Topology, Geometry and Physics

Harvard University, Cambridge MA

Investigators

Abstract

Abstract for DMS - 0104196 The three principle investigators on this proposal will be involved in research in the following areas: Barry Mazur will be involved in at least four major projects. The first concerns Euler systems and their applications in algebraic geometry, arithmetic and analytic number theory. The second project will study interpolations of modular eigenforms via special curves. The third concerns the ABC conjecture in number theory and the fourth concerns rational points on rationally connected varieties. Raoul Bott will conduct research at the interface between mathematics and physics including research topics that relate to computational complexity, equivariant cohomology, diffeomorphism groups and toric varieties. The major project of Clifford Taubes concerns the coding of smooth 4-manifold structures by the pseudo-holomorphic curves defined by their self-dual 2-forms. The first long range goal is to read off invariants of smooth 4-manifolds from these curves. The second goal is to use them to simplify a presentation of the manifold. Here are some comments of a less technical nature about this research: Barry Mazurs work concerns properties of the integers and relations between them. A typical question asks for integers, say a, b and c, that satisfy some given equation. An example is the equation of Pythagoreas, which asks that the sum of the squares of a and b equal that of c. There are very general and natural classes of such relations (such as those involved in the now proved Fermat conjecture) that probe to the heart of the theory of numbers; and the study of such equations forms the core of Mazurs research. Raoul Botts work probes the manner in which families of symmetries can be used to simplify computations in mathematics and theoretical physics. These symmetries relate different structures and often this relation equates a simple to understand property of one system to a mysterious and not understood property of another. Clifford Taubes major project probes the possibilities for the large scale structure of four dimensional universes. Our universe has four dimensions, these the usual three dimensions of space plus time as the fourth. This said, the research seeks to provide a complete list of the possible large scale structures for a universe such as ours. To put this in perspective, there is a conjectured list for all possible three dimensional spaces, but there is as yet no credible conjecture for a similar list for four dimensions.