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 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 is an emeritus mathematics professor and member of the Center for Advanced Study at the University of Illinois, Urbana-Champaign. His specialties include 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) retired from the University of Delft in the Netherlands 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 fields related to it, such as functional analysis and operator theory (MacTutor Archive), which would have attracted Paul Halmos. He also is interested in signal theory, combinatorial number theory, and the history of mathematics (RWTH).

Ralph S. Phillips (1913-1998) spent most of his career at Stanford University in Palo Alto, California. He earned his Ph.D. at the University of Michigan in 1939, and went on to advise 21 Ph.D. students of his own, the first four during a ten-year stint at the University of Southern California (USC), and the rest at Stanford University, where he was professor of mathematics from 1960 onward and retired as Grimmett Chair of Mathematics in 1978 (Mathematics Genealogy Project, Stanford University). Phillips was a functional analyst and a founding editor of the *Journal of Functional Analysis,* but perhaps is best known for work in mathematical physics done with Peter Lax and now known as Lax-Phillips theory (Stanford University). 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). According to the *Great Soviet Encyclopedia* (third edition, 1979), Szökefalvi-Nagy "graduated from the University of Szeged, Hungary, in 1936 and became a professor there in 1948. ... [His] principal works deal with the approximation of functions and with functional analysis---in particular, the expansion of operators in [Hilbert] space" (The Free Dictionary). Indeed, these are the topics of his papers and books listed in MathSciNet, along with articles on the lives and work of Hungarian mathematicians Frigyes Riesz, John von Neumann, and Alfred Haar. The Mathematics Genealogy Project shows Szökefalvi-Nagy advising two Ph.D. students at the Technical University of Budapest in 1956 and 1957. Additional information about the life and work of this Hungarian mathematician would be much appreciated.

Halmos photographed mathematicians Alberto Calderón (1920-1998) and Alexandra Bellow (whom Halmos identified with her nickname and first name "Gugu Alexandra" as well as with the two last names "Bellow Calderón," rather than his usual last name only) at San Francisco Airport (SFO) in January of 1990.

Alberto Calderón (1920-1998) 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 may have 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 again from 1959 to 1961, with Halmos moving to the University of Michigan in 1961. (Additional sources: "Alberto Pedro Calderón (1920-1998)" and "Singular Integrals: The Roles of Calderón and Zygmund," *AMS Notices* 45:9, October 1998)

Alexandra Bellow 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. She delivered the Association for Women in Mathematics (AWM) Emmy Noether Lecture at the 1991 Joint Mathematics Meetings in San Francisco. She and Calderón first met when they shared an office as visitors to the M.I.T. Mathematics Department (AWM). Bellow also appears in a photograph on page 5 of this collection.

Halmos photographed John "Ian" Cassels and his wife Constance Cassels 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 remains 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 report 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) either in 1959 or 1966. The famous differential geometer spent most of his career in the United States at the University of Chicago during the 1950s, and then from 1960 onward at the University of California, Berkeley. He was the first director of the Mathematical Sciences Research Institute (MSRI) in Berkeley from 1981 to 1984. Chern visited the Institute for Advanced Study in Princeton during 1943-45, when he worked on what would become known as Chern characteristic classes and "gave a now famous proof of the Gauss-Bonet formula" (MacTutor Archive). He also visited IAS during 1945-46, 1954-55, and 1964-65 (IAS) and he received the National Medal of Science in 1975 (MacTutor Archive). Chern and Halmos may have first met at the University of Chicago, where they were colleagues throughout the decade of 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. 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 interested 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 for American Mathematics, Dolph Briscoe Center for American History, University of Texas, Austin.