Floer Theories for 3-Manifolds

3 流形的 Floer 理论

基本信息

批准号：
2028658
负责人：
Ciprian Manolescu
金额：
$ 8.21万
依托单位：
Stanford University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-09-01 至 2021-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2028658&HistoricalAwards=false
关键词：
Floer Theories Manifolds

项目摘要

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.

该NSF资助的研究项目是拓扑，这是现代数学中的中央领域。主要目标是使用理论粒子物理学（量规理论）的思想开发用于研究三维几何形状的新工具。主要研究者还计划应用这些工具来研究可以将高维形状的问题进行三角剖分的问题，即分解为简单的零件（类似于表面分解为三角形）。三角形具有纯数学以外的应用程序，例如在计算机图形中。该项目还旨在支持研究生教育并向公众传播数学思想。在更多的技术术语中，该项目涉及与三个manifolds相关的漂浮理论及其拓扑应用。特别是，首席研究人员将研究Seiberg-Witten的浮动稳定同型类型，相关的PIN（2） - 等级的Seiberg-Witten Floer同源性和参与的Heegaard Floer同源性。这些理论可用于在三个维度上获取有关同源性共生组的信息。反过来，同源性的同源性使人深入了解至少五个维度的三角分类的分类。首席研究人员还建议了解Heegaard Floer同源性的3个磁盘覆盖物的行为，并从SL（2，c）平坦连接的模量空间中开发出3个manifold不变性。