Geometric Partial Differential Equations
During the academic year 1997-98, there was a full year program in Geometric PDE at the Institute. Karen Uhlebeck was in residence as Distinguished Visiting Professor for the year, and she served as primary organizer of the program.
The following is Karen Uhlenbeck's statement about the organization and goals of this program:
"Since most of my background is in geometric elliptic PDE and topology, it is likely that a lot of the participants will also have this background. Certainly, the program will reflect the interests of those who attend; nevertheless, I hope to branch out and have seminars running in integrable systems and evolution equations. Chuu-Lian Terng has agreed to come for the year. So the integrable systems seminar will offer an opportunity to learn more classical geometry as well as some good models for evolution equations. The interesting overlap between integrable systems and the various supersymmetric quantum field theory models which are at the heart of duality calculations in quantum field theory will not be neglected.
"The status of non-linear geometric hyperbolic equations inevitably surprises most geometers, since the subject is not nearly as well understood as the stationary elliptic theory. It seems as if the basic estimates in linear PDE are not yet adequate, and there are few useful stable models for blow-up. In particular, neither the conformally invariant problems, nor the convex problems above the critical dimension are understood. Klainerman's preliminary analysis of lower order terms is a convincing argument that the geometry of lower order terms is far richer in the hyperbolic than in the elliptic theory. We will run a weekly seminar inviting the few experts in geometric hyperbolic equations and singularity theory to explain their ideas on the subject, and see what happens. Because the theory is still rather undeveloped, one cannot tell what type of mathematics will be useful and relevant.
"I also would like to run a series of expository talks on numerical computation, as it seems about time that the geometric PDE people got a piece of this action. It is not clear to me how feasible it is to get useful expository information out of the scientific computation community, so the idea is to keep this to a modest once a month or so, and hope we can find the right expositors. Suggestions are very welcome!