Floer Theories for 3-Manifolds

这个由美国国家科学基金会资助的研究项目是拓扑学，这是现代数学的一个中心领域。主要目标是利用理论粒子物理学(规范理论)的思想来开发研究三维几何形状的新工具。首席研究员还计划应用这些工具来研究哪些高维形状可以被三角化的问题，即分解成简单的碎片(类似于将曲面分解成三角形)。三角测量的应用超越了纯数学，例如在计算机图形学中。该项目还旨在支持研究生教育和向公众传播数学思想。用更专业的术语来说，该项目涉及与三个流形相关的Floer理论及其拓扑应用。特别是，主要研究者将研究Seiberg-Witten Floer稳定同伦类型、相关的Pin(2)-等变Seiberg-Witten Floer同调和对合Heegaard Floer同调。这些理论可以用来获得关于三维同调余边群的信息。反过来，同调共边给出了至少五维流形的三角剖分的分类。主要研究者还建议理解3-流形覆盖的Heegaard Floer同调的行为，并从SL(2，C)平坦联络的模空间发展3-流形不变量。

This NSF funded research project is in topology, a central field in modern mathematics. The main goal is to use ideas from theoretical particle physics (gauge theory)to develop new tools for studying three-dimensional geometric shapes. The principal investigator also plans to apply these tools to study the problem of which high dimensional shapes can be triangulated, that is, decomposed into simple pieces (similar to a decomposition of a surface into triangles). Triangulations have applications beyond pure mathematics, for example in computer graphics. The project also aims to support graduate education and to disseminate mathematical ideas to the general public.In more technical terms, the project concerns Floer theories associated to three-manifolds, and their topological applications. In particular, the principal investigator will investigate Seiberg-Witten Floer stable homotopy types, the associated Pin(2)-equivariant Seiberg-Witten Floer homology, and involutive Heegaard Floer homology. These theories can be used to get information about the homology cobordism group in three dimensions. In turn, homology cobordism gives insight into the classification of triangulations for manifolds of dimension at least five. The principal investigator also proposes to understand the behavior of Heegaard Floer homology for coverings of 3-manifolds, and to develop 3-manifold invariants from the moduli spaces of SL(2,C) flat connections.