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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 Libuše Löwig and Henry F. J. Löwig (1904-1995) on May 22, 1972, during a visit to the University of Alberta in Edmonton, Alberta, Canada, where Henry Löwig was a mathematics professor. Heinrich Löwig earned his Ph.D. in 1928 from the German Technical University in Prague in what is now the Czech Republic with the dissertation “On Periodic Difference Equations.” During the 1930s he taught at the German University in Prague and at a variety of German-language schools. During World War II, he was interned in German labor camps. In 1948, he moved to Hobart, Australia to teach at the University of Tasmania and, in 1957, he joined the faculty at the University of Alberta, becoming Professor Emeritus in 1972. According to MathSciNet, he published mathematical papers from 1931 to 1981, on the topics of difference and differential equations during the first decade, and mainly on algebra and lattices from 1941 (when he became “Henry”) onward. (Sources: Introduction to *Forgotten Mathematician Henry Löwig (1904-1995)* (2012), Mathematics Genealogy Project, MathSciNet)

Eugene Lukacs (1906-1987), left, and Lipman (Lipa) Bers (1919-2004) were photographed in April of 1973 at an AMS Council meeting in New York. Two photos of Bers appear on page 5 of this collection, where you can read more about him.

Born in Hungary, Lukacs grew up in Vienna, Austria, and earned his Ph.D. in geometry in 1930 from the University of Vienna. After teaching secondary school and working as an actuary in Vienna, Lukacs emigrated to the U.S. in early 1939 after the Anschluss. There, his friend Abraham Wald interested him in statistics and probability research and, by 1942, he had made important advances in statistics and was publishing papers on probability with Otto Szasz. After teaching at various colleges in Cincinnati, Ohio, and working briefly for the National Bureau of Standards and the Office of Naval Research, Lukacs joined the faculty at the Catholic University of America in Washington, D.C., in 1955 and made CUA a center of statistical research. In 1972 he returned to Ohio to teach at Bowling Green State University. During the 1960s and 1970s, he held many visiting positions, mainly in Europe. (Source: MacTutor Archive)

Halmos photographed Gunter Lumer (1929-2005) in June of 1990 at Stanford University in Palo Alto, California. Lumer earned his Ph.D. in 1959 from the University of Chicago under advisor Irving Kaplansky (pictured on page 26 of this collection). He and Halmos first met at the University of Montevideo, Uruguay, where Halmos was a visiting professor during the 1951-52 academic year and Lumer one of two students who worked closely with Halmos during that year. Halmos later wrote (1985, p. 187):

Gunter Lumer was born in Germany, received some of his early education in France (where he was Guy Lumer), and went to university in Uruguay (which was the permanent haven his parents found away from Hitler). He ... [has] a wide grin, and a vivacious manner; he is always in motion. ... His positive personality was visible even in his mathematical attitudes, even when he was a young student. ... He was energetic and talented enough that it’s not at all clear I really did him any good; I am sure he would have become a mathematician no matter what I did.

According to the second source listed below, after receiving his Ph.D. from the University of Chicago in 1959, Lumer spent one year each at the University of California, Los Angeles, and Stanford University in Palo Alto, California, before joining the mathematics faculty at the University of Washington in Seattle in 1961, where he remained until 1974. In 1973, he became a member of the faculty at the Université de Mons-Hainaut, Belgium, and, in 1999, he added a position at the Solvay Institutes for Physics and Chemistry in Brussels, Belgium. He would serve in both of these positions until his death in 2005. Indeed, in 1985, Halmos wrote that Lumer had traveled to a 1980 conference at Oberwolfach from Belgium (p. 386) and that he currently was “doing hard Hardy spaces in Belgium” (p. 188). Lumer's primary research interests were functional analysis, partial differential equations, and evolution equations. Sources:

- Paul Halmos,
*I Want To Be A Mathematician,*Springer, 1985, pp. 187-9, 386; - "Life and Work of Günter Lumer,"
*Functional Analysis and Evolution Equations: The Günter Lumer Volume,*Birkhäuser, 2008, pp. ix-x (available for download from Springer).

Halmos photographed, left to right, Maxwell Reade, Roger Lyndon (1917-1988), and Frieda (?) Lyndon (Halmos identified the two on the right only as “Lyndon^{2}” on the back of the photograph) on June 7, 1967, at the University of Michigan in Ann Arbor (“A^{2}”). Reade, Roger Lyndon, and Halmos were faculty members at the University of Michigan at the time, with Reade serving from 1946 to 1986, Lyndon from 1953 to 1988, and Halmos from 1961 to 1968.

Complex analyst Maxwell Reade earned his Ph.D. in 1940 from the Rice Institute (now Rice University) in Houston, Texas. After teaching at Ohio State University and Purdue University, he joined the mathematics faculty at the University of Michigan in 1946, where he has been both a very popular teacher and a researcher in complex function theory throughout his career and where he became Professor Emeritus in 1986. Besides mathematical analysis, he and Halmos had photography as a common interest. The University of Michigan’s African American Music Collection includes the Maxwell O. Reade Collection of Early Jazz and Blues Recordings. For a description of mathematics faculty life at the University of Michigan from 1946 to 1960 from the perspective of a faculty spouse, see Marjorie Reade’s “What Was It Like Then? (Post War 1946-1960).” (Sources: Mathematics Genealogy Project, University of Michigan Faculty History Project, University of Michigan Library)

Group theorist Roger Lyndon earned his Ph.D. in 1946 from Harvard University under advisor Saunders Mac Lane, with the dissertation, “The Cohomology Theory of Group Extensions.” From 1946 to 1948, he worked in the Office of Naval Research and, from 1948 to 1953, he was on the mathematics faculty at Princeton University, where he became interested in combinatorial group theory. He also was interested in logic and model theory, and his three books, *Notes in Logic *(1966); *Combinatorial Group Theory**, *co-authored by Paul Schupp (1976); and *Groups and Geometry* (1985) represent his main mathematical interests well. The latter two were especially influential and authoritative. From 1953 onward, Lyndon was a mathematics professor at the University of Michigan, becoming Professor Emeritus in 1988. (Sources: MacTutor Archive, University of Michigan Faculty History Project)

Halmos photographed Saunders Mac Lane (1909-2005) in 1958. Mac Lane earned his Ph.D. in 1934 from the University of Göttingen under advisors Hermann Weyl and Paul Bernays. He was on the faculty at Harvard University from 1938 to 1947, and at the University of Chicago from 1947 to 1982, serving as department chair from 1952 to 1958, when this photo was taken. Halmos was on the faculty at Chicago from 1946 to 1961.

Mac Lane’s primary research areas were homological algebra and category theory, and he advised at least 41 Ph.D. students during his career, most of them at the University of Chicago. His first doctoral student was Irving Kaplansky (pictured on page 26 of this collection), who earned his Ph.D. in 1941 at Harvard and joined the Chicago faculty in 1945. Mac Lane may be best known for his textbook, *A Survey of Modern Algebra,* co-authored with Garrett Birkhoff in 1941. Mac Lane was MAA president during 1951-52 and AMS president during 1973-74. (Sources: MacTutor Archive, Mathematics Genealogy Project, MAA Presidents, AMS Presidents)

Halmos photographed Saunders Mac Lane (1909-2005) again in March of 1982 in Bloomington, Indiana.

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. Permission to reproduce photos must be obtained from the 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 32," *Convergence* (January 2012), DOI:10.4169/loci003801