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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 statistician Jerzy Neyman (1894-1981) on June 7, 1966, in Budapest, Hungary. Neyman was born into a Polish family in what was then Russia and is now Moldova, and received his undergraduate education and some preparation for an academic career at Kharkov University in what was then Russia and is now Ukraine. He earned his Ph.D. in 1924 from the University of Warsaw, Poland, with a dissertation on applying statistics to agricultural experiments written under Waclaw Sierpinski. After study in London and Paris, he returned to Warsaw to set up a Biometric Laboratory at the Nencki Institute for Experimental Biology (1928). Neyman returned to England in 1934 to join the faculty at University College, London, where his friend and collaborator, Egon Pearson, son of Karl Pearson, was head of the applied statistics department. In 1938, he accepted a position at the University of California, Berkeley, where he set up and directed a Statistics Laboratory and eventually a separate statistics department. His (at least) 40 Ph.D. students at Berkeley (and one more at the University of London) included George Dantzig (see his photograph on page 11 of this collection) in 1946. (Sources: MacTutor Archive, Mathematics Genealogy Project, UC Berkeley Memorial)

Marshall Stone (1903-1989), Shiing-Shen Chern (1911-2004), and Louis Nirenberg, left to right, were photographed by Halmos on May 22, 1968, in Chicago, Illinois. At the time, Stone was just ending his tenure at the University of Chicago (perhaps the event shown above was part of his retirement celebration?) and preparing to move to the University of Massachusetts; Chern was at the University of California, Berkeley; Nirenberg at the Courant Institute at New York University; and Halmos about to leave the University of Michigan to take up a position at the University of Hawaii. Halmos had been a faculty member at Chicago from 1946 to 1961 and Chern from 1949 to 1960, where both worked under department chair Stone until 1952 and then under department chair Saunders Mac Lane. Stone is pictured on page 4 of this collection, where you can read more about him. Another photograph of Chern appears on page 9 of this collection, where you can read more about him.

After undergraduate studies at McGill University in Montreal, Quebec, Canada, Louis Nirenberg earned his Ph.D. in 1949 from New York University, with the dissertation “The Determination of a Closed Convex Surface Having Given Line Elements,” which he published in 1953 as “The Weyl and Minkowski problems in differential geometry in the large.” Nirenberg remained at the Institute of Mathematical Sciences and Mechanics (which became the Courant Institute of Mathematical Sciences in 1964) at NYU after earning his Ph.D., advising at least 46 Ph.D. students of his own and directing the Institute for a short time. He became Emeritus Professor in 1999. O’Connor and Robertson of the MacTutor Archive quoted Joseph Cohn’s description of Nirenberg:

Louis Nirenberg is recognized throughout the world for his many important research contributions to mathematics, his marvelous lectures, and his lucid expository writing. His range of interest is very broad: differential equations, harmonic analysis, differential geometry, functional analysis, complex analysis, etc.

In 2010, Nirenberg won the first-ever Shiing-Shen Chern Medal, awarded by the International Mathematical Union, for “his role in the formulation of the modern theory of non-linear elliptic partial differential equations and for mentoring numerous students and postdocs in this area.” (Sources: MacTutor Archive, Mathematics Genealogy Project, ICM 2010 Prize Winners)

Halmos photographed Betty and Ivan Niven (1915-1999) on December 26, 1968 in Honolulu, Hawaii. Halmos spent the academic year 1968-69 at the University of Hawaii and he and his wife Virginia entertained many mathematical visitors at their home in Honolulu that year. Niven was a number theorist at the University of Oregon whose Erdős number was 1 and who wrote seven books in all, including the MAA books

*Numbers: Rational and Irrational*(1961, the first book in the New Mathematical Library series), and*Diophantine Approximations*(1963, Niven’s 1960 MAA Hedrick Lectures, now available from Dover Publications)

and the textbook

*An Introduction to the Theory of Numbers*(1960-1991, five editions).

After earning bachelors and masters degrees from the University of British Columbia in Vancouver, Canada, Niven earned his Ph.D. in 1938 from the University of Chicago with the dissertation, “A Waring Problem,” written under advisor Leonard Eugene Dickson. He studied for a year with Hans Rademacher at the University of Pennsylvania (1938-39), then spent three years at the University of Illinois (1939-42, probably just missing Halmos, who left Illinois in 1939) and five years at Purdue University (1942-47) before joining the faculty at the University of Oregon in 1947. In 1947, he also published a one-page proof of the irrationality of pi, the first such short proof. Ivan Niven was president of the MAA during 1983-84. The University of Oregon Mathematics Department honors his memory with its annual Ivan Niven Lectures. (Sources: University of Oregon obituary, Mathematics Genealogy Project, MAA Presidents)

Halmos photographed his Ph.D. student Eric Nordgren, left, and Robert Ellis on August 26, 1964, at the AMS Summer Meeting at the University of Massachusetts in Amherst. Nordgren earned his Ph.D. that year (1964) from the University of Michigan with the dissertation “Some Contributions to the Theory of Toeplitz Operators,” written under advisor Halmos. He is now Professor Emeritus of Mathematics at the University of New Hampshire. Bob Ellis most likely earned his Ph.D. in 1953 from the University of Pennsylvania and now is Professor Emeritus of Mathematics at the University of Minnesota with research interests in topological dynamics. However, it is possible that he instead received his Ph.D. in 1966 from Duke University and now is Professor Emeritus at the University of Maryland. (Sources: Mathematics Genealogy Project, University of New Hampshire Mathematics, Universities of Minnesota and Maryland Mathematics)

Halmos photographed Donald Ornstein in April of 1970 in Columbus, Ohio. Ornstein earned his Ph.D. in 1957 from the University of Chicago with the dissertation “Dual Vector Spaces,” written under advisor Irving Kaplansky. He spent the years 1956-58 at the Institute for Advanced Study in Princeton, New Jersey. In 1960, he published his first paper on ergodic theory and joined the faculty at Stanford University, where he has remained and is now Professor Emeritus of Mathematics. In the year this photo was taken, 1970, Ornstein published his first really big result in ergodic theory, “Bernoulli shifts with the same entropy are isomorphic.” During the next few years, he continued to publish important results in ergodic theory and collected them in the book, *Ergodic theory, randomness, and dynamical systems* (1974). Ornstein would have first met Halmos when he was a graduate student at Chicago during the 1950s, and, indeed, in his 1985 book, *I Want to Be a Mathematician,* Halmos confirmed that he taught “Don Ornstein in several courses and he began to learn ergodic theory from me (but wrote his thesis with Kaplansky on algebra)” (p. 202). Halmos was a faculty member at Chicago from 1946 to 1961. He also was a member of the Institute for Advanced Study during 1957-58, as was Ornstein. (Source: MacTutor Archive)

Halmos photographed Alexander Ostrowski (1893-1986), left, and Béla Szökefalvy-Nagy (1913-1998) on July 19, 1968, in Oberwolfach, Germany, where all three of them were attending a conference on approximation theory at the Oberwolfach Conference Center. This conference was one of a series of seven organized by Paul Butzer (page 9) from 1963 to 1983 on approximation theory and related topics in functional analysis. The 1968 conference was the first of the seven Halmos attended. Photos of Szökefalvy-Nagy appear on page 9, page 14, and page 36 of this collection; read more about him on pages 9 and 14.

Born in Kiev, Ukraine, Alexander Ostrowski earned his Ph.D. (Dr. phil.) at the University of Göttingen, Germany, in 1920 with a dissertation written under unofficial advisor David Hilbert and official advisors Edmund Landau and Felix Klein. He had studied previously with Dmitry Grave at the University of Kiev and with Kurt Hensel at the University of Marburg, and his article based on his doctoral dissertation was his 15^{th} publication! His dissertation also led to his solution of Hilbert’s 18^{th} Problem. After working with Erich Hecke at the University of Hamburg for two years, teaching at Göttingen for three more years, and visiting universities in Britain for a year (1925-26), Ostrowski accepted the Chair of Mathematics at the University of Basel, Switzerland, and was based there for the rest of his career and life. His work was mainly, but not exclusively, in algebra and number theory, and he had a great interest in numerical methods as well as abstract mathematics. (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 38," *Loci* (January 2012), DOI:10.4169/loci003801