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.

Index to the Collection

Halmos photographed operator theorist Ronald G. Douglas in 1966, possibly at the University of Michigan in Ann Arbor, where both were mathematics professors at the time. Douglas earned his Ph.D. at Louisiana State University in 1962 with the dissertation "Structure of L(p) Spaces." He has advised at least 24 Ph.D. students, most of them at the State University of New York, Stony Brook (Mathematics Genealogy Project), where he was a professor and administrator from 1969 to 1996. He is now Distinguished Professor of Mathematics at Texas A & M University in College Station, where he has held both administrative and faculty positions since 1996 (Texas A & M University).

MIT professors and probabilists Daniel Ray (d. 1979) and Richard Dudley were photographed in 1971. Ray earned his Ph.D. in 1953 from Cornell University under probabilist Mark Kac. He was professor of mathematics at the Massachusetts Institute of Technology from 1957 to 1979 (MIT). Dudley points out that the Ray-Knight theorems of probability were named for Ray (and for Frank Knight) and that, in addition to papers on probability, Ray published three papers on torsion on manifolds with his MIT colleague I. M. Singer. He reports that Ray died "suddenly and unexpectedly" in 1979. Dudley earned his Ph.D. in 1962 from Princeton under Gilbert Hunt and Edward Nelson (pictured on page 15 of this collection). He joined the MIT faculty in 1967, conducts research in both probability and statistics, has advised 31 Ph.D. students, and has published at least 120 books and papers (MathSciNet, MIT).

Halmos photographed Underwood Dudley in 1979 at Wabash College in Crawfordsville, Indiana, during a Wabash Seminar. Dudley earned his Ph.D. in 1965 from the University of Michigan under number theorist William LeVeque. He spent most of his career, from 1967 to 2004, at DePauw University in Greencastle, Indiana, where he is now emeritus. He has written a number theory textbook and several popular mathematics books (amazon), many of them published by the MAA (MAA Bookstore).

Bill Swift and his colleages at Wabash College began hosting the monthly Wabash Functional Analysis Seminar during the late 1960s, at the instigation of Halmos and other mathematicians from Indiana University, Purdue University, and the University of Illinois at Urbana-Champaign. Swift retired in 1990, but the Wabash Seminar still meets a few times each semester (Wabash College: "Remembering Bill Swift").

Halmos photographed William L. Duren (1905-2008) on Oct. 14, 1971, at the University of Virginia in Charlottesville, where he was University Professor in the Department of Applied Mathematics and Computer Science. Duren earned his Ph.D. at the University of Chicago in 1930 with a dissertation on calculus of variations written under G. A. Bliss. He was professor of mathematics at his alma mater, Tulane University, from 1931 to 1955; Dean of the College of Arts and Sciences at the University of Virginia from 1955 to 1962; and University Professor and founding member of the Department of Applied Mathematics and Computer Science at Virginia from 1962 to 1976 (University of Virginia). Duren was MAA president during 1955-56 and a founding member of the MAA Committee on the Undergraduate Program in Mathematics (CUPM) (MAA Presidents).

Peter Duren was photographed on Feb. 24, 1968, at the University of Michigan in Ann Arbor, where he and Halmos were mathematics professors. Duren is now Professor Emeritus of Mathematics at Michigan; Halmos spent the 1968-69 academic year at University of Hawaii and then moved to Indiana University. Duren writes, "I well remember Paul Halmos the photographer, not to mention the mathematician or the lecturer. At least once I attended a talk he gave away from home and saw him begin by taking a photo of his audience!"

Duren earned his Ph.D. at the Massachusetts Institute of Technology in 1960 under Gian-Carlo Rota (Mathematics Genealogy Project). He has advised 25 Ph.D. students at the University of Michigan. In addition to numerous books and articles on complex analysis, he edited the American Mathematical Society's three-volume set, *A Century of Mathematics in America,* published in 1988 (University of Michigan).

William L. Edge (1904-1997) is pictured during the British Mathematical Colloquium, in Dundee, Scotland, in April of 1965. Halmos was a plenary speaker on "Some recent progress in Hilbert space" at the 1965 BMC (BMC Speakers). Edge received his Ph.D. from Cambridge with a dissertation on projective geometry and spent most of his career at Edinburgh University, specializing in finite geometry (MacTutor Archive).

Halmos photographed Samuel Eilenberg (1913-1998) in 1970. Eilenberg earned his doctorate in point-set topology from the University of Warsaw in 1936 under advisors Kazimierz Kuratowski (pictured on page 29 of this collection) and Karol Borsuk (page 8). He moved to the U.S. in 1939, spending most of his career at Columbia University in New York City. He was a second-generation member of Bourbaki, invited in 1949 to write about homotopy groups for the project. Eilenberg is most famous for the books *Foundations of Algebraic Topology*, with Norman Steenrod, and *Homological Algebra,* with Henri Cartan (MacTutor Archive).

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.

Index to the Collection