Topology, Geometry and Physics

点击翻译按钮获取中文摘要

Abstract for DMS - 0104196 The three principle investigators on this proposal will beinvolved in research in the following areas: Barry Mazur will be involvedin at least four major projects. The first concerns Euler systems andtheir applications in algebraic geometry, arithmetic and analytic numbertheory. The second project will study interpolations of modulareigenforms via special curves. The third concerns the ABC conjecture innumber theory and the fourth concerns rational points on rationallyconnected varieties. Raoul Bott will conduct research at the interfacebetween mathematics and physics including research topics that relate tocomputational complexity, equivariant cohomology, diffeomorphism groupsand toric varieties. The major project of Clifford Taubes concerns thecoding of smooth 4-manifold structures by the pseudo-holomorphic curvesdefined by their self-dual 2-forms. The first long range goal is to readoff invariants of smooth 4-manifolds from these curves. The second goalis to use them to simplify a presentation of the manifold.Here are some comments of a less technical nature about thisresearch: Barry Mazurs work concerns properties of the integers andrelations between them. A typical question asks for integers, say a, band c, that satisfy some given equation. An example is the equation ofPythagoreas, which asks that the sum of the squares of a and b equal thatof c. There are very general and natural classes of such relations (suchas those involved in the now proved Fermat conjecture) that probe to theheart of the theory of numbers; and the study of such equations forms thecore of Mazurs research. Raoul Botts work probes the manner in whichfamilies of symmetries can be used to simplify computations in mathematicsand theoretical physics. These symmetries relate different structures andoften this relation equates a simple to understand property of one systemto a mysterious and not understood property of another. Clifford Taubesmajor project probes the possibilities for the large scale structure offour dimensional universes. Our universe has four dimensions, these theusual three dimensions of space plus time as the fourth. This said, theresearch seeks to provide a complete list of the possible large scalestructures for a universe such as ours. To put this in perspective, thereis a conjectured list for all possible three dimensional spaces, but thereis as yet no credible conjecture for a similar list for four dimensions.