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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 former Ph.D. student Donald Sarason (center, with Dan Halperin at left), in February of 1982 at the University of California, Davis (according to Halmos' notation on the back of the photograph; please let us know of any corrections or further identifications and information). Sarason earned his Ph.D. in 1963 from the University of Michigan with the dissertation “The *H ^{p}* Spaces of Annuli.” He has spent his career at the Univerity of California, Berkeley, where he has advised 40 Ph.D. students and became Professor Emeritus in January of 2012. His first book was the monograph

In his *I Want to Be a Mathematician: An Automathography* (Springer 1985), Halmos wrote of Sarason:

I still haven’t decided whether Don was my best student or Errett Bishop; fortunately I don’t have to. ... Don is intelligent and quick, and although he doesn’t waste words, he never fails to say what must be said. ... [He] is a quiet man; he never uses eight words when seven will do. He is one of the smoothest and clearest lecturers I know and he has an extraordinary sense of time. He knows almost to the minute how long it will take him to explain something. (pp. 276-277)

For a photograph of Erret Bishop, see page 6 of this collection, where you can read more about him. (Sources: Mathematics Genealogy Project, UC Berkeley Mathematics)

Isaac J. Schoenberg (1903-1990) was photographed by Halmos in 1978. Born in Galati, Romania, Iso Schoenberg earned his Ph.D. in analytic number theory in 1926 from the Alexandru Ioan University of Iasi (or Jassy), Moldavia, under advisors Simeon Sanielevici and Issai Schur. Schur was at the University of Berlin at the time, but Schoenberg had studied in Berlin with Schur and in Göttingen with Edmund Landau from 1922 to 1925. From 1928 to 1930, Schoenberg visited Hebrew University of Jerusalem and, from 1930 onward, he held various positions in the U.S., including professorships at the University of Pennsylvania from 1941 to 1966 and at the University of Wisconsin at Madison from 1966 onward. While doing ballistics research at the Aberdeen (Maryland) Proving Ground from 1943 to 1945, he introduced the theory of splines, publishing his first two papers on the subject in 1946. Splines really took off during the 1960s when computers became available for scientific research and engineering design, and Schoenberg not only became well known for their invention but continued to make advances in both theory and practice. He continued to work on approximation theory for the rest of his life. Do you recognize anyone in the crowd? (Sources: MacTutor Archive, Mathematics Genealogy Project)

Alan H. Schoenfeld, left, and David Blackwell (1919-2010) were photographed by Halmos in December of 1985 in Berkeley, California. Schoenfeld wrote recently:

I remember that meeting - Paul, David Blackwell, Leon Henkin and I had gotten together to discuss the mathematics content (specifically problem solving) that ought to be highlighted in AAAS's Project 2061 mathematics report.

Two more photographs of Blackwell appear on page 1 and page 7 of this collection, where you can read more about him. Schoenfeld earned his Ph.D. in 1973 from Stanford University with the dissertation “Topological and Measure-Theoretic Studies on Cantor Sets and Peano Spaces.” He has spent his career at the University of California, Berkeley, where he holds the Conner Chair in Education, is an Affiliated Professor of Mathematics, and continues to advise Ph.D. students in mathematics education. He is a national leader in the field of mathematics education, specializing in cognition and development or, as his UC Berkeley biography puts it, “Schoenfeld’s research deals with thinking, teaching, and learning. His book *Mathematical Problem Solving* characterized what it means to think mathematically ....” Among his many accomplishments, he was lead author for grades 9-12 of the groundbreaking *Principles and Standards for School Mathematics,* produced by the National Council of Teachers of Mathematics (NCTM) in 2000. In 2011 Schoenfeld was awarded the Felix Klein Medal for lifetime achievement by the International Commission on Mathematical Instruction. (Sources: Mathematics Genealogy Project, UC Berkeley Graduate School of Education)

Halmos photographed his former Ph.D. student Morris Schreiber (d. 1988) on May 29, 1966, in Irvine, California. Schreiber earned his Ph.D. in 1955 from the University of Chicago with the dissertation “Unitary Dilations of Operators.” In *I Want to Be a Mathematician: An Automathography* (Springer 1985), Halmos wrote of Schreiber’s thesis work:

He gave me one of the rare deep surprises that thesis advisors get. I suggested to him that he study the unitary power dilations of strict contractions and look at them as spectral invariants; what information about a contraction can we infer from knowledge of its unitary power dilation? I didn’t believe Moe the first time when he, hesitantly, told me the answer. None, he said. All those unitary power dilations are the same. ... Schreiber’s theorem (as it quickly became called) is that the spectral information conveyed by the unitary power dilation ... says something about everybody but it doesn’t say anything about anybody. (page 222)

Schreiber was on the faculty at Cornell University in Ithaca, New York, during the 1960s, and he was a professor at Rockefeller University in New York City at the time of his death. In 1990, Schreiber’s sister Hilda Schreiber established a fund at Princeton University to support graduate fellowships in mathematics. (Sources: Mathematics Genealogy Project, *New York Times* obituary, Princeton University)

Halmos photographed Jacob T. Schwartz (1930-2009) on May 8, 1978, at the University of California, Santa Barbara. Born in New York City, Jack Schwartz earned his Ph.D. in 1952 from Yale University with the dissertation “Linear Elliptic Differential Operators.” He remained on the faculty at Yale until 1957, when he joined the soon-to-be Courant Institute at New York University, where he remained for the rest of his career and life. The first of his at least 28 Ph.D. students was Gian-Carlo Rota (Yale, 1956), who is pictured on page 45 of this collection, with his remaining Ph.D. students at NYU. O’Connor and Robertson of the MacTutor Archive quote Schwartz’s NYU colleague Martin Davis (see page 11 of this collection) on Schwartz’s research *modus operandi:*

Jack's style has been to enter a new field, master quickly the existing research literature, add the stamp of his own forceful vision in a series of research contributions, and finally, leave behind an active research group that continues fruitful research for many years along the lines he has laid down.

At first, these research fields were in mathematical analysis, but in 1964 Schwartz founded the computer science department at NYU and became its first chair. His interests in computer science included parallel computing, compiler optimization, and robot design. By the time of his death, he was doing research in molecular biology. (Sources: MacTutor Archive, Mathematics Genealogy Project)

Halmos photographed Laurent Schwartz (1915-2002) in 1978. Born in Paris and educated there through the Agrégation de Mathématiques from the École Normale Supérieure in 1937, Schwartz earned his Ph.D. in 1943 from the Université Louis Pasteur – Strasbourg with the dissertation “Sommes de Fonctions Exponentielles Reelles.” He spent his career at various universities in France, including Grenoble (1944-45), Nancy (1945-53), and Paris (1953-59, 1980-83). He was on the faculty at the École Polytechnique in Paris from 1959 to 1980. Schwartz’s best known contribution to mathematics, the one for which he won the Fields Medal in 1950, was the theory of distributions. In his *I Want to Be a Mathematician* (Springer 1985), Halmos described Schwartz’s difficulties in obtaining a visa to enter the U.S. to attend the 1950 International Congress of Mathematicians (ICM) to be held at Harvard University in Cambridge, Massachusetts, describing Schwartz only partly tongue-in-cheek as a “known Trotskyite activist” (page 163). Schwartz was eventually allowed to enter the U.S. and to speak at the ICM and receive his Fields Medal there. Halmos also wrote, “The last time I saw Schwartz was in Berkeley in the 1970’s” (*ibid*), so this photograph was quite possibly taken in Berkeley, California. (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 47," *Convergence* (June 2013), DOI:10.4169/loci003801