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 Alex Rosenberg (1926-2007), right, in 1980. Our first question this week is, who is the man on the left? Our second question: Who is the woman on the right?

Rosenberg earned his Ph.D. in 1951 from the University of Chicago with the dissertation “Subrings of Simple Rings with Minimal Ideals,” written under advisor Irving Kaplansky (pictured on page 26 of this collection). Halmos was on the Chicago faculty from 1946 to 1961 and he and Rosenberg would have first met there. After one year at the University of Michigan and nine years at Northwestern University, Rosenberg joined the faculty at Cornell University in Ithaca, New York, in 1961 and remained there until 1986, advising 17 of his 20 Ph.D. students there. After serving on the University of California, Santa Barbara, faculty from 1986 to 1994, he retired for good and, a few years later, moved to Germany, land of his birth, from whence his family had fled the Nazis in 1939, eventually settling in Canada. (Sources: Mathematics Genealogy Project; Spelman College obituary)

Rosenberg was editor of the MAA’s *American Mathematical Monthly* during 1974-1978. In preparation for the American bicentennial in 1976, Halmos agreed to assemble a team to prepare a presentation for the 1976 Winter Meetings on developments in American mathematics since 1940. In his *I Want to Be a Mathematician,* Halmos reported that the project was not without its challenges and that he and his co-authors breathed a “justified sigh of relief” (p. 399) when Rosenberg accepted their article, “American mathematics from 1940 to the day before yesterday,” for publication in the *Monthly* (83:7, August-September 1976, pp. 503-516). Halmos and one of his team members, John Ewing, would become *Monthly* editors themselves, during 1982-86 and 1992-96, respectively. (Sources: *I Want to Be a Mathematician: An Automathography,* Springer 1985, pp. 397-399; History of MAA Journals)

Halmos photographed George Rosenstein, Jr., on September 26, 1966, in Cleveland, Ohio, at Case Western Reserve University, where Rosenstein was on the faculty. Rosenstein earned his Ph.D. in 1963 from Duke University with the dissertation “Generalizations and an Extension of Lebesgue’s Covering Theorem.” After serving on the faculty at Case Western during the 1960s, he has spent most of his career at Franklin and Marshall College in Lancaster, Pennsylvania, where he is now Professor Emeritus of Mathematics. At Franklin and Marshall, he has been interested in the teaching of calculus, serving as Chief Reader for the College Board’s AP Calculus program during the late 1980s, writing the “reform” text *Discovering Calculus* with colleague Alan Levine in 1994, and writing the article, “One Hundred and Fifty Years of Teaching Calculus” in 2002. (Sources: Mathematics Genealogy Project, Franklin & Marshall College Mathematics)

Halmos photographed his Ph.D. student Peter Rosenthal, left, and another Ph.D. student, Robert Kopp, on September 10, 1964, in Ann Arbor, Michigan. Rosenthal earned his Ph.D. from the University of Michigan in 1967 with the dissertation “On Lattices of Invariant Subspaces.” He has spent most of his career as a mathematician at the University of Toronto, specializing in operators on Hilbert spaces. Since 1992, Rosenthal has had a parallel career as an attorney. He currently is Professor Emeritus of Mathematics at UT, an adjunct on the UT Faculty of Law, and a lawyer for a Toronto law firm, specializing in litigation for social change. (Sources: Mathematics Genealogy Project, University of Toronto Mathematics, UT Faculty of Law)

Rosenthal was the last of Halmos’ six Ph.D. students at the University of Michigan. In his *I Want to Be a Mathematician* (Springer 1985), Halmos recalled Rosenthal’s difficulty in addressing him as “Paul” rather than “Professor Halmos” after he had completed his Ph.D.:

I treasure a letter I got from Peter. Once we were finished with the student-teacher business, I told him to lay off the “Professor Halmos” stuff – but he found it hard to promote our relation to a first-name basis. In a letter there is no way out – you do, or you don’t – so he began it this way: “Dear P-P-P-P-Paul: There! I’ve said it!” (p. 278)

Robert Kopp earned his Ph.D. from the University of Michigan in 1966 with the dissertation “A Class of Banach Spaces of Analytic Functions on the Unit Disk,” written under advisor Allen Shields. He was one of Shields’ 29 Ph.D. students at Michigan. More information about Robert Kopp would be appreciated. (Source: Mathematics Genealogy Project)

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John Barkley Rosser (1907-1989), second from left, Paul Halmos, second from right, and Rosser’s family were photographed (possibly by Virginia Halmos) in 1958 on the Queen Elizabeth, perhaps en route to or from the International Congress of Mathematicians held in Edinburgh, Scotland, in August of that year.

After receiving bachelors and masters degrees from the University of Florida, John Barkley Rosser earned his Ph.D. in 1934 from Princeton University with the dissertation “A Mathematical Logic without Variables,” written under advisor Alonzo Church. His first academic posts were at Princeton and Harvard, but by 1941 he was on the faculty at Cornell University and, by about 1969, he had moved to the University of Wisconsin at Madison, where he directed the U.S. Army Mathematics Research Center and spent the rest of his career and life. His best known work may be his book, *Logic for Mathematicians,* originally published in 1953. Besides symbolic logic, he became an expert in numerical analysis, ballistics, and rocket development. (Sources: Mathematics Genealogy Project, Archives of American Mathematics biography)

Gian-Carlo Rota (1932-1999) was photographed by Halmos in August of 1961. Rota earned his Ph.D. in 1956 from Yale University with the dissertation “Extension Theory of Differential Operators,” written under advisor Jacob T. Schwartz (he was the first of Schwartz’s 28 Ph.D. students), but soon changed his research area from functional analysis to combinatorics. The American Mathematical Society cited his 1964 paper, “On the foundations of combinatorial theory,” as

the single paper most responsible for the revolution that incorporated combinatorics into the mainstream of modern mathematics (quoted in O’Connor’s and Robertson’s MacTutor Archive biography of Rota).

Rota’s first two academic posts were at New York University (1956-57) and Harvard University (1957-59). He spent the rest of his career at the Massachusetts Institute of Technology (MIT), where he was Professor of Applied Mathematics and, from 1972 onward, Professor of Applied Mathematics and Philosophy. He advised at least 50 Ph.D. students throughout the course of his career, most of them at MIT. His Ph.D. students included Peter Duren (page 13 of this collection), Kenneth Lange, and Richard Stanley. (Sources: MacTutor Archive, Mathematics Genealogy Project)

Halmos photographed Pierce Ketchum, left, and Lee A. Rubel in 1959.

Pierce Waddell Ketchum (1903-1993) earned his Ph.D. in 1926 from the University of Illinois at Urbana-Champaign with the dissertation “Functions of Hypercomplex Variables.” He immediately joined the faculty at Illinois and spent his entire career there, advising 20 Ph.D. students from 1947 to 1968 and becoming Professor Emeritus of Mathematics in 1969. Ketchum was Halmos’ first graduate analysis instructor at the University of Illinois during 1935-36. Somewhat surprisingly, Halmos struggled with analysis during his final year as an undergraduate at Illinois and most of his first year of graduate school there:

Matters did not immediately improve when I started graduate school. Pierce Ketchum tried to teach me some analysis, but I just didn’t catch on (

I Want to Be a Mathematician,p. 47).

Halmos credited his friend and fellow graduate student Warren Ambrose (pictured on page 1 of this collection) with showing him the light.

The day when the light dawned ... Ambrose and I were talking ... and something he said was the last candle that this blind camel needed. I suddenly understood epsilons and limits, it was all clear, it was all beautiful, it was all exciting. ... I had become a mathematician (p. 48).

(Sources: Mathematics Genealogy Project; Judy Green, Jeanne LaDuke, *Pioneering Women in American Mathematics,* AMS-LMS, 2009, pp. 218-219; Paul R. Halmos, *I Want to Be a Mathematician: An Automathography,* Springer 1985, pp. 47-48)

Lee A. Rubel (d. 1995) earned his Ph.D. in 1954 from the University of Wisconsin at Madison with the dissertation “Entire Functions and Ostrowski Sequences,” written under advisor R. Creighton Buck (for a photograph of Buck, see page 8 of this collection; for photos of Alexander Ostrowski, see page 38 and page 40). Rubel spent most of his career, from 1958 onward, at the University of Illinois at Urbana-Champaign, where his later research was in theoretical computer science; more specifically, the development of “mathematics to improve analog computers” (*New York Times* obituary). According to a computer scientist quoted in the same obituary,

Rubel’s theoretical blueprint has been as important a contribution to analog computing as Alan Turing’s machine was to digital computing (Jonathan W. Mills, Indiana University).

(Sources: Mathematics Genealogy Project, *New York Times* obituary)

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.