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

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

In late August, SciFri writer Annette Heist sent out a call for photographs of women in mathematics. The article, titled “Picture Another Mathematician”, featured two photos from the Halmos Collection of Olga Taussky-Todd (pictured) and Mary Ellen Rudin.

Heist wrote:

Laura McHugh of the Mathematical Association of America wrote to tell me about mathematician and photographer Paul Halmos. Throughout his career, Halmos snapped thousands of photos of his fellow mathematicians. After his death, Halmos’s wife donated the photos to the University of Texas’s Archives of American Mathematics. The photos are in the process of being digitized and made available online according to archivist Carol Mead, who sent the photos below.

The photo shows Lloyd Lininger (left) and Sir Michael Atiyah (right) in Ann Arbor, Michigan on April 3, 1968.  Halmos was a faculty member at the University of Michigan from 1961 through 1968.  Atiyah had won the Fields Medal in 1966 and published his book K-theory, which included discussion of the Atiyah-Singer Index Theorem, in 1967 (MacTutor History of Mathematics Archive).  He was knighted in 1983.  According to the Mathematics Genealogy Project, Lininger had earned his Ph.D. in 1964 at the University of Iowa with the dissertation “Some Results on Crumpled Cubes” under Steve Armentrout, whose photograph appears on page 1 of this collection.  Ken Millett (University of California, Santa Barbara) suggests that the young man in the background facing the camera may be topologist William Thurston, who would have been in his first year of graduate school at the University of California, Berkeley.  Bill Thurston would win the Fields Medal in 1982.

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

Photo Caption: Emma and Dick Lehmer, May 1986 

“Emma Lehmer: A collaborator of Dick Lehmer’s in more senses than one, she is also a known and respected mathematical translator; we must be grateful to her, in particular, for translating Pontrjagin’s Topological groups.” —Paul R. Halmos, I Have a Photographic Memory

Emma Lehmer wrote around 60 papers on different aspects of number theory, about 20 of these being joint publications with her husband, Derrick Henry Lehmer. Their initial collaboration began as a three-way one with Lehmer working with both her husband and her father-in-law, Derrick Norman Lehmer. Her collaborations were both over deep mathematical results, as well as developing computers and computational methods to assist in solving number theory problems.

Emma Lehmer 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