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 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

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.

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: Amir Moez, 1967

“Ali has been a linear algebra enthusiast for much of his professional life.” — Paul R. Halmos, I Have a Photographic Memory

Ali Reza Amir-Moez earned his BA at the University of Teheran in 1942, and served as a Math Instructor at Teheran Technical College from 1942 - 46.

He immigrated to the United States in 1947. His first love was drama and the performing arts, however, he was forced to study math to receive an extension on his visa, and thus he continued his education earning his MA in 1951, and PhD in 1955, both from UCLA. He served as a Professor of Math at the University of Idaho; Queens College, New York City; Purdue University; University of Florida, Gainesville; Clarkson College, Potsdam, New York; and Texas Tech University, Lubbock.

Amir-Moez was dedicated to mathematics research and established scholarships at both Texas Tech University and the University of California at Los Angeles. In 1975, he was awarded the medal of Pro Mundi Beneficio, Academia Brasileira De Ciencias Humanas.

He was the author of books including, Elements of Linear Space; Extreme Properties of Linear Transformations and Geometry in a Unitary Space; Classes Residues et Figure ance Ficelli; and plays including Kaleeleh and Demneh and Three Persian Tales. His writings included over 150 papers, articles, and books, and he was often featured in Highlights for Children. a

Ali Reza Amir-Moez Obituary (Texas Tech University, August 25, 2007)

Dr. Ali Reza Amir-Moez (Lubbock Online, August 27, 2007)

Photo Caption: Budapest 1931

“My family lived in a third floor apartment, in Budapest, that faced out on a busy street (now called Lenin Boulevard). It was an exciting street—colorful, crowded, noisy. There were many shops—a glamorous hardware store displaying shiny knives behind its huge plate glass front, several bookstores with books of many colors piled and strewn around, coffee houses with grouchily servile waiters carrying white napkins on their black left sleeves, and stores full of toys and candy and crutches and clothes and shoes and watches. The sidewalk was broad, and milling, crowds of people separated the shop windows from teh curb-side trees and scales (your weight for a penny) and newspaper kiosks and taxi stands. The crowds seemed always to be there—they were there when went to school early in the morning and they were there on the rare occasions when I was brought home late at night from an excursion or from a movie. Later, when I grew up, went to Hungary as an American tourist, and was out real late at night, the crowds were still there. The lights were bright and gypsy music could be heard from the coffee houses.” —Paul R. Halmos, I Want to Be a Mathematician…

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