Photos by Paul Halmos

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: 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

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: Mary Cartwright 3 June 68 Cambridge

“She became Mistress of Girton College (Cambridge), and, later, Dame Mary; she is just an outstanding complex analyst.” —Paul R. Halmos, I Have a Photographic Memory

Dame Mary Cartwright (1900-1998)
Mary Lucy Cartwright was the first woman mathematician elected to the Royal Society of London. While at Cambridge University, under the supervision of G.H. Hardy and E.C. Titschmarsh, her thesis on zeros of integral functions generated a series of papers and eventually led to her book on integral functions. Although she did important work with Dirichlet series, Abel summation, analytic functions regular on the unit circle, integral functions, and cluster sets, she is best known for her work with Littlewood on van der Pol’s equation and nonlinear oscillators. Cartwright served as Mistress of Girton College and as president of the British Mathematical Association and the London Mathematical Society. She was a recipient of the Sylvester Medal from the Royal Society and the De Morgan Medal from the London Mathematical Society. She authored nearly 100 articles and books. She was a very effective administrator at Cambridge University and ambassador for several mathematical and scientific organizations. In 1969, Queen Elizabeth II elevated her to Dame Mary Cartwright, the female equivalent of a knighthood.

Mary Cartwright

Photo Caption: McLaughlin, Ann Arbor, 1963 

“Jack is a hard-working and broadly informed algebraist, but the sort of algebra he writes papers about is not the sort I am fond of. Example: he discovered one of the notorious sporadic simple groups. I am a hard-working and quite knowledgeable operator-theorist, and the kind of operator theories that interest me leave Jack completely cold. It turned out, however, that there is a part of mathematics we both know and like. It is a small subject, not considered deep. It has, they say, no intrinsic importance; it is merely a useful tool and an occasional source of examples in other subjects. Both Jack and I tend to be shamefaced about admitting we like it; it is a little like admitting that you read westerns. The subject I am talking about is linear algebra.” —Paul R. Halmos, I Want to Be a Mathematician. 

Jack McLaughlin’s research ranged widely, encompassing several subfields of algebra—lattice theory, finite groups, and commutative algebra. He discovered one of the sporadic finite simple groups, that of order 898,128,000, which now bears his name. He also participated in the discovery of a module of finite projective dimension with a negative intersection multiplicity. His work on group cohomology, most of which passed on through the writings of his students, has had an important impact on the field. He was well respected by his colleagues in the field. Paul Halmos, his UM colleague, once said that there are a number of ways to tackle a mathematical problem, but when all else fails, ask McLaughlin.

“Jack McLaughlin (1923 – 2001)”, ContinuUM, The Newsletter of the Department of Mathematics at the University of Michigan (pdf)  

Photo Caption: 1931 

Happy Birthday, Halmos! (March 3, 1916)

Math DL: On This Day… 

Important events from the history of math

Photo Caption: Anneli Lax 

“Anneli is an outstanding mathematical editor (look at any volume of the New Mathematical Library).” —Paul R. Halmos, I Have a Photographic Memory

Anneli Lax Cahn’s greatest contribution to mathematical literature was triggered by a very different sort of event. The launch of the Soviet satellite Sputnik in 1957 was a shock to the American scientific community, a shock felt at every level. Much thought was devoted to the education of a new generation who would accelerate the pace of American scientific productivity. It was at this point that Lax realized the major contribution that could be made in mathematics education. Out of this endeavor grew the New Mathematical Library. The idea was to make accessible to interested high school students deep results in mathematics described by research mathematicians.

Anneli Lax Cahn