Topology, Geometry, and Physics

奖项：DMS 1708310，首席研究员：克利福德·H·陶布斯该项目的近期目标是开发新的工具和方法来研究三维和四维空间的结构。关于这些空间的问题是当前许多几何学和拓扑学研究的核心(更不用说物理学和天体物理学/宇宙学--四维是相关的，因为空间与时间是四维的)。研究这些空间已经有了强大的数学工具，但还需要更多的工具。对于四维空间来说尤其如此，因为这里的无知是广泛而深刻的。这种信息的缺乏促使该项目开发新工具并改进旧工具，同时牢记更多地了解所涉及的空间的长期目标。即将开展的研究可以在拓扑学、微分几何、辛几何和数学物理等领域之间架起桥梁。这个项目的数学领域的概念现在被广泛地(和集中地)用于凝聚态物理、高能物理以及重力和宇宙学理论。这个项目的长期目标是通过以下主要研究领域来了解光滑、低维流形的结构：(1)将在四维流形上研究Seiberg-Witten方程的某些推广的解序列的行为。这方面的最终目标是确定非收敛序列是否具有阻碍解计数定义的不变量的病理，这些不变量可以区分不同的四维流形，或者当四维流形有边界时，区分边界上的不同结或环。(2)研究自旋丛的调和截面的零点轨迹的结构和拓扑意义，该零点轨迹仅定义在截面不为零的地方。第二个研究焦点对第一个有影响，因为在研究某些感兴趣的方程的解序列时，会出现所描述的那种调和部分。(3)第三个研究重点是寻找能够区分具有反自对偶Weyl曲率的4-流形上黎曼度量空间的路分量的不变量。特别令人感兴趣的是区分具有双曲三维流形的圆的乘积的不变量，因为这样做的不变量可能区分某些对同胚但不一定是微分同胚的四维流形，这些流形不能用现有的光滑结构工具来分离。所有这些主题都使用了微分几何和分析的方法来构建研究低维流形的微分拓扑的新工具。

Award: DMS 1708310, Principal Investigator: Clifford H. TaubesThe immediate goal for the project is to develop new tools and methods to study the structure of three and four dimensional spaces. Questions about these spaces are central to much of current research in both geometry and topology (not to mention physics and astrophysics/cosmology - four dimensions is relevant because space with time is four dimensional). There exist powerful mathematical tools for studying these spaces, but more tools are needed. This is especially true for four dimensional spaces because the ignorance here is broad and deep. This lack of information motivates the project to develop new tools and sharpen the old ones while keeping in mind the long term goal of understanding more about the spaces involved. The research to be carried out could construct bridges between the fields of topology, differential geometry, symplectic geometry and mathematical physics. Concepts from the mathematical fields of the project are now used extensively (and centrally) in condensed matter physics, high energy physics, and theories of gravity and cosmology.The long term goal of this project is to understand the structure of smooth, low dimensional manifolds through the following primary areas of study: (1) The behavior of sequences of solutions to certain generalizations of Seiberg-Witten equations will be studied on four dimensional manifolds. The ultimate goal in this regard is to determine whether non-convergent sequences have pathologies that obstruct solution counting definitions of invariants that can distinguish different 4-manifolds or, when the four dimensional manifold has a boundary, distinguish different knots or links on the boundary. (2) The structure and topological significance of the zero locus of a harmonic section of a spin bundle that is defined only where the section is not zero will be investigated. This second research focus has implications for the first because harmonic sections of the sort described arise in the study of sequences of solutions to certain equations of interest. (3) The third research focus will look for invariants that could distinguish path components of the space of Riemannian metrics on a 4-manifold with anti-self dual Weyl curvature. Of particular interest would be an invariant that distinguishes the products of a circle with hyperbolic three-dimensional manifolds because an invariant that does this might distinguish certain pairs of homeomorphic, but not necessarily diffeomorphic, four dimensional manifolds that can not be separated with existing tools for smooth structures. All of these topics use methods from differential geometry and analysis to construct new tools for studying the differential topology of low dimensional manifolds.