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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.

Halmos photographed Phillip Jones, Bartel van der Waerden, and Theophil Hildebrandt on April 2, 1968, in Ann Arbor, Michigan. That spring, Halmos was still a faculty member at the University of Michigan in Ann Arbor, but he would move to the University of Hawaii in Honolulu for the 1968-69 academic year and then to Indiana University in Bloomington in the fall of 1969.

Phillip S. Jones (1912-2002) earned his Ph.D. in 1948 from the University of Michigan, where he had earned bachelors and masters degrees in mathematics ten years earlier, with a dissertation on the history of geometry and linear perspective written under the mathematics historian Louis Karpinski. He became a faculty member at Michigan in 1947 and remained there for the rest of his career, specializing in mathematics history and education. He was a national leader in both of his specialties and was perhaps best-known for combining the two: using mathematics history as a mathematics teaching tool and writing the history of mathematics education in the U.S. (Source: Phillip S. Jones (1912-2002) (pdf file), *History and Pedagogy Newsletter* 64, March 2007, 1-4)

Bartel van der Waerden (1903-1996) earned his Ph.D. in 1926 from the University of Amsterdam with the dissertation, “The algebraic foundations of the geometry of numbers,” after studying also at the University of Göttingen, Germany, with Emmy Noether (algebra) and Hellmuth Kneser (topology). After studying for a semester with Emil Artin at the University of Hamburg, van der Waerden began writing his most famous book, *Moderne Algebra,* basing Volume I (1930) on work of Noether and Artin and Volume II (1931) on his own work in algebra. He was professor of mathematics at the University of Leipzig from 1931 through the end of World War II in 1945 and at the University of Zürich, Switzerland, from 1951 onward. Although he was interested in mathematics history throughout his career, he published most of his work in this field later in his career. (Source: MacTutor Archive)

Theophil H. Hildebrandt (1888-1980) earned his Ph.D. in 1910 from the University of Chicago under advisor E. H. Moore. He joined the mathematics faculty at the University of Michigan in 1909 and spent his career there, specializing in functional analysis and integration theory. Hildebrandt is best known for giving the first general proof of the principle of uniform boundedness for Banach spaces and for serving as president of the American Mathematical Society during 1945-1946. (Source: Mathematics Genealogy Project, MacTutor Archive: Moore, AMS Presidents)

Halmos photographed Kirsti Ore Hille and Carl Einar Hille (1894-1980) on October 31, 1965, in their home in Irvine, California. Einar Hille had started out as a classical analyst but, long before 1965, had shifted to the same general field as Halmos, functional analysis. Born in the U.S. and raised in Stockholm, Sweden, Hille studied under Ivar Bendixson, Helge von Koch, and Marcel Riesz (and was influenced by Gösta Mittag-Leffler) at the University of Stockholm, earning his Ph.D. in 1918 for the dissertation “Some Problems Concerning Spherical Harmonics.” He moved to the United States in 1920, working with G. D. Birkhoff at Harvard for two years, then moved to Princeton, and then in 1933 moved to Yale. He spent most of his career at Yale, retiring in 1962, and then became a founding faculty member of the University of California, Irvine, which opened in October of 1965. About the occasion of the photograph, the Hilles' son Harald speculates, "The University was probably celebrating its recent opening and various people were invited, including perhaps Halmos." Kirsti Hille, who married Einar Hille in 1937, was the sister of Yale mathematician Oystein Ore. Einar Hille served as AMS president in 1947-48. (Sources: MacTutor Archive, AMS Presidents, UC Irvine)

Halmos photographed topologist Peter Hilton (1923-2010) in 1958. This was the year Hilton became professor of mathematics at Birmingham University. Four years later he moved to the U.S. to join the mathematics faculty at Cornell University in Ithaca, New York, and eventually ended up (in 1982) at the State University of New York (SUNY), Binghamton, where he became emeritus in 1993. As J. J. O'Connor and E. F. Robertson point out in their MacTutor Archive biography of him, “Hilton has made a remarkable contribution both as a research mathematician and as an author of textbooks.” They also point out that Hilton met the mathematicians who would influence him most as a topologist while doing wartime decoding work at Bletchley Park; namely, Henry Whitehead, who became his Ph.D. thesis advisor at Oxford University (Ph.D. awarded in 1950, with another earned at Cambridge in 1952); Max Newman, who employed him as a lecturer at the University of Manchester (1948-1952, 1955-1958); and Shaun Wylie, with whom he worked at Cambridge University from 1952 to 1955 and wrote perhaps his best-known book, *Homology Theory: An Introduction to Algebraic Topology,* published in 1960. (Source: MacTutor Archive)

Group theorist Kurt Hirsch (1906-1986) was photographed by Halmos in 1984. Hirsch earned a Ph.D. in 1933 from the University of Berlin with a dissertation on the philosophy of mathematics and another in 1937 from Cambridge University with the dissertation “A Class of Infinite Soluble Groups,” written under advisor Philip Hall. After stints at the University of Leicester and at King’s College, Newcastle, in 1948 he joined the mathematics faculty at Queen Mary College, University of London, where he spent the rest of his career. (Source: MacTutor Archive)

Halmos photographed analyst Isidore Hirschman (1922-1990) in June of 1960. Hirschman earned his Ph.D. in 1947 from Harvard with the dissertation “Some Representation and Inversion Problems for the Laplace Transform,” written under David Widder. After writing ten papers together, Hirschman and Widder published the book *The Convolution Transform* in 1955 (Princeton University Press; now available from Dover Publications). Hirschman spent most of his career (1949-1978) at Washington University in St. Louis, Missouri, where he published mainly in harmonic analysis and operator theory. Washington University hosts the I. I. Hirschman Lecture Series, inaugurated in February of 2007 with a lecture by Richard Askey (photographed on page 2 of this collection). When Askey was a senior at Washington University, Hirschman gave him a problem on ultraspherical polynomials. Askey recalls, "This led to a joint paper, and was what started my interest in special functions." (Sources: Mathematics Genealogy Project, Washington University Mathematics Department, MathSciNet, MacTutor Archive: Askey)

Topologist Friedrich Hirzebruch (1927-2012) was photographed in July of 1984 at the Edinburgh Mathematical Society Colloquium in St. Andrews, Scotland. At the time, Hirzebruch was professor of mathematics at the University of Bonn, Germany, where he spent most of his career (1956-1993). A doctoral student of Heinrich Behnke at the University of Münster, where he earned his Ph.D. in 1950, and of Heinz Hopf at ETH in Zürich, Hirzebruch is perhaps best known for his formulation and proof of the Riemann-Roch Theorem for algebraic varieties in 1954 and, more generally, for “outstanding work combining topology, algebraic and differential geometry, and algebraic number theory” (from his 1988 Wolf Prize citation, quoted in the MacTutor Archive biography of Hirzebruch. This biography was our source for this caption. See also “Friedrich Hirzebruch (1927-2012)” at the AMS website, an article identifying Hirzebruch as “the leading German mathematician of the second half of the twentieth century.”)

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 23," *Convergence* (January 2012), DOI:10.4169/loci003801