Emmy Noether Lectures

1988 Lecturer: Karen Uhlenbeck

Moment Maps in Stable Bundles: Where Analysis Algebra and Topology Meet

Karen Keskulla born on August 24, 1942 in Cleveland, Ohio, graduated from the University of Michigan in 1964 and received her PhD from Brandeis University in 1968, under the direction of Richard Palais. She has taught at many universities and has held the Third Sid W. Richardson Foundation Regents’ Chair in Mathematics at the University of Texas at Austin since 1987.

Her numerous honors include a MacArthur Fellowship (1983) and election to the American Academy of Arts and Sciences (1985) and to the National Academy of Sciences (1986). She served as Vice-President of the American Mathematical Society, has sat on the editorial boards of ten research journals, and regularly serves as consultant to mathematics departments and foundations. In 1988, she received an honorary DSc degree from Knox College. In addition to her Noether Lecture, Uhlenbeck has delivered numerous invited lectures at major research centers, She presented the Colloquium Lectures of the American Mathematical Society at the Joint Summer Meetings in 1985. In Kyoto, Japan in 1990, she became the second woman to give a Plenary Lecture at an International Congress of Mathematics. The first woman to have this honor was Emmy Noether, who gave a lecture on algebra in the 1932 Congress.

Uhlenbeck’s mathematical interests include the calculus of variations, nonlinear partial differential equations, differential geometry, gauge theory, topological quantum field theory, and integrable systems. “This is a time of finding interrelationships within mathematics,” she said in an article on her Noether Lecture in the May-June 1988 issue of the AWM Newsletter. “As a student, I had many crushes on various mathematical subjects, but sequentially. Now I can study them all at once.” She points to the gauge-theoretic study of stable holomorphic bundles on complex Kahler manifolds as a prototype of such crossroads. This can be approached as pure geometry, but more insight is gained if modern techniques in partial differential equations are used. As algebra, it becomes an infinite dimensional version of invariant theory. Symplectic geometry also plays an important role: the equations for the moduli space are the Yang-Mills equations from physics, and the topology of the moduli space is studied via Morse theory for the Yang-Mills functional.

“The list of possible applications is formidable,” she says. Donaldson’s invariants for four-manifolds are probably the best known, but nearly as important are the calculations of Atiyah-Bott on the topology of the space of stable bundles over curves. This machinery has also been used to classify flat bundles, to study Hodge structures, and to investigate the interaction of magnetic monopoles. “One can always guess this might be of use in string theory, which seems able to absorb every kind of mathematics.”

“This is a lot of mathematics! Our graduate courses are not really designed to teach it, but graduate students seem to manage (better than I do, in fact) to absorb the necessary ideas.” On a more personal level, she says, “I find graduate students and contact with young mathematicians very rewarding. I have had ten graduate students and have worked with a number of excellent young mathematicians. One of the most disappointing aspects of my career is the fact that I have had only two women students, and too infrequently interact with young women mathematicians.” In the past few years, Uhlenbeck has become active in educational outreach programs through her involvement with the Utah-based Regional Geometry Institute (now the Park City-Institute for Advanced Study Mathematics Institute). Outside of mathematics, she is interested in her cats, gardening, hiking and the “intellectual world in general.”