Applications of Gauge Theory and Floer Homology to Low-Dimensional Topology

规范理论和Floer同调在低维拓扑中的应用

基本信息

批准号：
1811111
负责人：
Daniel Ruberman
金额：
$ 21.9万
依托单位：
Brandeis University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-09-01 至 2023-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1811111&HistoricalAwards=false
关键词：
Applications Gauge Theory Floer Homology

项目摘要

The understanding of the structure of the four-dimensional universe in which we live is a key topic of investigation in modern mathematics and physics. Many of the questions posed by geometers and topologists have to do with the nature of three-dimensional spaces sitting in a four-dimensional space, and with the restrictions that the global properties of larger space places on such sub-spaces. The research in this National Science Foundation funded project uses modern tools of analysis and geometry to shed light on the local nature of such subspaces, including new methods for showing that singularities in such spaces cannot be smoothed. Related analytical techniques will be used to explore the global topology of four-dimensional spaces, including an investigation of their symmetries.Daniel Ruberman will carry out research in geometric topology, using Seiberg-Witten gauge theory, Heegaard-Floer homology, and more traditional topological techniques. The first parts of the project, joint with Jianfeng Lin and Nikolai Saveliev, are concerned with the smooth topology of four-manifolds that homologically resemble a product of a three-dimensional manifold with a circle. Central questions concern the interpretation of the classical Rohlin invariant and multi-signature invariants in terms of gauge theory; solutions of the main problems will decide the existence of manifolds predicted by high dimensional surgery theory. The PI will work with Adam Levine on embeddings of punctured three-manifolds in four-space, using refined techniques from Heegaard Floer theory to find obstructions. An ongoing project with David Auckly, Hee Jung Kim, and Paul Melvin is concerned with the topology of the diffeomorphism group of a four-dimensional manifold and how it is affected by stabilization of the manifold. Finally, the PI will work with Saveliev and Demetre Kazaras on a new technique to obstruct the existence of positive scalar curvature cobordisms between even-dimensional manifolds.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

对我们生活的四维宇宙结构的理解是现代数学和物理学研究的关键主题。几何学家和拓扑家提出的许多问题与位于四维空间中的三维空间的性质有关，以及限制了此类子空间上较大空间的全球性能。这项国家科学基金会资助的项目的研究使用现代的分析工具和几何学工具来阐明此类子空间的当地性质，包括新方法，表明无法平滑此类空间中的奇异性。相关的分析技术将用于探索四维空间的全球拓扑，包括对它们的对称性的研究。DanielRuberman将使用Seiberg-Witten仪表理论，Heegaard-loer同源性和更多传统拓扑技术进行几何拓扑研究，并进行几何拓扑。该项目的第一部分与Jianfeng Lin和Nikolai Saveliev的关节关注的是，四个manifolds的平滑拓扑结构在杂题上类似于带有圆圈的三维歧管的产物。中心问题涉及对仪表理论的古典罗林不变和多签名不变的解释；主要问题的解决方案将决定由高维手术理论预测的流形的存在。 PI将使用Heegaard Floer理论中的精炼技术与Adam Levine一起在四个空间的刺穿三个序列的嵌入中使用，以发现障碍物。 Hee Jung Kim和Paul Melvin与David Auckly一起进行的一个正在进行的项目关注的是四维流形的差异群体的拓扑，以及如何受歧管的稳定影响。最后，PI将与Saveliev和Demetre Kazaras合作采用一种新技术，以妨碍均匀的歧义歧管之间存在积极的标量曲率恢复性。该奖项反映了NSF的法定任务，并认为使用基金会的知识分子和更广泛的影响审查Criteria，它被认为是值得通过评估的支持。