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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 Nicolaas Kuiper (1920-1994) in April of 1961 in Chicago, Illinois. Kuiper earned his Ph.D. in differential geometry in 1946 from the University of Leiden in Holland. He then visited the U.S. for three years, spending most of his time at the Institute for Advanced Study in Princeton, where he began a collaboration with differential geometer Shiing-Shen Chern (whose photograph appears on page 9 of this collection) and where he may have first met Halmos, who was at IAS during 1947-48. Kuiper then taught at the Agricultural Institute in Wageningen, Holland, until 1962, when he became a professor at the University of Amsterdam. In 1971, he moved to Paris to become Director of the Institut des Hautes Études Scientifiques (IHES), a position he held until 1985. He retired from IHES in 1991 and moved back to Holland. Among differential geometers, he is best known for his work on tight and taut manifolds. (Sources: MacTutor Archive, IAS)

Left to right: Carl Pearcy, Kazimierz Kuratowski, Frederick Gehring, and Frank Harary were photographed by Halmos in September of 1963 in Ann Arbor, presumably on the University of Michigan campus. Halmos, Pearcy, Gehring, and Harary were faculty members at Michigan at the time.

Carl Pearcy earned his Ph.D. in operator theory in 1960 from Rice University. He was a Ph.D. student of Arlen Brown (photographed on page 8 of this collection), who in turn had been a Ph.D. student of Halmos and Irving Kaplansky (page 26) at the University of Chicago, earning his degree in 1952. Pearcy and Brown wrote many joint papers during the 1960s and 1970s. The two also co-authored *An Introduction to Analysis,* in the Springer Graduate Texts in Mathematics series, in 1995. Pearcy remained at Rice long enough to advise (with Brown) one Ph.D. student, Donald Deckard, with whom he published papers throughout the 1960s. Pearcy joined the faculty at the University of Michigan in 1963 and, when he retired in 1990, moved to Texas A & M University. He is now Professor Emeritus of Mathematics at both Michigan and Texas A & M, and has advised at least 31 Ph.D. students at Rice, Michigan, and Texas A & M. (Sources: Mathematics Genealogy Project, MathSciNet, University of Michigan Memoir, Texas A & M Department of Mathematics)

Kazimierz Kuratowski (1896-1980) earned his Ph.D. in 1921 from the University of Warsaw in the then-new discipline of topology. He was a mathematics professor at the Technical University of Lvov from 1927 to 1934, when he returned to the University of Warsaw. His Ph.D. students included Stanislaw Ulam (Lvov, 1933) and Samuel Eilenberg (Warsaw, 1936), with Eilenberg (pictured on page 13 of this collection) co-advised by Karol Borsuk (page 8). During the brutal German occupation of Poland during World War II (1939-1945), Kuratowski taught in the underground university. After the war, he led the rebuilding of the Polish mathematical education and research systems. In 1949, he became founding director of the Mathematical Institute of the Polish Academy of Sciences. He worked in topology and set theory, and is best known today for Kuratowski’s Theorem, which states that a graph is planar if and only if it does not contain a subgraph homeomorphic to the complete graph *K*_{5} or the complete bipartite graph *K*_{3,3}. (Sources: MacTutor Archive, Mathematics Genealogy Project)

Born and raised in Ann Arbor, Michigan, complex analyst and geometer Frederick Gehring earned his Ph.D. at Cambridge in 1952, taught at Harvard for three years, and, since 1955, has been a mathematics professor at the University of Michigan in Ann Arbor, where he has specialized in quasiconformal mappings and advised at least 29 Ph.D. students. He was appointed T. H. Hildebrandt Distinguished University Professor in 1987 and, in 1995, he added Emeritus to his title. (T. H. Hildebrandt is photographed on page 23 of this collection.) In 1996, Michigan established the Gehring Collegiate Professorship in his honor, and appointed complex analyst John Erik Fornaess as its first holder. During 2001-02, UM held the Gehring Special Year in Complex Analysis. Gehring and Halmos were colleagues at UM from 1961 to 1968, when Halmos was a faculty member there. Gehring also is photographed on page 16 of this collection. (Sources: Mathematics Genealogy Project, UM Memoir, UM: Fornaess, ContinuUM)

Frank Harary (1921-2005) earned his Ph.D. in 1948 from the University of California, Berkeley, with the dissertation, “The Structure of Boolean-like Rings.” He joined the faculty at the University of Michigan in 1948 and retired in 1986, becoming Professor Emeritus. Harary is considered the “father” of modern graph theory. He developed and shaped the field in his own courses, numerous papers, even more numerous presentations, and influential books, especially his 1969 text, *Graph Theory,* and by helping found and run journals in combinatorics and graph theory. He was interested in applications of graph theory to a wide variety of disciplines and was a member of the Institute for Social Research at Michigan from 1950 to 1982. In 1987, he became Distinguished Professor of Computer Science at New Mexico State University in Las Cruces. (Sources: Mathematics Genealogy Project, UM Memoir, UM Memorial)

Halmos photographed Serge Lang (1927-2005), probably between 1953 and 1955 when Lang taught at the University of Chicago. Halmos was a professor at Chicago from 1946 to 1961. Born near Paris, France, Lang earned his Ph.D. in 1951 from Princeton University under advisor Emil Artin, with the dissertation “On Quasi Algebraic Closure.” He remained at Princeton and the Institute for Advanced Study for two years, then joined the University of Chicago faculty for two years. He was a professor at Columbia from 1955 to 1971 and at Yale from 1972 onward. He did significant research, mainly in algebra and number theory, but he is best known for his many highly regarded textbooks for which he won the AMS Steele Prize for Mathematical Exposition in 1999. The AMS cited in particular two of his graduate texts, *Algebra* (1965) and *Algebraic Number Theory* (1970), as having changed the way the subjects were taught. Jay Jorgenson and Steven G. Krantz, editors of a 2006 AMS tribute to Lang, reported that Lang wrote 61 books in all, not counting multiple editions and translations, and they repeated the mathematical riddle:

Q: Why did Bourbaki stop writing? A: They discovered that Serge Lang is one person.

(Sources: MacTutor Archive; “1999 Steele Prizes,” *AMS Notices* 46:4 (April 1999), 457-462; “Serge Lang: 1927-2005” (Part 1), *AMS Notices* 53:5 (May 2006), 536-553)

Robert Langlands was photographed by Halmos on Pi Day (March 14), 1971, in DeKalb, Illinois. Born in British Columbia, Canada, Langlands earned his Ph.D. in 1960 from Yale University, with the dissertation “Semi-Groups and Representations of Lie Groups.” He was a faculty member at Princeton from 1960 to 1967 and Yale from 1967 to 1972 before becoming a permanent faculty member at the Institute for Advanced Study, where he is now emeritus. After completing his Ph.D., he turned his attention to the theory of automorphic forms and to Eisenstein series. He is best known for the “Langlands program,” a series of conjectures originally presented in a 17-page handwritten letter to André Weil in 1967. According to Bill Casselman (quoted by J. J. O’Connor and E. F. Robertson in their MacTutor Archive article), the "Langlands conjectures" were a “collection of far-reaching and uncannily accurate conjectures relating number theory, automorphic forms, and representation theory.” (Sources: “2005 Steele Prizes,” *AMS Notices* 52:4 (April 2005), 439-442; IAS; MacTutor Archive)

Halmos photographed Anneli Lax (1922-1999) at the AMS Summer Meeting, held August 18-22, 1975, at Western Michigan University in Kalamazoo. Born in Katowice, Poland, Anneli Cahn Lax earned her Ph.D. in partial differential equations in 1955 from the Institute of Mathematics at New York University under advisor Richard Courant. (The Institute was renamed for Courant in 1964.) Lax joined the faculty at NYU in 1961 and retired in 1992. She is best known as founding editor of the New Mathematical Library, which began in 1958 as a series of monographs for high school students with talent and interest in mathematics. The NML was part of a nationwide response to the USSR's successful launch of its Sputnik satellite in 1957. It was intended to be temporary, but it proved to be popular and the MAA took it over in 1975. Lax served as its editor until her death in 1999 and, in 2000, the MAA renamed the series the Anneli Lax New Mathematical Library. Anneli and Peter Lax, a fellow NYU math grad student, were married in 1948. Richard Courant and Peter Lax are pictured together on page 10 of this collection. (Sources: MacTutor Archive, MAA Online: “The New Mathematical Library Records”)

Halmos photographed Peter Lax on November 4, 1969, in Bloomington, Indiana. Born in Budapest, Hungary, Peter Lax has spent most of his career at the Courant Institute at New York University, earning his Ph.D. there in 1949, becoming a faculty member in 1951, and directing the Institute during the 1970s. Now Professor Emeritus at Courant, Lax is an applied mathematician with wide-ranging interests, mostly centered around differential equations and numerical methods for solving them. He may be best known for developing scattering theory with Ralph S. Phillips; see page 9 of this collection for a photo of Phillips. Lax attributes his interest in applied mathematics and computing to his participation in the Manhattan Project at the Los Alamos National Laboratory in New Mexico during 1945-46 and 1950. He was president of the AMS during 1979-80. Another photograph of Lax appears on page 10 of this collection. (Sources: MacTutor Archive, AMS Presidents)

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