For more information about Paul R. Halmos (1916-2006) and about the Paul R. Halmos Photograph Collection, please see the introduction to this article on page 1. A new page featuring six photographs will be posted at the start of each week during 2012.
Index to the Collection
Halmos photographed, from left to right, Stephen Smale, William Browder, and Franklin Peterson on August 27, 1968, at the Joint Summer Mathematics Meetings in Madison, Wisconsin. Halmos affixed his Honolulu, Hawaii, address label to the back of the photo, indicating that he had already moved to Hawaii in preparation for his 1968-69 post as chair of the University of Hawaii Mathematics Department. Another photograph of William Browder appears on page 8 of this collection, where you can read more about him.
Stephen Smale is best known for having won the Fields Medal in 1966 for his work in differential geometry -- more specifically, for having proved in 1961 the higher dimensional Poincaré Conjecture for n=5 and up -- and for having completed at least part of this work on the beaches of Rio de Janeiro. Smale earned his Ph.D. in 1957 from the University of Michigan with the dissertation “Regular Curves on Riemannian Manifolds,” written under advisor Raoul Bott. (For photographs of Bott, see page 8 and page 10 of this collection.) He spent 1956-58 as an instructor at the University of Chicago, where he probably first met Halmos, who was on the faculty there from 1946 to 1961. He spent the first 18 months of a two-year 1958-60 NSF Fellowship at the Institute for Advanced Study in Princeton, New Jersey, and the last six months at the Instituto de Mathematica Pura e Aplicada in Rio de Janeiro, Brazil. Smale has spent most of his career at the University of California, Berkeley, where he became Professor Emeritus of Mathematics in 1995. He now is based at the City University of Hong Kong. (Sources: MacTutor Archive, University of California, Berkeley)
Franklin Peterson (1930-2000) earned his Ph.D. in 1955 from Princeton University with the dissertation “Generalized Cohomotopy Groups,” written under advisor Norman Steenrod. He remained on the Princeton faculty until 1958, when he moved to the Massachusetts Institute of Technology (MIT). He was an algebraic topologist who specialized in cohomology operations, and he advised 23 Ph.D. students at MIT. Peterson also served the American Mathematical Society (AMS) as treasurer for 25 years, from 1974 to 1998. (Sources: Mathematics Genealogy Project, MITnews; see also “Franklin P. Peterson (1930-2000),” AMS Notices 48:10 (November 2001), pp. 1161-1168)
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)
Halmos photographed Robert Phelps, left, and Andrew Bruckner in June of 1985 in Santa Barbara, California. The event was a conference in honor of Ky Fan on his retirement from UCSB. Halmos spent 1975-1977 at UCSB and would have first met both Fan and Bruckner then, if not earlier.
Bob Phelps (1926-2013) earned his Ph.D. in 1958 from the University of Washington with the dissertation “Subreflexive Normed Linear Spaces,” written under advisor Victor Klee. He then spent two years at the Institute for Advanced Study in Princeton, New Jersey, and two years at the University of California, Berkeley. In 1960, he and Errett Bishop, a Ph.D. student of Halmos whose photograph appears on page 6 of this collection, proved what quickly became known as the Bishop-Phelps Theorem. A survey article on this result is available from Phelps’ website. In 1962, Phelps joined the faculty at the University of Washington, where he continued to carry out research in functional analysis. At Washington, he was an active member of the John Rainwater Functional Analysis Seminar and was the primary author of Rainwater’s work; see the Topology Atlas link below for details. (Sources: Mathematics Genealogy Project, University of Washington Mathematics: Robert Phelps (including "The Bishop-Phelps Theorem"), Topology Atlas, Seattle Times obituary)
Andy Bruckner earned his Ph.D. in 1959 from the University of California, Los Angeles, with the dissertation “Minimal Superadditive Extensions of Superadditive Functions,” written under advisor John W. Green. (For a photograph of Green, see page 18 of this collection.) Bruckner joined the faculty at the University of California, Santa Barbara, in 1959. He has spent his career at UCSB, becoming Professor Emeritus of Mathematics in 1994. He has written both monographs and textbooks on real analysis, the latter with co-authors Judith Bruckner and Brian Thomson. (Sources: Mathematics Genealogy Project, UCSB Mathematics, Classical Real Analysis)
Bruckner wrote recently (Oct. 2012) of Halmos:
Seeing the pictures brought back all sorts of fond memories of Halmos' two years in Santa Barbara. One of many such memories illustrated his desire to be precise. Our department used to have "guess your own speed races." We ran around the UCSB track, without a watch, predicting our times just before the race began. Halmos walked, but did so in the fifth lane of the track so that he wouldn't get in the way of the runners when they lapped him. (He walked very fast with his cane, about 12 minutes per mile.) During one of our colloquium lectures I sat next to him and wondered about the notes he was taking of the lecture. So I took a look. There I saw a picture of the track and some scribbling that calculated just how much further he had to walk per lap in the 5th lane at the next day's race. I had told Paul that each lap out from the inside lane adds seven and a third yards, so if he walks at 12 minutes per mile in the 5th lane he would have to take 12 seconds longer each lap. He didn't believe me, I guess, and had to check it out for himself. I resisted the temptation to tell him that, to be precise, he had to walk exactly 12 inches from the line separating the 4th and 5th lanes.
Ralph S. Phillips (1913-1998) was photographed by Halmos on February 16, 1970, at Indiana University in Bloomington. At the time, Phillips was a functional analyst at Stanford University. Another photograph of Phillips appears on page 9 of this collection, where you can read more about him.
Halmos photographed George Pólya (1887-1985) and Alexander Ostrowski (1893-1986) at the International Congress of Mathematicians in Edinburgh, Scotland, 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 by Adolf 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)
Halmos photographed Evgenii Mishchenko (1922-2010), left, and Lev Pontryagin (1908-1988) at the International Congress of Mathematicians (ICM) in Edinburgh, Scotland, in 1958, where Pontryagin was one of the main speakers.
Evgenii Mishchenko earned his Ph.D. in general topology in 1953 from Moscow State University under renowned topologists Pavel Aleksandrov and Lev Pontryagin. In 1961, Mishchenko co-authored The Mathematical Theory of Optimal Processes with Pontryagin and two other students of Pontryagin, V. G. Boltyanskii and R. V. Gamkrelidze. (For a photo of Gamkrelidze, see page 16 of this collection.) In 1952, Pontryagin had shifted his research program from topology to differential equations and control theory, and Mishchenko seems to have followed suit. He has been on the faculty of Moscow State University and of the Moscow Institute of Physics and Technology. (Sources: MacTutor Archive: Pontryagin, Mathematics Genealogy Project, Steklov Mathematical Institute)
Lev Semenovich Pontryagin (or, as Halmos spelled it, Pontrjagin) was blinded by an explosion at age 14, and learned mathematics primarily by listening to his mother, Tat’yana Andreevna Pontryagina, a seamstress, read mathematical books and papers to him. In 1925, he entered the University of Moscow and soon was working on topology with Pavel Aleksandrov. In 1929, he earned the equivalent of the Ph.D. and joined the Faculty of Mechanics and Mathematics of the University of Moscow. In 1934 he was appointed to the prestigious Steklov Mathematical Institute, where just one year later he became head of the Department of Topology and Functional Analysis. According to O’Connor and Robertson of the MacTutor Archive:
By 1927, although he was still only 19 years old, Pontryagin had begun to produce important results on the [James] Alexander duality theorem. His main tool was to use link numbers which had been introduced by [L. E. J.] Brouwer and, by 1932, he had produced the most significant of these duality results when he proved the duality between the homology groups of bounded closed sets in Euclidean space and the homology groups in the complement of the space.
This work led to his development of a theory of characters for commutative topological groups and his proof of Hilbert’s Fifth Problem for Abelian groups, both by 1934. He described all of this work in his 1938 book, Topological Groups, which was to become an enduring classic of topological algebra.
In his I Want to Be a Mathematician: An Automathography (Springer 1985), Halmos described Pontryagin’s Topological Groups, which he read shortly after it was translated to English by Emma Lehmer (whose photograph appears on page 30 of this collection) in 1939, as “an eye opener, a revelation, a thriller,” which he read “almost as I would read a detective story, to find out whodunit.” The “it” of the “whodunit” was the “big duality theorem” and its implications and applications (p. 93). (Sources: MacTutor Archive: Pontryagin, Steklov Mathematical Institute)
For an introduction to this article and to the Paul R. Halmos Photograph Collection, please see page 1. Watch for a new page featuring six new photographs each week during 2012.
Regarding sources for this page: Information for which a source is not given either appeared on the reverse side of the photograph or was obtained from various sources during 2011-12 by archivist Carol Mead of the Archives of American Mathematics, Dolph Briscoe Center for American History, University of Texas, Austin.
Index to the Collection