Emmy Noether Lectures

1994 Lecturer: Lesley Sibner

Analysis in Gauge Theory

Lesley Sibner is a native New Yorker. After a brief career in the theatre, she received her bachelor’s degree in Fine Arts from the City College of New York in 1959. She came to mathematics quite late, only discovering her enthusiasm for it after taking a required calculus course designed for liberal arts majors. While at City College, she met her husband, Robert Sibner, who at the time was a graduate student in mathematics and was teaching as a lecturer in the Physics Department. She went on to study at the Courant Institute of Mathematical Sciences, receiving a PhD in 1964 under the joint supervision of Lipman Bers and Cathleen Morawetz.
Lesley Sibner was an instructor at Stanford University (l965-l966), a Fulbright Scholar at the Institut Henri Poincare’ in Paris (1966-1967), an awardee of the Visiting Professorships for Women Program of the National Science Foundation (1986-1987), and a Bunting Science Scholar at Radcliffe College (1990-1991). Brooklyn Polytechnic University has been her home base since 1967.
Starting with her doctoral thesis on equations of mixed type, Sibner’s love of analysis, especially partial differential equations, has remained constant throughout her career. A few years after her PhD, she and her husband were led to study nonlinear Hodge theory following an informal discussion with Bers, in which he conjectured the existence of compressible flows on a Riemann surface. This added a new ingredient, differential geometry, to her mathematical interests. While a member of the Institute for Advanced Study during 1971-1972, she met and was greatly influenced by Michael Atiyah and Raoul Bott. She then pursued geometric problems which could be solved using analysis, and produced an “integral equations” proof of the Riemann-Roch Theorem during this period.
About this time, she met Karen Uhlenbeck, whose work on nonlinear variational problems had a significant effect on Sibner. Uhlenbeck suggested studying some of the analytic questions having to do with the Yang-Mills equations, which had just captured the interest of mathematicians and mathematical physicists. During a sabbatical at the Massachusetts Institute of Technology during 1979-1980, Sibner met Clifford Taubes, then a graduate student at Harvard, and learned from him a great deal about gauge field theory. Sibner went on to prove results about removing point singularities in the Yang-Mills and Yang-Mills-Higgs equations.
Her interest in singularities soon brought her deeper into geometry, leading to a classification of singular connections and to a condition for removing two-dimensional singularities. Realizing that instantons could under certain circumstances be viewed as monopoles, the Sibners and Uhlenbeck constructed non-minimal unstable critical points of the Yang-Mills functional over the four-sphere.
In her Noether Lecture, Lesley Sibner gave an overview of the way in which analytic techniques are used in gauge theory, with an emphasis on the Yang-Mills model and the Higgs model.