Topology, Geometry and Physics

对于DMS-0104196，这项提议的三个主要研究人员将参与以下领域的研究：巴里·马祖尔将参与至少四个主要项目。第一部分是关于欧拉系统及其在代数几何、算术和解析数论中的应用。第二个项目将研究通过特殊曲线的模本征式的插补。第三个是关于数论中的ABC猜想，第四个是关于有理连通族的有理点。拉乌尔·博特将在数学和物理之间进行研究，包括与计算复杂性、等变上同调、微分同胚群和环变簇有关的研究课题。Clifford Taubes的主要项目是利用由其自对偶2-形式定义的伪全纯曲线来编码光滑的4-流形结构。第一个长期目标是从这些曲线读出光滑4-流形的不变量。第二个目标是使用它们来简化流形的表示。以下是关于这项研究的一些不太技术性的评论：Barry Mazur的工作涉及整数的性质以及它们之间的关系。一个典型的问题是要求满足某个给定方程的整数，比如a，带c。毕达哥拉斯的方程就是一个例子，它要求a和b的平方和等于c的平方和。这种关系有非常一般和自然的类(如现已证明的费马猜想中涉及的那些)，它们探讨了数论的核心；对这些方程的研究构成了马祖尔研究的核心。拉乌尔·博茨的工作探索了对称族可以用来简化数学和理论物理中的计算的方式。这些对称性涉及不同的结构，这种关系往往将一个系统的简单可理解的性质等同于另一个系统的神秘而不被理解的性质。克利福德·陶贝斯的重大项目探索了我们维度宇宙大规模结构的可能性。我们的宇宙有四个维度，这是通常的三个空间维度加上时间的第四个维度。也就是说，这项研究试图提供像我们这样的宇宙可能存在的大尺度结构的完整清单。正确地说，对于所有可能的三维空间，都有一个猜想列表，但对于四个维度的类似列表，至今还没有可信的猜测。

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.