Photos by Paul Halmos

Functional analyst Billy James Pettis (1913-1979) was photographed by Halmos in August of 1975 at the Joint Summer Mathematics Meetings in Kalamazoo, Michigan. Pettis earned his Ph.D. in 1937 from the University of Virginia with the dissertation “Integration in Vector Spaces,” written under advisor Edward J. McShane. In fact, Pettis was McShane’s first Ph.D. student. (McShane is pictured on page 34 of this collection.) Pettis was a faculty member at Tulane University in New Orleans, Louisiana, and, from 1957 onward, at the University of North Carolina, Chapel Hill. (Sources: Mathematics Genealogy Project; A Guide to the B. J. Pettis Papers, 1938-1980, Archives of American Mathematics)

Who’s That Mathematician? Images from the Paul R. Halmos Photograph Collection

Halmos photographed George Pólya (1887-1985) and Alexander Ostrowski (1893-1986) in 1958. Another photo of Ostrowski appears on page 38 of this collection, where you can read more about him.

Born in Budapest, Hungary, George (György) Pólya entered the University of Budapest (now Eötvös Loránd University) in 1905. After studying law, languages, literature, philosophy, and, finally, physics and mathematics, he received his Ph.D. in mathematics in 1912 with a thesis in geometric probability written under Leopold (Lipót) Fejér. He then spent a year studying at the University of Göttingen, Germany, with its who’s who of eminent mathematicians, and then another few months studying in Paris, before being invited byAdolf Hurwitz, then Chair of Mathematics at Eidgenössische Technische Hochschule (ETH) in Zürich, Switzerland, to join the faculty there, which he did in 1914. Pólya worked closely with Hurwitz until Hurwitz’s death in 1919.

Although he may be best known today for his contributions to mathematics teaching and learning, Pólya was a prolific and formidable researcher who made important contributions in complex analysis, probability, combinatorics, geometry, and mathematical physics. Besides writing many papers (O’Connor and Robertson of the MacTutor Archive pointed out that he published 31 papers just from 1926 to 1928), he also wrote influential books. In 1925, after years of work, Pólya and Gábor Szegő published Problems and Theorems in Analysis, Volumes I, II (Springer), and in 1924 Pólya began to work with G. H. Hardy and J. E. Littlewood (page 31 of this collection) on the book Inequalities (Cambridge, 1934). In 1940, Pólya moved to the United States and, after short stints at Brown University and Smith College, he joined the faculty at Stanford University in Palo Alto, California, where Szegő had been based since 1938. He and Szegő continued their collaboration, producing another influential book, Isoperimetric Inequalities in Mathematical Physics (Princeton, 1951).

In 1945, Pólya published what may be his best known book, and certainly is the one that established him as a leader in mathematics teaching and learning, How to Solve It: A New Aspect of Mathematical Method (Princeton), which has been translated into 17 languages. Other books on mathematical reasoning and surveys/textbooks include:

• Mathematics and Plausible Reasoning: Volume I, Induction and Analogy in Mathematics; Volume II, Patterns of Plausible Inference (Princeton, 1954);
• Mathematical Discovery: On understanding, learning, and teaching problem solving: Volume I (Wiley, 1962), Volume II (1965);
• Complex Variables, with Gordon Latta (Wiley, 1974);
• Mathematical Methods in Science, with Leon Bowden (MAA, 1977); and
• Notes on Introductory Combinatorics, with Robert Tarjan and Donald Woods (Birkhäuser Boston, 1983).

Pólya advised at least 30 Ph.D. students at ETH and Stanford, plus one more at England’s Cambridge University, Imre Lakatos, who received his Ph.D. in 1961. Lakatos’ Ph.D. dissertation, titled “Essays in the Logic of Mathematical Discovery,” eventually became the book Proofs and Refutations: The Logic of Mathematical Discovery (Cambridge University Press, 1976). (Sources: MacTutor Archive, Mathematics Genealogy Project, MathSciNet, WorldCat)

Who’s That Mathematician? Images from the Paul R. Halmos Photograph Collection

Richard Rado (1906-1989), Robert Rankin (1915-2001), and Hans Reimann, left to right, were photographed by Halmos in April of 1965 at the British Mathematical Colloquium in Dundee, Scotland. Halmos was one of three main speakers at this conference (I Want to Be a Mathematician, Springer, 1985, pp. 290-292). Another photograph of Rankin appears on page 7 of this collection, where you can read more about him. 

Born in Berlin, Germany, Richard Rado earned doctoral degrees from the University of Berlin in 1933 and from Cambridge University in 1935. At the University of Berlin, he wrote the dissertation, “Studies on combinatorics,” under advisor Issai Schur and at Cambridge, he wrote the dissertation, “Linear Transformations on Bounded Sequences,” under advisor G. H. Hardy. Although he would write papers in both fields, his research throughout his career was primarily in combinatorics. In 1934, Rado met Paul Erdős, who had earned his Ph.D. in Budapest that year and accepted a fellowship at the University of Manchester in England, and the two began to collaborate. Erdős described the strengths each brought to their collaboration as follows:

I was good at discovering perhaps difficult and interesting special cases and Richard was good at generalising them and putting them in their proper perspective (quoted by O’Connor and Robertson in their MacTutor Archive biography of Rado).

After spending 1935-36 at Cambridge University, Rado was on the mathematics faculty at the University of Sheffield, England, from 1936 to 1947, then at King’s College, London, from 1947 to 1954, and finally at the University of Reading in England from 1954 onward. Much like another couple featured in this collection, Leonard and Reba Gillman (see page 17), Richard Rado and his wife, Luise Zadek Rado (d. 1990), were highly accomplished musicians, he as a pianist and she as a singer, and gave both public and private concerts. (Sources: MacTutor Archive, Mathematics Genealogy Project) 

Hans-Martin Reimann earned his Ph.D. in 1969 at the Eidgenössische Technische Hochschule (ETH) in Zürich, Switzerland. If our identification is correct (based on the notation “Reimann (Swiss)” by Halmos), Reimann would have been a beginning graduate student at the time this photograph was taken. He has spent most of his career at the University of Bern, Switzerland, becoming Professor Emeritus in 2006, and lists his research interests as complex analysis, quasiconformal mappings, Lie groups, symplectic geometry, and wavelets. (Sources: Mathematics Genealogy Project, Universität Bern Mathematics)

Who’s That Mathematician? Images from the Paul R. Halmos Photograph Collection

Halmos photographed Martin Gardner (1914-2010) in New York City on Oct. 26, 1974. The magazine Scientific American paid tribute to Gardner, the “Mathematical Gamester,” as follows: “For 25 years, he wrote Scientific American’s Mathematical Games column, educating and entertaining minds and launching the careers of generations of mathematicians.” The magazine also credits Gardner with “single-handedly populariz[ing] recreational mathematics in the U.S.” Gardner wrote his first article for Scientific American in 1956 and was immediately invited to write the magazine’s “Mathematics Games” column, which he did from 1957 to 1981. The 15 books containing all of his columns are among the over 100 books and pamphlets he published during his career. (Sources: Scientific American,MacTutor Archive)

Who’s That Mathematician? Images from the Paul R. Halmos Photograph Collection

James Tanton will present the MAA Carriage House and Gathering for Gardner event “Weird Ways to Work with Pi” on December 5, 2012, at the MAA Carriage House in Washington, D.C. 

Martin Gardner thought deeply about the number pi, wrote about our attempts to come to terms with this troublesome number, and shared with the world a multitude of surprising puzzles whose solutions involve circles and use of the number pi. But who said the concept of “pi” applies only to circles? What is the value of pi for a square? What interesting non-circular problems can be solved with non-circular pi-values?

In this 2012 Celebration of Mind event at the MAA we shall explore some weird and wonderful ways to play with pi for shapes that might or might not be circles. This talk will be lively and accessible to all—students and teachers, mathematics professionals and mathematics enthusiasts alike—and chock-full of insight and gotchas! and ahas! Let’s continue to roam the exciting mathematical landscapes that Martin Gardner shared with the world and enjoy, in his honor, the jewels still to be found in them.

Halmos photographed Natalie Davis and Alfréd Rényi (1921-1970) in August of 1961.

Natalie Zemon Davis, wife of mathematician Chandler Davis, is a noted social and cultural historian, primarily of early modern France. Her best known book is The Return of Martin Guerre (1983), also the title of a popular film released at the same time. Natalie and Chandler Davis were victims of the “Red scare” in the United States during the 1950s, with Chandler Davis losing his job at the University of Michigan in 1954 and even being imprisoned for six months. They moved to Toronto, Canada, in the early 1960s, at about the time this photograph was taken. Chandler Davis is now Professor Emeritus of Mathematics at the University of Toronto (Wikipedia, University of Michigan History, University of Toronto Mathematics)

Born in Budapest, Hungary, Alfréd Rényi earned his doctoral degree in 1945 from the University of Szeged, Hungary, under advisor Frigyes (Frédéric) Riesz. According to O’Connor and Robertson of the MacTutor Archive, this was after graduating from the University of Budapest, where he studied from 1940 to 1944 under Lipót Fejér and Paul Turán, escaping from a forced-labor camp, hiding out to avoid capture, and rescuing his parents from the Budapest ghetto by impersonating a soldier. After a postdoctoral year in Russia (1946-47) during which he obtained important results on the Goldbach Conjecture, Rényi continued to obtain results in number theory, probability, and analysis as a professor at the University of Budapest and a member of the Hungarian Academy of Sciences and director of its Institute for Applied Mathematics before dying suddenly at age 48. Rényi’s wife was the mathematician Katalin (Kató) Rényi, and possibly she and/or Chandler Davis were among the assembled party as well. We will search for photographic evidence! (Source: MacTutor Archive)

Who’s That Mathematician? Images from the Paul R. Halmos Photograph Collection

Sisters Julia Robinson (1919-1985), left, and Constance Reid (1918-2010) were photographed by Halmos in July of 1984 in Eugene, Oregon. Constance Reid was the well-known author of popular books about mathematics, most notably From Zero to Infinity: What Makes Numbers Interesting (MAA, 1961), and of biographies of mathematicians, including E. T. Bell (or John Taine), Richard Courant (photographed on page 10 of this collection), David Hilbert, Jerzy Neyman (photographed on page 38 of this collection), andJulia Robinson. Another photograph of Julia Robinson appears on page 30 of the collection, where you can read more about her. (Sources: MacTutor Archive, MAA obituary: Constance Reid). 

Who’s That Mathematician? Images from the Paul R. Halmos Photograph Collection

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: 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: 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: 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