Paul R. Halmos

Photo Caption: Lancaster 1984 - Sheldon Axler and Halmos

“[This photo*] was taken at a conference in Lancaster (England) in 1984, and it represents four mathematical generations. I am at right, next to me is Don (D.E. Sarason), my student, next to him is Sheldon, his Ph.D. student, and next to Sheldon is Pam (Axler), who is, of course, Sheldon’s Ph.D. student.” –Paul R. Halmos, I Have a Photographic Memory

 *This photo is a slightly different angle than the photo featured in I Have a Photographic Memory


Photo Caption: Sheldon Axler August 75


Sheldon Axler is Dean of the College of Science & Engineering at San Francisco State University.  In 1996, he received the Lester R. Ford Award for expository writing from the Mathematical Association of America for his article Down with Determinants!He also served as an Associate Editor of The American Mathematical Monthly.

Sheldon Axler Homepage

Sheldon Axler Biography by MSRI  

Photo Caption: M Atiyah 29 Mar 69 

“The Atiyah-Singer index theorem was the toughest hurdle for me, but, somehow, we conquered it too. (To be sure, after it appeared in print, Singer told me that it didn’t come out quite right—the relation with the Riemann-Roch theorem was unclear or perhaps even misstated—but there it was, and I feel sure that my fellow ignoramuses and I learned something worth knowing that we hadn’t known before.)”–Paul R. Halmos, I Want to Be a Mathematician

Michael Francis Atiyah contributed to a wide range of topics in mathematics centering on the interaction between geometry and analysis. His work showed how the study of vector bundles on spaces could be regarded as the study of cohomology theory, called K-theory. He was awarded the Fields Medal in 1966.

The ideas which led to Atiyah being awarded a Fields Medal were later seen to be relevant to gauge theories of elementary particles.

The theories of superspace and supergravity and the string theory of fundamental particles, which involves the theory of Riemann surfaces in novel and unexpected ways, were all areas of theoretical physics which developed using the ideas which Atiyah was introducing. 

In addition to the Fields Medal, Atiyah received many honors during his career including the Feltrinelli Prize from the Accademia Nazionale dei Lincei in 1981, the King Faisal International Prize for Science in 1987, the Benjamin Franklin Medal, and the Nehru Medal. In 2004, he and Isadore Singer were awarded the Neils Abel prize of £480 000 for their work on the Atiyah-Singer Index Theorem.

Michael Francis Atiyah Biography

Photo Caption: WVD Hodge and ML Cartwright (1950)

“Bill Hodge (later Sir William) did algebraic geometry; there is something called a Hodge variety. His book with Pedoe was a large and difficult step forward when it came out.” — Paul R. Halmos, I Have a Photographic Memory

William Vallance Douglas Hodge was a Scottish mathematician, specifically a geometer.

“Hodge returned to Cambridge in 1932. He was appointed as a university lecturer in the following year and, in 1935, was elected to a fellowship at Pembroke College, Cambridge. During this period he developed the relationship between geometry, analysis and topology and produced some of his best remembered work on the theory of harmonic integrals. For these contributions Hodge won the Adams Prize in 1937 and Weyl described this contribution as ’… one of the great landmarks in the history of science in the present century.’

Hodge published a polished account of his important theory in 1941. This work marked an important change in direction for the Cambridge school of geometry which, under Baker’s leadership, had become somewhat isolated from other areas of mathematics.” Read More

William Vallance Douglas Hodge Biography

William Valance Douglas Hodge Obituary by M.F. Atiyah

Related entry: Dame Mary Cartwright

Photo Caption: Nash 

“John is a non-cooperative game theorist (non-associative phrase) who has also written about cooperative games, and is famous for his imbedding theorem for Riemannian manifolds.” —Paul R. Halmos,I Have a Photographic Memory 

In 1949, while studying for his doctorate, John Nash wrote a paper which 45 years later was to win a Nobel Prize for economics. During this period Nash established the mathematical principles of game theory. P Ordeshook wrote: 

The concept of a Nash equilibrium n-tuple is perhaps the most important idea in noncooperative game theory. … Whether we are analysing candidates’ election strategies, the causes of war, agenda manipulation in legislatures, or the actions of interest groups, predictions about events reduce to a search for and description of equilibria. Put simply, equilibrium strategies are the things that we predict about people.

John Nash

Photo Caption: Ed Begle

“Ed started out as a topologist, a student of Lefschetz’s at Princeton, but then became famous for two other reasons. He was, for one thing, Secretary of the AMS between 1951 and 1956, and, as one of the prime movers of the SMSG (School of Mathematics Study Group) he was also one of the prime movers of the “new math”. A lot of people liked the SMSG and worked hard for it, but, in the interests of historical honesty, I must report that some of the others referred to it as Some Mathematics, Some Garbage.” –Paul R. Halmos, I Have a Photographic Memory

Begle was awarded a thesis in 1940 for his thesis Locally Connected Spaces and Generalized Manifolds. In his thesis, Begle started with the concepts of a realization and a partial realization of finite complex on a space which had been by Lefschetz in a 1936 paper. He gave new definitions of these concepts which allowed him to use other techniques and simplify the study of generalized manifolds. He used Vietoris cycles throughout his thesis.

Edward Griffith Begle Biography

Photo Caption: Doob, Aug 1, 1974

“Ambrose and I were blasé graduate students; we knew everything about the department, we knew everyone, and we could be trusted to deal with anything that was likely to come up that was likely to come up one early September afternoon–the department secretary left us in charge to watch over the main office. This boy came in, looking like a new graduate student, crew cut, shirt sleeves, and all. He was 25 years old at the time, I later learned, but he looked 19 or 20. What do you want?, we asked. Is Coble here?, he wanted to know; my name is Doob, D,O,O,B. Ambrose and I had heard of the bright young hot shot who was coming to Illinois from a fellowship at Columbia; we yanked our feet off the desk and told him who we were.” —Paul R. Halmos, I Want to Be a Mathematician

Joseph Leo Doob’s work was in probability and measure theory, in particular he studied the relations between probability and potential theory. Building on the work by Paul Lévy, Doob developed basic martingale theory and many of its applications during the 1940s and 1950s. His work has become one of the most powerful tools available to study stochastic processes. In the introduction to his Stochastic Processes (1953),Doob states that:

“… [a stochastic process is] any process running along in time and controlled by probabilistic laws … [more precisely] any family of random variables {xt| t T [where] a random variable is … simply a measurable function." 

Joseph Leo Doob