Singer entered the University of Chicago in January 1947, still intending to undertake research in physics. However, still thinking that he would be in a better position to make advances in physics if he had a strong mathematical background, he took a Master's degree in mathematics. He was awarded an M.S. in 1948 but by this time was so excited by mathematics that he decided to continue with research in that topic. He was awarded a Ph.D. in 1950. His thesis advisor at Chicago was Irving Segal and his thesis was entitled Lie Algebras of Unbounded Operators. R S Doran writes:-
At the time the department [at Chicago] was under the visionary leadership of Marshall Stone, who had assembled one of the finest mathematical faculties in the world. Senior members among this group included S S Chern, S Mac Lane, A Weil, and A Zygmund. Irving Kaplansky, Irving Segal, and Paul Halmos were active young researchers in the department with interests, among many other things, in operator theory and operator algebras. In addition, a large number of distinguished visitors and extraordinary graduate students came to Chicago to study with this illustrious faculty. This is the exciting initial environment underlying the work [Singer carried out].After the award of his doctorate, Singer was appointed C L E Moore Instructor at the Massachusetts Institute of Technology in 1950. After two years at MIT he was appointed as an Assistant Professor of Mathematics at the University of California, Los Angeles. In 1954 he went to the Columbia University where he spent the academic year 1954-55 as a Visiting Assistant Professor. The next academic year was also sent in a visiting position, this time at the Institute for Advanced Study at Princeton. His publications in these early years of his career include: (with Richard V Kadison) Some remarks on representations of connected groups (1952); Uniformly continuous representations of Lie groups (1952); ( with Warren Ambrose) A theorem on holonomy (1953); (with Richard Arens) Function values as boundary integrals (1954).
In 1953 Kaplansky asked Singer what derivations the algebra of continuous complex-valued functions on a compact Hausdorff space possessed. Singer had answered the question by the following day, showing the answer was 0, and this led to considerable work on the topic including further work by Singer, jointly with J Wermer, published in the paper Derivations on commutative normed algebras (1955) in which they showed that all bounded derivations on a semisimple commutative normed algebra are 0.
Singer returned to the Massachusetts Institute of Technology in 1954 where he was appointed as an Assistant Professor. He was promoted to Associate Professor in 1958, and then to full professor in 1959. In 1961 he married Sheila Ruff and they had a daughter Natascha Singer (who is a reporter for the NYTimes). That marriage broke up in the mid 70's and eventually he married Rosemarie and they had two daughters, Emily and Annabelle, in Berkeley. He was named Norbert Wiener Professor of Mathematics at MIT in 1970. In 1977 Singer went to the University of California at Berkeley where he spent two years as a Visiting Professor. Offered a permanent post at Berkeley, he resigned from MIT in 1979 taking up the professorship at Berkeley. He was named Miller Professor at Berkeley in 1982 but left in the following year to return to MIT where he was named John D MacArthur Professor of Mathematics. He was appointed as an Institute Professor in 1987.
Singer is justifiably famous among mathematicians for his deep and spectacular work in geometry, analysis, and topology, culminating in the Atiyah-Singer Index theorem and its many ramifications in modern mathematics and quantum physics.In the citation for the Steele Prize for Lifetime Achievement which Singer received in 2001, his work on the Atiyah-Singer Index theorem is highlighted :-
Singer's series of five papers with Michael F Atiyah on the Index Theorem for elliptic operators (which appeared in 1968 - 71) and his three papers with Atiyah and V K Patodi on the Index Theorem for manifolds with boundary (which appeared in 1975 - 76) are among the great classics of global analysis. They have spawned many developments in differential geometry, differential topology, and analysis ...In his reply to receiving the Steele Prize, Singer spoke of his collaboration with Michael Atiyah:-
My collaboration with Sir Michael Atiyah for more than twenty years has been very exciting, and our work continues to have great impact. Sir Michael is a remarkable human being who - mathematics aside - has devoted much time and energy in the support of science throughout the world. I've been fortunate in having many collaborators in mathematics and physics with whom I have enjoyed working and who have become close friends. It is a pleasure to acknowledge them, over thirty in number - too many to list here.The citation also mentions other outstanding contributions by Singer:-
However, [the Index Theorem] represents only a small part of his contributions to geometry and analysis. Other significant contributions to geometry were his work with D B Ray on analytic torsion, the precursor of much modern work on "determinant" invariants in geometry, and an influential textbook joint with J A Thorpe, Lecture Notes on Elementary Topology and Geometry .... Moreover, in addition to his work in pure mathematics, Singer has laboured for two decades to bring together mathematicians and theoretical physicists. This has been not simply a matter of interpersonal relations and seminar talks, but has entailed a long effort to understand, rework, and make available to mathematicians the deepest results of modern theoretical physics. This renaissance of serious interaction between mathematicians and physicists, which dates from the mid-1970s, has had a dramatic effect on mathematics, and Singer has played a major role in this development.Singer has been honoured with election to the National Academy of Sciences and to the American Academy of Arts and Sciences. Among the many awards which Singer has received, in addition to the Steele Prize, we mention the Bôcher Prize Memorial Prize from the American Mathematical Society (1969), the Eugene Wigner Medal (1988), the National Medal of Science (1983), the Award for Distinguished Public Service from the American Mathematical Society (1992), the Abel Prize (2004), and the James Rhyne Killian Faculty Achievement Award (2005). He served on the Council of the National Academy of Sciences, the Governing Board of the National Research Council, and the White House Science Council. Singer was vice president of the American Mathematical Society during 1970-72.
The 2004 Abel Prize was a joint award to Atiyah and Singer:-
... for their discovery and proof of the index theorem, bringing together topology, geometry and analysis, and their outstanding role in building new bridges between mathematics and theoretical physics.Teaching has also played a large role in Singer's life. He said :-
For me the classroom is an important counterpart to research. I enjoy teaching undergraduates at all levels, and I have a host of graduate students, many of whom have ended up teaching me more than I have taught them.His liking for teaching "at all levels" was emphasised in the citation for the Killian Award :-
He is perhaps the only American mathematician to hold a Distinguished University Professorship who regularly teaches ordinary (as opposed to Honours) first semester calculus.As to interests outside mathematics, Singer list literature, hiking and tennis. He said :-
I love to play tennis, and I try to do so two or three times a week. That refreshes me, and I think that it has helped me work hard in mathematics all these years.
Article by: J J O'Connor and E F Robertson