### In Memoriam

#### Walter Rudin

1921-2010

Walter Rudin

Vilas Professor Emeritus Walter Rudin died after a long illness on May 20, 2010.

Walter Rudin was born in Vienna on May 2, 1921 to a well-established Jewish family. In the 1830’s, Walter’s great-grandfather Aron Pollak had become wealthy manufacturing matches, and was known for his charitable activities, including the building of a residence hall for needy students at the Technical University in Vienna. As a result he was knighted by Emperor Franz Joseph in 1869 and took the name Aron Ritter Pollak von Rudin. The family prospered, and Walter’s father Robert was a factory owner and electrical engineer, with a particular interest in sound recording and radio technology. After the Anschluss in 1938, the situation for Austrian Jews became impossible, and the Rudin family left Vienna.

Walter Rudin served in the British Army and Navy during the Second World War, and rejoined his parents and sister Vera in New York in late 1945. He entered Duke University, obtaining a B.A. in 1947 and a Ph.D. in mathematics in 1949. He was a C.L.E. Moore Instructor at the Massachusetts Institute of Technology, and began teaching at the University of Rochester in 1952. While on leave visiting Yale in 1958, Rudin received a call from RH Bing at the University of Wisconsin-Madison, asking if he would be interested in teaching summer school. Rudin said that since he had a Sloan Fellowship he wasn’t interested in summer teaching, but then, as he writes in his autobiography The Way I Remember It, “my brain slipped out of gear but my tongue kept on talking and I heard it say `but how about a real job?’” Walter Rudin joined the Department of Mathematics at UW-Madison in 1959, where he remained until his retirement as Vilas Professor in 1991. He and his wife Mary Ellen (Estill) Rudin, were popular teachers at both the undergraduate and graduate level and served as mentors for many graduate students. They lived in Madison in a house designed by Frank Lloyd Wright, and its intriguing architecture and two-story high living room made it a center for social life in the department.

Walter Rudin was one of the preeminent mathematicians of his generation. He worked in a number of different areas of mathematical analysis, and he made major contributions to each. His early work reflected his classical training, and focused on the study of trigonometric series and holomorphic functions of one complex variable. He was also very influenced by the then relatively new study of Banach algebras and function algebras. One of his important results in this area, building on the work of Arne Beurling, is the complete characterization of the closed ideals in the disk algebra in 1956.

Another major area of Rudin’s interest was the general theory of harmonic analysis on locally compact Abelian groups. In the late 1950’s and 1960’s this was a very active and popular area of research, and only partially in jest, Rudin suggested that mathematicians introduce a new word, `lgbalcag’, to replace the phrase “Let G be a locally compact Abelian group” which is how almost every analysis seminar began at that time. One of Rudin’s major achievements in this area was his 1959 work with Helson, Kahane and Katznelson, which characterized the functions that operate on the Fourier transforms of the L1-algebra. Rudin synthesized this aspect of his mathematical career in his 1962 book, Fourier Analysis on Groups.

Rudin’s interests changed again in the late 1960’s, and he began to work on problems in several complex variables. At that time the study of the analytic aspects of complex analysis in several variables was a relatively new and unexplored, and it was not even clear what the right several-variable generalization should be of the one-dimensional unit disk. There are at least two candidates: the polydisk and the ball. Rudin did important work with both. For example, he showed for the polydisk (1967) and the unit ball (1976) that, in contrast with the one-dimensional situation, the zero sets of different Hp classes of functions are all different. His work on the “Inner function conjecture” led to a tremendous amount of research, and after the solution by Alekasandrov and Hakim-Sibony-Løw (1981), Rudin made additional important contributions to this question. Much of Rudin’s work in several complex variables is presented in three of his advanced books. The first, published in 1969 is Function Theory in Polydiscs. The second, published in 1980, is Function Theory in the Unit Ball of Cn. His work on inner functions was summarized in a series of NSF-CBMS lectures, which were then published in 1986 as New Constructions of Functions Holomorphic in the Unit Ball of Cn.

Walter Rudin is also known to generations of undergraduate and graduate students for his three outstanding textbooks: Principles of Mathematical Analysis (1953), Real and Complex Analysis (1966), and Functional Analysis (1973). In 1993 he was awarded the American Mathematical Society Leroy P. Steele Prize for Mathematical Exposition. He received an honorary degree from the University of Vienna in 2006.

In addition to his widow Mary Ellen, Walter is survived by his four children: Catherine Rudin, Professor of Modern Languages and Linguistics at Wayne State College, Nebraska; Eleanor Rudin, an engineer working for 3M in St. Paul, Minnesota; Robert Rudin of Madison Wisconsin; and Charles Rudin, Professor of Oncology at the Johns Hopkins University in Baltimore. He is also survived by four grandchildren: Adem, Deniz, Sofia, and Natalie.

#### Philip Miles

1929-2011

Philip Miles

Emeritus Professor Philip E. Miles Jr. of the University of Wisconsin-Madison died in Madison on November 8, 2011.

Philip Miles was born in Madison on December 27, 1929 and after graduating from Madison's West High School attended Harvard University, graduating in mathematics in 1951. He worked at the Johns Hopkins Applied Physics Laboratory for a year before starting graduate school. Phil's graduate study began at Harvard but was interrupted by service in the army. In 1961 Phil completed his PhD at Yale, working with Charles Rickart. His thesis was in Functional Analysis, a strong mathematical field both at Yale and at the University of Wisconsin-Madison.

Loving Madison from his childhood, Phil wanted to return here after finishing graduate school. He joined the mathematics faculty of the University of Wisconsin-Madison in 1960 and was an active faculty member for 40 years. He spent the years 1964-1966 on leave in California as Executive Director of the Committee on Educational Media for the Mathematical Association of America. In a student interview from 1993, Phil described that project as making instructional mathematical movies for college undergraduates and others, and we can see in them issues we still find pressing: Pre-service training of elementary teachers, films for teaching Calculus, and automated learning. Phil had a deep interest in mathematics education, and was an early advocate of the use of new kinds of technology in teaching. However, while he was always open to new ideas and approaches, he coupled these interests with very high standards. He continued after retiring to be a very active participant and leader in the department's mathematics education seminar.

Professor Miles took on a demanding and difficult job as Associate Chair of the mathematics department for many years. The chair position rotates every few years, and by continuing as Associate Chair Phil provided continuity and great knowledge of how the university works. His knowledge and his commitment to excellence were of great assistance in making a big department with many diverse interests and backgrounds work smoothly.

In a 1976 interview conducted as a part of the UW Oral History Project, Phil discussed the ways that the university had already changed since he joined the department. When he came, the department was entering on a period of rapid expansion. (There had been a time in the recent past when the whole department could have gone down to Curt commons and sat around a single table and solved all their business over lunch.) The rapidly growing department was changing its view of itself, moving from a situation in which people thought in terms of face-to-face dealings to thinking in terms of group dealings. If you wanted to talk about a matter of department interest, you'd have a meeting for that purpose and do the talking at that meeting, as opposed to dropping into somebody else's office and saying "I've talked to X and he says this..."

In addition to his lifelong interest in mathematics and education, Phil loved to enjoy the natural world through canoeing, gardening, birding and walking. He was a devotee of classical music and opera, an avid film-goer, and an insatiable reader with wide-ranging tastes but a particular fondness for murder mysteries. He liked to travel, and shared many adventures around the world with his family. He was a prolific photographer, and leaves behind many boxes of slides. His dry and sometimes subtle sense of humor was enjoyed by family, friends, and colleagues, and persisted through his last days.

Phil is survived by his four children, Sara Miles Schiller, Katherine Miles Duren, Elizabeth Miles Waring, and Charles Erskine Miles, together with three stepchildren, nine grandchildren, and eight great-grandchildren, in addition to other close relatives and friends.