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关于拓扑学的，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高维形状可以三角化的问题，也就是说，分解成简单的部分（类似于将一个表面分解成三角形）。三角剖分法在纯数学之外也有应用，例如在计算机图形学中。该项目还旨在支持研究生教育，并向公众传播数学思想。在更专业的术语中，该项目涉及与三流形相关的花理论及其拓扑应用。特别是，首席研究员将研究Seiberg-Witten Floer稳定同伦类型，相关的Pin(2)-等变Seiberg-Witten Floer同伦和对合Heegaard Floer同伦。这些理论可以用来在三维空间中得到关于同调配群的信息。反过来，同调协定给出了对至少五维流形的三角剖分分类的见解。作者还提出理解3流形覆盖的Heegaard花同调的行为，并从SL（2，C）平连接的模空间发展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.