Topology, Geometry and Physics

摘要建议：DMS-9803241主要研究人员：C.Taubes，R.Bott和B.MazurBott教授的研究将集中在两个主题上。第一类是由Witten，Axelrod-Singer和Kontsevich提出的基于点的位形空间上的积分的嵌入不变量。一个特别的目的是理解卡森不变量作为位形空间积分，并了解这些积分在手术中的表现。第二个主题旨在将Duistermaat-Heckmann定理推广到对称对的变分完备轨道。Mazur教授的研究集中在四个项目上。第一个项目研究模本征形式及其L函数的p-进插值问题。第二部分研究椭圆曲线的Shararevich-Tate群中元素的可表现性问题。第三章从一般理据的角度研究了柯利维林的欧拉系统。第四个项目将研究与ABC猜想有关的某些“圆法类问题”。陶贝斯沃克教授围绕两个主题展开研究。第一部分是关于四维流形上的奇异辛型的伪全纯曲线理论，它是由自对偶调和二维型产生的。将分析这些曲线的规律性，并考虑它们在定义辛形式存在的障碍方面的用途。第二个项目研究了Seiberg-Witten方程的各种推广的紧性问题。这里的目标是确定这些一般化是否可以用来获得流形不变量。更通俗地说，Bott教授将首先研究一系列新的不变量，这些不变量处理将环或曲面拟合到更高维空间的不同方式。这类问题最近在一些新的量子场理论中从物理学中出现。博特教授项目的第二部分研究了空间的对称性如何制约其全球结构。Mazur教授将首先研究abc猜想。这是一个有限断言，它可以支配一系列方程的解的数量。(例如，出现在费马大定理中的方程。)其次，Mazur教授计划研究模形式的傅立叶系数。这是一个对许多其他数学分支(群论、复函数论，令人惊讶的是，理论物理学)都有使用结果的领域。陶布斯教授计划研究我们维度空间的行为，并开发技术来区分这些空间。例如，包括时间在内，我们的宇宙是4维的，其大规模的拓扑结构尚不清楚。在这种背景下，陶布斯教授的研究关注的是对那些可能的结构进行分类。

AbstractProposal: DMS-9803241Principal Investigators: C. Taubes, R. Bott, and B. MazurProfessor Bott's research will center on two topics. The first dealswith imbedding invariants based on integrals on configuration spacesof points which were introduced by Witten, Axelrod-Singer andKontsevich. A particular aim is to understand the Casson invariant asa configuration space integral, and to understand how these integralsbehave under surgery. The second topic aims to extend theDuistermaat-Heckmann theorem to variationally complete orbits ofsymmetric pairs. Professor Mazur's research centers on four projects.The first studies the p-adic interpolation of modular eigenforms andtheir L-functions. The second studies the question of therepresentability of elements in the Shararevich-Tate group of ellipticcurves. The third studies the Euler systems of Kolyvagin in a generalmotivic context. The fourth project will study certain "circlemethod-type questions" related to the ABC conjecture. Professor Taubeswork centers on two topics. The first deals with the theory ofpseudo-holomorphic curves for the singular symplectic forms on4-manifolds which arise as self-dual harmonic 2-forms. The regularityof these curves will be analyzed, their use in defining obstructionsto symplectic form existence will be considered. The second projectstudies the compactness question for various generalizations of theSeiberg-Witten equations. The goal here is to determine whether thesegeneralizations can be used to obtain manifold invariants.More colloquially, Professor Bott will study, first, a series of newinvariants which deal with the different ways in which a loop orsurface can be fitted into a higher dimensional space. These sorts ofquestions have arisen recently from physics in some novel quantumfield theories. The second part of Professor Bott's project studiesthe ways in which the symmetries of a space constrain its globalstructure. Professor Mazur will be studying, first, the ABCconjecture. This is a finiteness assertion that may govern the numberof solutions to a broad collection of equations. (For example, theequations which occur in Fermat's last theorem.) Second, ProfessorMazur plans to study the Fourier coefficients of modular forms. Thisis an area with results of use to quite a number of other branches ofmathematics (group theory, complex function theory, and, surprisingly,theoretical physics.) Professor Taubes plans to study the behavior offour dimensional spaces and to develop techniques to distinguish suchspaces from each other. For example, with time included, our universeis 4-dimensional and its large scale topological structure is notknown. In this context, Professor Taubes' research concerns theclassification of those structures which are possible.