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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 himself was photographed in Varenna, Italy, in June of 1960, apparently caught in the act of setting up a photograph. Written on the back of this photo, in a very neat hand not belonging to Halmos, are the words “Best regards, Eberhard.” Assuming not only that Eberhard was a mathematician, but also that he was the best-known mathematician named “Eberhard” of the 20th century, the writer (and presumably photographer) may have been Eberhard Hopf (1902-1983). Born in Salzburg, Austria, Hopf received his Ph.D. in 1926 from the University of Berlin, Germany, under advisors Erhard Schmidt and Issai Schur. He worked in topology and ergodic theory, holding faculty positions mainly in the U.S., except for professorships at the universities of Leipzig and Munich, Germany, from 1936 to 1947. Hopf’s longest faculty appointment was at Indiana University from 1949 to 1972, when he retired, but he remained a research faculty member until his death in 1983. Halmos was a faculty member at Indiana University from 1969 to 1985. (Sources: MacTutor Archive, Mathematics Genealogy Project)

Israel Halperin (1911-2007) was photographed by Halmos in 1967. Halperin earned his Ph.D. from Princeton University in 1936 with the dissertation, “Adjoints and Closures of Linear Differential Operators,” written under Salomon Bochner and John von Neumann. After short stints at Yale and Harvard, he was on the mathematics faculty of Queen’s University in Kingston, Ontario, Canada, from 1939 to 1966 and the University of Toronto, Ontario, from 1966 to 1976. He was a pioneer in the field of operator algebras and also became known as a human rights activist following his own “Red scare” imprisonment in Canada in 1946-47. (Sources: Mathematics Genealogy Project, *Toronto Globe and Mail* (pdf file), *Princeton Alumni Weekly*)

Halmos photographed number theorist Helmut Hasse (1898-1979) and his wife, identified only as Mrs. Hasse, in Honolulu, Hawaii, in January of 1967. Hasse earned his Ph.D. in 1922 at the University of Marburg under advisor Kurt Hensel, and taught at a number of German universities with his longest tenure, 1950-1966, at Hamburg University, from which he retired. He is famous for his contributions to many areas of number theory, including class field theory, Brauer groups of algebraic number fields, elliptic curves, and perhaps above all local methods in algebraic number theory. He also served as editor of *Crelle’s Journal* for 50 years. Halmos would return to Honolulu as a professor at the University of Hawaii during the 1968-1969 academic year. (Sources: MacTutor Archive, Mathematics Genealogy Project)

Mathematical logician Louise Hay (1935-1989) was photographed in November of 1978 at a colloquium at the University of Illinois, Chicago Circle (which became the University of Illinois at Chicago in 1982). Hay was a mathematics professor at Chicago from 1968 until her death in 1989 and Mathematics Department Head from 1980 to 1989. She earned her Ph.D. in 1965 from Cornell University with a dissertation on recursion theory written under Anil Nerode. She was a founding member of the Association for Women in Mathematics (AWM), which in 1990 established the Louise Hay Award for Contributions to Mathematics Education to honor her success as a teacher and mentor. (Sources: MacTutor Archive, University of Illinois at Chicago, Biographies of Women Mathematicians (Agnes Scott College), Mathematics Genealogy Project)

Halmos photographed complex analyst Walter Hayman in 1968, possibly in London, England. Hayman earned his Ph.D. in about 1947 from Cambridge under advisor Mary Cartwright. Although he spent some years at the University of Exeter (1947-1956) and York University (1985-1993), he spent most of his career at Imperial College, London (1956-1985, 1995-present). In 1955, he proved the asymptotic Bieberbach Conjecture and in 1956 he became a fellow of the Royal Society as well as the first-ever Professor of Pure Mathematics at Imperial College. He and his wife, mathematics teacher Margaret Crann Hayman, founded the British Mathematical Olympiad in 1966 and first sent a UK team to the International Mathematical Olympiad in 1967. (Sources: MacTutor Archive, Mathematics Genealogy Project)

Halmos photographed Walter Hayman again at the Oberwolfach Conference Center in Germany in August of 1983. For two more photos of Hayman at Oberwolfach, see the Oberwolfach Photograph Collection.

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