Topology, Geometry and Physics

主要研究人员：C. Taubes， R. Bott和B. mazurbott教授将围绕两个主题进行研究。第一部分讨论了Witten、Axelrod-Singer和kontsevich在点的位形空间中引入的基于积分的嵌入不变量。一个特别的目标是理解卡森不变量作为位形空间积分，并理解这些积分在手术下的行为。第二个主题是将duistermaat - heckmann定理推广到对称对的变完全轨道。马祖尔教授的研究集中在四个项目上。第一部分研究了模特征型及其l函数的p进插值。第二章研究椭圆曲线Shararevich-Tate群中元素的可表示性问题。第三章在广义的背景下研究柯利瓦金的欧拉系统。第四个项目将研究一些与ABC猜想有关的“圆方法型问题”。Taubeswork教授主要研究两个主题。第一部分讨论了4流形上奇异辛形式的伪全纯曲线理论，这些形式是自对偶调和2形式。我们将分析这些曲线的规则性，并考虑它们在定义辛形式存在的障碍时的用途。第二个项目研究了塞伯格-威滕方程的各种推广的紧性问题。这里的目标是确定这些推广是否可以用来获得流形不变量。更通俗地说，博特教授将首先研究一系列新的不变量，这些不变量处理环路或曲面拟合到高维空间的不同方式。这类问题最近在物理学中出现在一些新颖的量子场理论中。博特教授项目的第二部分研究的是空间的对称性如何限制其整体结构。Mazur教授将首先研究abc猜想。这是一个有限的断言，可以控制一个广泛的方程集合的解的数量。（例如，出现在费马最后定理中的方程。）其次，mazur教授计划研究模形式的傅里叶系数。这个领域的结果对许多其他数学分支（群论、复函数理论，以及令人惊讶的理论物理）都有帮助。Taubes教授计划研究四维空间的行为，并开发区分这些空间的技术。例如，考虑到时间，我们的宇宙是四维的，它的大规模拓扑结构是未知的。在这种背景下，Taubes教授的研究涉及到那些可能的结构的分类。

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.