Proposal: DMS-9803241 Principal Investigators: C. Taubes, R. Bott, and B. Mazur
Professor Bott's research will center on two topics. The first deals with imbedding invariants based on integrals on configuration spaces of points which were introduced by Witten, Axelrod-Singer and Kontsevich. A particular aim is to understand the Casson invariant as a configuration space integral, and to understand how these integrals behave under surgery. The second topic aims to extend the Duistermaat-Heckmann theorem to variationally complete orbits of symmetric pairs. Professor Mazur's research centers on four projects. The first studies the p-adic interpolation of modular eigenforms and their L-functions. The second studies the question of the representability of elements in the Shararevich-Tate group of elliptic curves. The third studies the Euler systems of Kolyvagin in a general motivic context. The fourth project will study certain "circle method-type questions" related to the ABC conjecture. Professor Taubes work centers on two topics. The first deals with the theory of pseudo-holomorphic curves for the singular symplectic forms on 4-manifolds which arise as self-dual harmonic 2-forms. The regularity of these curves will be analyzed, their use in defining obstructions to symplectic form existence will be considered. The second project studies the compactness question for various generalizations of the Seiberg-Witten equations. The goal here is to determine whether these generalizations can be used to obtain manifold invariants.
More colloquially, Professor Bott will study, first, a series of new invariants which deal with the different ways in which a loop or surface can be fitted into a higher dimensional space. These sorts of questions have arisen recently from physics in some novel quantum field theories. The second part of Professor Bott's project studies the ways in which the symmetries of a space constrain its global structure. Professor Mazur will be studying, first, the ABC conjecture. This is a finiteness assertion that may govern the number of solutions to a broad collection of equations. (For example, the equations which occur in Fermat's last theorem.) Second, Professor Mazur plans to study the Fourier coefficients of modular forms. This is an area with results of use to quite a number of other branches of mathematics (group theory, complex function theory, and, surprisingly, theoretical physics.) Professor Taubes plans to study the behavior of four dimensional spaces and to develop techniques to distinguish such spaces from each other. For example, with time included, our universe is 4-dimensional and its large scale topological structure is not known. In this context, Professor Taubes' research concerns the classification of those structures which are possible.
