4]:-

... his raucous asthmatic delivery gave him a somewhat forbidding manner, yet he was most kind and sympathetic to his students and always willing to spend time coaching the weaker ones among them. ... At other people's lectures, he was noted for the quickness with which he grasped new ideas; and his courteous interventions were often the makings of what would otherwise have been rather dull seminars.

As to his character, Tyrrell notes that [4]:-

... he displayed an enigmatic combination of the modest and the flamboyant ... Du Val was a remarkable man to know and a most amusing companion. His many friends were saddened to hear that his last years were troubled by illness, both of himself and of his wife, who survived him by little more than a year. They leave a son Nicholas and two daughters, Paula and Belinda.

Article by: J J O'Connor and E F Robertson

February 2010

MacTutor History of Mathematics