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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.

Paul Halmos photographed Donald Burkholder (1927-2013) in March of 1990 in Urbana, Illinois, quite probably in the home of Joseph and Elsie Doob. On page 1, Burkholder appears in a photo taken the same day of a group gathered on the porch of the Doob home. This group also included Colin Blyth, Elsie and Joseph Doob, Warren Ambrose, Paul Halmos, and David Blackwell. Burkholder was an emeritus mathematics professor and member of the Center for Advanced Study at the University of Illinois, Urbana-Champaign, when this article appeared. His specialties included probability theory, stochastic processes, functional analysis, and Fourier analysis. He edited *Joseph Doob: A Collection of Mathematical Articles in His Memory*, which comprised the first four issues of the *Illinois Journal of Mathematics* (and over 1000 pages!) in 2007 (WorldCat).

From left to right, mathematicians Adriaan Zaanen, Paul Butzer, Ralph Phillips, and Béla Szökefalvi-Nagy are pictured in August of 1983 at the Oberwolfach Conference Center in Germany.

Adriaan "Aad" Cornelis Zaanen (1913-2003) earned his doctorate from the University of Leiden in the Netherlands in 1938. He retired as Chair of Mathematics from Leiden in 1982 (with a retirement symposium titled "From A to Z"), but remained active in functional analysis, integration theory, and Riesz space theory. He wrote both important series of papers and influential books on these topics (MacTutor Archive).

Paul Leo Butzer has spent most of his career at Rheinisch Westfälische Technische Hochschule (RWTH) in Aachen, Germany, where he is professor emeritus of mathematics. He organized seven conferences at the Oberwolfach Conference Center from 1963 to 1983 on his own specialty of approximation theory and related fields such as functional analysis and operator theory (MacTutor Archive), which would have attracted Paul Halmos. Indeed, Butzer says Halmos first attended his 1968 conference at Oberwolfach. Butzer also is interested in signal theory, combinatorial number theory, and the history of mathematics (RWTH). (Additional source: P. L. Butzer, "A retrospective on 60 years of approximation theory and associated fields," *Journal of Approximation Theory* (2009), doi:10.1016/j.jat.2009.05.004)

Ralph S. Phillips (1913-1998) earned his Ph.D. at the University of Michigan in 1939, taught at the University of Southern California (USC) for ten years, and from 1960 onward was professor of mathematics at Stanford University, where he retired as Grimmett Chair of Mathematics in 1978. Phillips was a functional analyst and a founding editor of the *Journal of Functional Analysis,* but perhaps is best known for his work in mathematical physics with Peter Lax, now known as Lax-Phillips theory (Stanford University) or scattering theory (MacTutor Archive). He and Paul Halmos would have first met in 1939 at the Institute for Advanced Study in Princeton, where both were members during the 1939-40 academic year. (Additional source: "Ralph Phillips (1913-1998)," *AMS Notices* 47:5, May 2000)

Béla Szökefalvi-Nagy (1913-1998) was Butzer's co-organizer for the last four of the seven Oberwolfach conferences held during 1963-1983 (MacTutor Archive). A faculty member at the University of Szeged, Hungary, from 1940 onward, Szökefalvi-Nagy was "one of the founders of modern operator theory, and one of its major contributors," according to the editors of *Recent Advances in Operator Theory and Related Topics: The Béla Szökefalvi-Nagy Memorial Volume* (Birkhäuser, 2001). Butzer describes him as "the greatest mathematician from Hungary from ca. 1950 till his death." Ergodic theorist Alexandra Bellow (see next photo) remembers that when she was a student in Romania and the U.S. during the 1950s, "Bela Sz.-Nagy ... was regarded as one of the leading European mathematicians. ... [H]is book with Frederic Riesz, *Leçons d'Analyse Fonctionnelle* [1952], was ... the book from which we all learnt Functional Analysis; it became a classic." Sz.-Nagy also appears in a photo on page 14 of this collection, where you can read more about him.

Butzer reports that because Zaanen, Phillips, Sz.-Nagy, and Blagovest Sendov (University of Sofia, Bulgaria) all turned 70 in 1983, they were honored with tributes to their lives and work in the *Anniversary Volume on Approximation Theory and Functional Analysis,* P. Butzer, R. Stens (RWTH), and B. Sz.-Nagy, eds., ISNM Vol. 65, Birkhäuser, Basel, 1984.

Halmos photographed mathematicians Alberto Calderón (1920-1998) and Alexandra Bellow at the San Francisco Airport in January of 1991.

Alberto Calderón spent most of his career at the University of Chicago, where he earned his Ph.D. in 1950. He and his thesis advisor Antoni Zygmund (1900-1992) collaborated throughout their careers and are most famous for their Calderón-Zygmund theory of singular integral operators (MacTutor Archive). Calderón and Halmos probably first met in the late 1940s when Calderón arrived at the University of Chicago, where Halmos was a professor, to complete his Ph.D. work. He and Halmos were colleagues at Chicago only from 1959, when Calderón returned to Chicago from MIT, to 1961, when Halmos moved to the University of Michigan. Calderón was at Chicago 1959-72, MIT 1972-75, and Chicago 1975-85 and from 1989 onward. (Additional sources: "Alberto Pedro Calderón (1920-1998)" and "Singular Integrals: The Roles of Calderón and Zygmund," *AMS Notices* 45:9, October 1998. Much of the biographical material for the AMS volume, *Selected Papers of Alberto P. Calderón with Commentary* (2008), co-edited by Bellow, Carlos Kenig, and Paul Malliavin, is available here.)

Alexandra Bellow, who also appears in a photograph on page 5 of this collection, has spent most of her career at Northwestern University in Evanston, Illinois, where she is now emeritus. An ergodic theorist, she earned her Ph.D. at Yale University in 1959 under the direction of Shizuo Kakutani. In the photo above, Bellow and her husband Calderón were in San Francisco in January of 1991 for the Joint Mathematics Meetings, where she delivered the Association for Women in Mathematics (AWM) Emmy Noether Lecture, "Almost Everywhere Convergence: The Case for the Ergodic Viewpoint." Bellow recalls that "conversation with Paul Halmos was never dull or predictable," and offers an example from the 1991 JMM, when she said to him, "I need to ask you a question, Professor Halmos," and he replied, "The answer is yes. Now what was the question?" According to the brochure for the AWM Noether Lecture, Bellow and Calderón first met in 1974 when they shared an office at MIT. Bellow explains that she arrived at MIT as a visiting professor. Calderón, then a professor at MIT, "had a magnificently large office, in keeping with his mathematical stature. There was a shortage of office space in the Math Department and the Chairman asked Alberto if he would mind sharing his office with a visitor. Alberto, always a gentleman, agreed." They were married in 1989.

Halmos photographed John "Ian" Cassels (1922-2015) and his wife Constance Cassels (d. 2000) at Macquarie University in Sydney, Australia, in July of 1989. J. W. S. Cassels retired from Cambridge University, England, as Sadleirian Professor of Pure Mathematics and Head of Department in 1984, but remained active in number theory. He began his career with research on the geometry of numbers and Diophantine approximation and added new topics and interests over the years. J. J. O'Connor and E. F. Robertson reported that "Cassels has worked on every aspect of the theory of numbers, particularly on the theory of rational quadratic forms and local fields," and that he received both the Sylvester and DeMorgan medals for this work (MacTutor Archive).

Halmos photographed Shiing-Shen Chern (1911-2004) in 1959. The famous differential geometer spent most of his career in the United States at the University of Chicago during the 1950s, and from 1960 onward at the University of California, Berkeley. He received the National Medal of Science in 1975, and was the first director of the Mathematical Sciences Research Institute (MSRI) in Berkeley from 1981 to 1984. Chern first visited the Institute for Advanced Study in Princeton during 1943-45, where he developed what would become known as Chern characteristic classes and "gave a now famous proof of the Gauss-Bonet formula" (MacTutor Archive). Chern and Halmos may have first met at the University of Chicago, where they were colleagues throughout the 1950s. Chern was a faculty member at Chicago from 1949 to 1960 and Halmos from 1946 to 1961. (Additional sources: "Interview with Shiing Shen Chern," *AMS Notices* 45:7, August 1998; "Shiing-Shen Chern (1911-2004)," *AMS Notices* 58:9, October 2011)

Halmos photographed his Ph.D. student Lewis Coburn in 1969 at Yeshiva University in New York City, where Coburn was a faculty member. Coburn remembers that his department "had some extra cash" and used it to invite three senior faculty members, including Halmos, to visit for two weeks each. He adds, "I don't remember the event at which this picture was taken, but it must have been something special because I was wearing a tie." Coburn earned his Ph.D. from the University of Michigan in 1964 with a dissertation titled, "Function Algebras and Hilbert Spaces" (Mathematics Genealogy Project). He is professor of mathematics at the State University of New York (SUNY) at Buffalo, where he is a functional analyst working in operator theory, C*-algebras, and quantum mechanics (SUNY Buffalo).

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.

Janet Beery (University of Redlands) and Carol Mead (Archives of American Mathematics, University of Texas, Austin), "Who's That Mathematician? Paul R. Halmos Collection - Page 9," *Convergence* (January 2012), DOI:10.4169/loci003801