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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.

Our first photograph this week is of Armand Borel (1923-2003) and Raoul Bott (1923-2005) in 1958, possibly at the International Congress of Mathematicians held that year in Edinburgh, Scotland.

Armand Borel is best known for his work on Lie groups, beginning with his Ph.D. dissertation, defended at University of Paris in 1952. Largely through his influence, Lie groups and algebras would become important in many areas of mathematics and would help connect many areas of mathematics. A member of the younger generation of the Bourbaki, he was the main contributor to the Bourbaki’s book on Lie groups and algebras. Borel’s work on Lie groups led him to work on algebraic groups and arithmetic groups as well. Born in Switzerland, Borel spent most of his career at the Institute for Advanced Study (IAS) in Princeton, where he became a permanent professor in 1957. He had been a student at ETH = Eidgenössische Technische Hochschule Zürich (German) = École Polytechnique Fédérale de Zurich (French) = Swiss Federal Institute of Technology (English) between 1942 and 1947 (including military service), then an instructor there for two years, before spending a formative year in Paris (1949-50), learning from and working with the group of mathematicians known as the Bourbaki. He would return to ETH as a professor during 1955-57 and 1983-86. (Sources: MacTutor Archive and IAS)

Another photograph of Raoul Bott appears below; a review of his career accompanies that photo.

Paul Halmos photographed Karol Borsuk (1905-1982) in the spring of 1968 at Indiana University in Bloomington. (Halmos would accept a professorship there one year later.) Borsuk earned his Ph.D. at the University of Warsaw in 1931 for a dissertation in topology “in which he invented the theory of retracts” (MacTutor Archive). His other great contribution to topology would be his theory of shape. At the time he earned his Ph.D., Borsuk had already begun teaching at the University of Warsaw. The Mathematics Genealogy Project shows that he advised at least eight Ph.D.s there through 1936, with the last one earned by Samuel Eilenberg for his dissertation “On the Topological Applications of Maps onto a Circle” with Borsuk as second advisor to Kazimierz Kuratowski. After attempting to help take the university “underground” during World War II (MacTutor Archive), Borsuk and Kuratowski set about rebuilding the mathematics program after the war ended in 1945. The Mathematics Genealogy Project shows Borsuk advising two Ph.D.s in 1958, one each at the Polish Academy of Science's Institute of Mathematics and Moscow State University; and then, beginning in 1965, six more at the University of Warsaw, including the one Krystyna Kuperberg began there in 1966 and finally completed at Rice University in Houston, Texas, in 1974.

Halmos photographed Raoul Bott (1923-2005) in April of 1980 in Bloomington, Indiana, where Halmos was a professor at Indiana University. Bott was an engineer-turned-mathematician who produced many important results in topology and differential geometry throughout his career and who advised 24 Ph.D. students, including two Fields medalists. In fact, his first Ph.D. student was future Fields Medalist Stephen Smale, who earned his degree from the University of Michigan in 1957 with the thesis “Regular Curves on Riemannian Manifolds.” Bott spent most of his career at Harvard, where one of his early Ph.D. students (1964) was future Fields Medalist Daniel Quillen. Bott’s own Ph.D. was from Carnegie-Mellon University, earned in 1949 for his dissertation on “Electrical Network Theory,” reflecting his interest up to that time in electrical engineering. (Sources: MacTutor Archive, Mathematics Genealogy Project)

William Browder was photographed by Halmos in April of 1990 at an AMS Meeting in Albuquerque, New Mexico. The well-known topologist is most famous for his invention of surgery on manifolds. He has advised at least 32 Ph.D. students, three at Cornell and the rest at Princeton, where he has spent most of his career. His own thesis, “Homology of Loop Spaces,” was written at Princeton in 1958. He became a faculty member at Princeton immediately following his first Institute for Advanced Study (in Princeton) appointment in 1963-64 and remains a professor of mathematics there today. He served as AMS president in 1989-90. Browder and Halmos may have first met when Browder spent part of the academic year 1959-60 at the University of Chicago, where Halmos was a faculty member. (Sources: MacTutor Archive, Mathematics Genealogy Project, AMS Presidents, IAS, Princeton University Mathematics Department)

Halmos photographed H. Arlen Brown (Herbert Arlen Brown), in 1952, the year Brown earned his Ph.D. at the University of Chicago under advisors Paul Halmos and Irving Kaplansky for the dissertation “Two Classes of Non-Normal Operators.” The photograph, however, was taken in East Lansing, Michigan, perhaps at Michigan State University. Brown was one of at least nine Ph.D. students Halmos advised during his tenure at University of Chicago (1946-61). The Mathematics Genealogy Project shows Brown advising Ph.D. theses at Rice University in 1957 and 1960, at University of Michigan in 1965 (he was on the faculty there from 1963 to 1967), and at Indiana University in 1969. Halmos began his 1969-1985 professorship at Indiana University in 1969, following his 1968-69 academic year at University of Hawaii. Brown was active in operator theory research through the 1970s, including two joint papers with Halmos during the early 1960s and many joint papers with one of his own Ph.D. students, Carl Pearcy, during the 1960s and 1970s. He and Pearcy co-authored *An Introduction to Analysis,* in the Springer-Verlag Graduate Texts in Mathematics series in 1995. Pearcy is emeritus professor of mathematics at both the University of Michigan and Texas A & M University, and has advised at least 31 Ph.D. students. (Sources: Brown: Mathematics Genealogy Project, MathSciNet, University of Michigan Faculty History Project; Pearcy: Mathematics Genealogy Project, University of Michigan Department of Mathematics, Texas A & M University Department of Mathematics)

Halmos photographed R. Creighton Buck (1920-1998) in July of 1984 at Carleton College in Northfield, Minnesota. Buck earned his Ph.D. from Harvard in 1948 with Ralph P. Boas, Jr., pictured on page 7, as his second advisor. Buck taught first at Brown, and then, beginning in 1950, enjoyed a long and successful career at the University of Wisconsin. He worked in approximation theory, complex analysis, operations research, topological algebra, and the history of mathematics. In fact, many know of Buck either through his historical article, “Sherlock Holmes in Babylon,” in *The* *American Mathematical Monthly* in 1980 and *Cryptologia* in 1981, or through his textbook, *Advanced Calculus* (McGraw-Hill, 1956, 1965, 1978). He also served both the AMS and MAA in various roles. (Sources: University of Wisconsin Department of Mathematics, Mathematics Genealogy Project, MathSciNet)

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.

Janet Beery (University of Redlands) and Carol Mead (Archives of American Mathematics, University of Texas, Austin), "Who's That Mathematician? Images from the Paul R. Halmos Photograph Collection," *Loci* (March 2012)