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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.

Paul Halmos photographed Donald Albers and George Dantzig (1914-2005) in Menlo Park, California, in February of 1985.

Albers served as MAA Director of Publications from 1991 to 2006, then as Editorial Director of MAA Books from 2006 to 2011, and now is Senior Acquisitions Editor for MAA Publications. He was the founding editor of the MAA's magazine for students, *Math Horizons,* and may be best known for his interviews and profiles of mathematicians, many of them published in MAA journals and in the books *Mathematical People* and *More Mathematical People,* co-authored with Gerald Alexanderson, whose photograph appears on page 1 of this collection, and Constance Reid. In 1985, Albers was a mathematics professor at Menlo College.

Dantzig earned his Ph.D. from the University of California, Berkeley, in 1946 and then returned to his wartime job as a statistican for the U.S. Air Force. He continued to develop what would become known as linear programming and, in 1947, he invented the simplex method of optimization. In 1952, he moved to the RAND Corporation, where he was able to to implement linear programming on computers. He became a professor of operations research at Berkeley in 1960, but in 1966 he moved across the bay to Stanford University (MacTutor Archive). Dantzig advised at least 50 Ph.D. theses, 11 of them at Berkeley and 40 at Stanford (Mathematics Genealogy Project). He received the National Medal of Science in 1975. (Sources: MacTutor Archive, Mathematics Genealogy Project; see also AMS Notices 54:3 March 2007.)

Halmos photographed Martin Davis at the American Mathematical Society (AMS) Hilbert Problems Conference in DeKalb, Illinois, on May 14, 1974. Davis contributed to the solution of Hilbert's Tenth Problem during the late 1950s and early 1960s (MacTutor Archive). "The crucial final step in the solution of the problem," as Davis describes it, was provided by Russian mathematician Yuri Matiyasevich (or Matijasevic) in 1970. "A paper by Hilary Putnam, Julia Robinson, and me ... had reduced the problem to finding a single equation that satisfied a certain exponential growth condition. What Yuri's beautiful proof did was to provide just such an equation."

In 1973, Davis published a widely read paper explaining the proof (MacTutor Archive), but he himself points to the more forward-looking "Hilbert's tenth problem. Diophantine equations: Positive aspects of a negative solution," co-authored with Matiyasevich and Robinson, in the AMS volume *Mathematical Developments Arising from Hilbert Problems* (1976). The solution to "H10" is known as the "MRDP Theorem" in recognition of the four mathematicians who contributed significantly to its proof. Davis earned his Ph.D. in mathematical logic at Princeton in 1950 with the dissertation, "On the Theory of Recursive Unsolvability." He then taught at several universities, including Yeshiva University in New York City from 1960 to 1965, but spent most of his career at the Courant Institute of Mathematical Sciences, New York University, where he is now emeritus (MacTutor Archive).

Louis de Branges and his wife Tatiana Jakimov de Branges are photographed at Stanford University in Palo Alto, California, in April of 1985. De Branges earned his Ph.D. at Cornell University in 1957 with a dissertation on "Local Operators on Fourier Transforms" and has spent most of his career at Purdue University, where he is Edward C. Elliott Distinguished Professor of Mathematics. From the late 1950s onward, he has worked on Hilbert spaces of entire functions, always with an eye toward proving the Riemann Hypothesis. In 1984 this work led not to a proof of the Riemann Hypothesis, but to a proof of the Bieberbach Conjecture, which, as O'Connor and Robertson of the MacTutor Archive point out, should now rightly be known as "de Branges' Theorem" (MacTutor Archive). De Branges' current work on the Riemann Hypothesis is posted at his Purdue University webpage.

Keith Devlin is pictured at Stanford University in September of 1987. Devlin had just begun a 1987-89 visit to Stanford University as a professor of mathematics and philosophy, while photographer Halmos had joined the faculty of nearby Santa Clara University in 1985. Devlin also became a researcher at Stanford University's Center for the Study of Language and Information in 1987. He would return to the Bay Area for good in 1993, taking positions at St. Mary's College and Stanford University. He currently is director of the Human-Sciences and Technologies Advanced Research Institute (H-STAR) at Stanford University. Originally trained as a logician, with a Ph.D. from the University of Bristol, U.K., in 1971, his main interest today is to use "different media to teach and communicate mathematics to diverse audiences." He is a prolific author of popular mathematics books, and popularizes mathematics on television and radio as well (Stanford University webpage).

Halmos photographed Joseph Diestel at Kent State University in Kent, Ohio, on June 1, 1975. Diestel earned his Ph.D. in functional analysis from the Catholic University of America in 1968. He has spent most of his career at Kent State University in Kent, Ohio, where his first Ph.D. student (in 1974) was Barbara Faires, current MAA Secretary. According to the Mathematics Genealogy Project, Diestel has advised at least 23 Ph.D. dissertations in functional analysis, operator theory, and measure and integration theory. Six of his Ph.D. students earned their degrees at universities in Spain, Venezuela, South Africa, and Italy. He is now professor emeritus of mathematics at Kent State (Kent State emeritus faculty webpage).

Halmos photographed Jean Dieudonné (1906-1992) and Carol Hughes on May 1, 1962. Dieudonne earned his Ph.D. at the École Normale Supérieure Paris in 1931. Soon after, he became a founding member of the Bourbaki and credited the group with expanding his mathematical horizons beyond the classical analysis in which he had been trained. According to O'Connor and Robertson of the MacTutor Archive, Dieudonné "was the leading influence in a group whose whole object was to avoid anyone taking on this role," and they quote Armand Borel's and Pierre Cartier's descriptions of him as the final drafter of the first 30 Bourbaki volumes. Dieudonné worked in general topology, topological vector spaces, algebraic geometry, invariant theory, and classical groups. He also wrote histories of functional analysis, algebraic geometry, and algebraic and differential topology, and edited the works of Camille Jordan. (Source: 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.

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 11," *Loci* (January 2012), DOI:10.4169/loci003801