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.

Index to the Collection

Halmos photographed number theorist Louis Mordell (1888-1972) in April of 1961 in Chicago, Illinois. Halmos left Chicago after the 1960-61 academic year. Mordell had retired from the Sadleirian Chair at Cambridge University in 1953, but he was still a very active mathematician and, according to O’Connor and Robertson of the MacTutor Archive, an enthusiastic traveler who “delighted in” accepting visiting professorships and invitations to speak at conferences and universities around the world. (Source: MacTutor Archive)

Louis Mordell was photographed again by Halmos on June 6, 1968, in Cambridge, England. As a theretofore self-taught and self-funded 18-year-old mathematician from Philadelphia, Pennsylvania, Mordell had traveled to Cambridge in December of 1906 to compete (successfully) for a scholarship to study mathematics at Cambridge University. Although he was disappointed with himself for finishing as only the Third Wrangler and although Cambridge did not offer a Ph.D. degree at the time, Mordell remained at Cambridge to begin his research career. From the start, he studied the indeterminate equations for which he may be best known, those of the form *y*^{2} = *x*^{3} + *k*, now sometimes called Mordell equations. He also is known for proving conjectures of Ramanajan (on the tau function) and Poincaré (the finite basis theorem), and for making the Mordell Conjecture, proved by Gerd Faltings in 1983. Mordell held positions at Birkbeck College in London (1913-20, with war work for the Ministry of Munitions), Manchester College of Technology (1920-22), University of Manchester (1922-45), and Cambridge University (1945 onward), where he succeeded G. H. Hardy in the Sadleirian Chair. His first Ph.D. student at Cambridge was number theorist J.W.S. (Ian) Cassells, who earned his Ph.D. in 1949 and whose photograph appears on page 9 of this collection. (Sources: MacTutor Archive, Mathematics Genealogy Project)

Halmos photographed Jürgen Moser (1928-1999) on January 23, 1969, at the Joint Mathematics Meetings in New Orleans, Louisiana, and managed to catch himself as well in the mirror. After surviving the Russian siege of Königsberg, Germany, in 1945, and several harrowing escapes from and re-entries into Königsberg (or Kaliningrad, as it was renamed by the Russians in 1946) in 1946-47, Moser made his way to Göttingen, Germany, in 1947. He earned his Ph.D. from Göttingen University in 1952 with a dissertation on the spectral theory of differential equations. He spent 1952-53 at Göttingen, 1953-54 at New York University, 1954-55 at Göttingen, and 1955 onward at New York University with a few years at the Massachusetts Institute of Technology (MIT) in the late 1950s. Moser directed the Courant Mathematical Institute at New York University in the late 1960s, but he was best known as a leading researcher in differential equations, spectral theory, celestial mechanics, and stability theory. In 1980, the Eidgenössische Technische Hochschule (ETH) in Zurich, Switzerland, wanting to have the leading dynamical systems researcher in the world on its faculty, enticed Moser to take a position there and eventually to become director of its Mathematics Research Institute (1984-1995). Moser was president of the International Mathematical Union from 1983 to 1986. In 1989, he delivered his last major address to the International Congress of Mathematicians in Berlin on “Dynamical Systems: Past and Present.” (Sources: MacTutor Archive, Mathematics Genealogy Project)

Benjamin Muckenhoupt and his family were photographed by Halmos in August of 1980 at the Oberwolfach Conference Center in Germany. From left to right in the photo are Carl, Benjamin, Meg, and Mary Kay Muckenhoupt. Ben Muckenhoupt earned his Ph.D. in 1958 from the University of Chicago with the dissertation, “On Certain Singular Integrals,” written under Antoni Zygmund. He and Halmos probably first met at Chicago, where Halmos was on the faculty from 1946 to 1961. In 1980, at the time of this photo, both were attending a conference on approximation theory and related areas of functional analysis co-organized by Paul Butzer and Béla Szökefalvy-Nagy (see page 9 of the collection for a photograph of Butzer and Szökefalvy-Nagy). Muckenhoupt spent most of his career at Rutgers University in New Brunswick, New Jersey, where he is now Professor Emeritus of Mathematics and lists his interests are harmonic analysis and weighted norm inequalities. (Sources: Mathematics Genealogy Project, Rutgers University Mathematics)

Halmos photographed logician John R. Myhill (1923-1987) in 1958. Myhill earned his Ph.D. in philosophy from Harvard University in 1949 with the dissertation, “A Semantically Complete Foundation for Logic and Mathematics,” written under advisors Willard Quine and Lynn Loomis. He was on the philosophy faculty at the University of California, Berkeley, from 1954 to 1960 and at Stanford University from 1960 to 1963. After having spent 1957-59 and 1963-64 at the Institute for Advanced Study in Princeton and 1964-66 as Visiting Professor of Mathematics at the University of Illinois, he joined the mathematics faculty at the State University of New York, Buffalo, in 1966 and spent the rest of his career there. He may be best known for the Russell-Myhill Paradox, about which you can read more at the Internet Encyclopedia of Philosophy. (Sources: Mathematics Genealogy Project; “John R. Myhill (1923-1987),” *History and Philosophy of Logic* 8:2 (1987), pp. 243-244 (available from Taylor & Francis Online); Internet Encyclopedia of Philosophy)

Halmos photographed Ivan Singer (left) and Béla Szökefalvy-Nagy (1913-1998) on July 22, 1968, at the Oberwolfach Conference Center in Germany. Singer, Szökefalvy-Nagy, and Halmos were participating in the conference, "Abstract spaces and approximation," organized by Szökefalvy-Nagy and Paul Butzer (see page 9 of this collection for a photograph of Butzer). Similar conferences were held at Oberwolfach every three years until 1983. Singer wrote recently that, while he was a Visiting Professor at the University of Iowa in 1969, he traveled during during the Winter Break to the West Coast of the U.S. to give a series of talks and to Honolulu, Hawaii, for one week to work with Halmos, who was department chair at the Univeristy of Hawaii at the time. Watch for photographic evidence of this visit on page 58 or so!

Singer now is Research Professor and Honorary Member of the Institute of Mathematics of the Romanian Academy in Bucharest. He is a functional analyst currently researching geometry in semimodules, and also interested in optimization, convex and discrete geometry, and operations research. Much of his research during the 1960s and 1970s, as evidenced by his published papers and monographs during this time, was on approximation theory. Photographs of Szökefalvy-Nagy also appear on page 9 and page 14 of the collection, where you can read more about him. (Sources: Institute of Mathematics of the Romanian Academy, Ivan Singer (Dec. 2012))

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.

Index to the Collection