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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 his first Ph.D. student, statistician Herman Rubin, in Bloomington, Indiana, on July 31, 1974. Rubin earned his Ph.D. in 1948 from the University of Chicago, where Halmos was a faculty member from 1946 to 1961, with the dissertation “Systems of Linear Stochastic Equations.” Both Rubin and Halmos spent the academic year 1947-48 at the Institute for Advanced Study in Princeton, New Jersey. Rubin advised Ph.D. students of his own at Stanford University in Palo Alto, California, during the 1950s, the University of Oregon and Michigan State University during the 1960s, and Purdue University in West Lafayette, Indiana, where he is Professor of Statistics and Mathematics, from the 1970s onward. (Sources: Mathematics Genealogy Project, Institute for Advanced Study, Purdue University Department of Statistics)

Topologist Mary Ellen Rudin (1924-2013) was photographed by Halmos at the Joint Mathematics Meeting at Cornell University in Ithaca, New York, on September 1, 1965. Another photograph of Rudin appears on page 25 of this collection, where you can read more about her; see page 44 for one more photo of her.

Halmos photographed complex analyst Walter Rudin (1921-2010) at a conference in Lancaster, England, in July of 1984. Born in Vienna, Austria, Rudin earned his Ph.D. in 1949 from Duke University in Durham, North Carolina, with the dissertation “Uniqueness Theory for Laplace Series.” His first two academic posts were at Duke (1949-50) and MIT (1950-1952). He was on the faculty at the University of Rochester from 1952 to 1959, and in 1959 moved to the University of Wisconsin at Madison, where he spent the rest of his career. He is best known for his analysis textbooks, *Principles of Mathematical Analysis* (1953), *Real and Complex Analysis* (1966), and *Functional Analysis* (1973). Do you recognize anyone in the crowd behind Rudin? (Sources: University of Wisconsin obituary, MacTutor Archive (Mary Ellen Rudin), Mathematics Genealogy Project)

Mathematical physicist David Ruelle was photographed by Halmos in Bloomington, Indiana, in April of 1980. Born in Ghent, Belgium, Ruelle earned his Ph.D. in physics jointly from the Free University of Brussels, Belgium, and Eidgenössische Technische Hochschule (ETH) in Zürich, Switzerland. He was a faculty member at ETH from 1960 to 1962 and a member of the Institute for Advanced Study in Princeton, New Jersey, from 1962 to 1964 (and again in 1970-71). In 1964, he joined the faculty at the Institut des Hautes Études Scientifiques (IHES), near Paris, where he became Honorary Professor in 2000. He also is a Distinguished Visiting Professor at Rutgers University in New Brunswick, New Jersey. A sampling of Ruelle’s book and monograph titles from the 1960s through the 1990s gives a representative outline of his important contributions to mathematical physics: *Statistical Mechanics *(1969),* Thermodynamic Formalism *(1978),* Elements of Differentiable Dynamics and Bifurcation Theory *(1989),* Chance and Chaos *(1991), and *Turbulence, Strange Attractors, and Chaos* (1995). His best known book today may be his fascinating *The Mathematician’s Brain* (2007). (Sources: Mathematics Genealogy Project, IHES Mathematics)

Halmos photographed operations researcher Thomas Saaty in November of 1962. Saaty earned his Ph.D. in 1953 from Yale University with the dissertation “On the Bessel Tricomi Equation,” written under advisor C. Einar (Carl) Hille. (For a photograph of Hille, see page 23 of this collection, where you can read more about him.) Saaty’s mathematical models of epidemics, urban design, arms control and disarmament, and the workings of the human brain, among other applications, have used linear programming, queuing theory, graph theory, nonlinear mathematics, game theory, and decision theory, among other mathematical tools. His research interests today are decision-making and conflict resolution – his dream is to establish an International Center for Conflict Resolution – and he is Distinguished University Professor in the University of Pittsburgh’s Katz School of Business. Before joining the Pitt faculty, he was on the faculty of the University of Pennsylvania’s Wharton School of Business. Before that, he worked in the U.S. State Department’s Arms Control and Disarmament Agency and, before that, for various U.S. government agencies in positions influencing scientific funding. At about the time the photo above was taken, Saaty, in his position at the Office of Naval Research, commissioned Halmos’ series of talks on operator theory and the subsequent publication of these talks in his 1963 survey, *A glimpse into Hilbert space*. (Sources: Mathematics Genealogy Project; University of Pittsburgh Katz School of Business; Paul Halmos, *I Want to Be a Mathematician: An Automathography,* Springer 1985, p. 390)

Halmos photographed algebraic geometer Pierre Samuel (1921-2009) in 1959. Born in Paris, Samuel earned the Agrégé de mathematique at the École Normale Supérieure before traveling to the U.S. to earn his Ph.D. at Princeton University in 1947 with the dissertation “Ultrafilters and Compactification of Uniform Spaces,” written under advisor Oscar Zariski. He then returned to France, joining the National Center for Scientific Research in 1947, the Faculty of Science at Clermont-Ferrand in 1949, and the Université Paris Sud XI Orsay in 1961. According to O’Connor and Robertson of the MacTutor Archive, even before earning his Ph.D., Samuel had become a second-generation member of the Bourbaki. He is best-known for his and Oscar Zariski’s two-volume *Commutative Algebra* (1958, 1960). (Sources: MacTutor Archive, Mathematics Genealogy Project)

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 46," *Convergence* (June 2013), DOI:10.4169/loci003801