Equivariant Floer Homology, Concordance, and Homology Cobordism

这个项目将研究拓扑学和几何学中的现象，使用现代数学的几个部分的工具。 这个项目的激励问题是关于四维几何形状必须满足什么样的约束。 解决这些问题的主要技术工具，数学规范理论，源于物理学，本项目的一部分将是关于进一步发展数学规范理论的技术。 另一个目标是支持研究生教育，并传播数学成果。更技术性的是，该项目将研究与结和三流形相关的Floer和Khovanov理论，并推导拓扑应用。 在弗洛尔理论中，最近有几个增强的强大和充分研究的弗洛尔同调的三维流形。 一方面，Hendricks-Manolescu定义了对合Floer同调，另一方面，Seiberg-Witten Floer谱也有了一些新的发展。 这些新的构造为Floer同调群的结构和应用提供了新的见解，但这些新的理论在实践中特别难以计算。 这个项目的重点是发展我们对这些和相关不变量的理解，并将它们应用于几何和拓扑学中的问题，特别是同调配边。 主要目标包括在Seiberg-Witten Floer同伦理论中证明手术精确三角形以及确定Seifert空间的Seiberg-Witten Floer同伦类型。 此外，主要研究者提出开发计算对合Floer同调的技术，以便在3-流形的同调配边群中发现新的现象。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project will study phenomena in topology and geometry, using tools from several parts of modern mathematics. The motivating questions for this project are about what constraints a four-dimensional geometric shape must satisfy. The main technical tool to approach such questions, mathematical gauge theory, originates from physics, and part of this project will be about further developing techniques in mathematical gauge theory. An additional goal is to support graduate education, and to disseminate mathematical results.More technically, this project will investigate Floer and Khovanov theories associated to knots and three-manifolds, and deduce topological applications. In Floer theory, there have recently been several enhancements of the powerful and well-studied Floer homologies of three-manifolds. On one hand, Hendricks-Manolescu have defined involutive Floer homology, and on the other there have been several recent developments for Seiberg-Witten Floer spectra. These new constructions have provided new insight into the structure and applications of Floer homology groups, but these new theories have been particularly difficult to compute, in practice. This project is centered on developing our understanding of these and related invariants, and applying them to questions in geometry and topology, especially on homology cobordism. Primary goals include proving surgery exact triangles in Seiberg-Witten Floer homotopy theory as well as determining the Seiberg-Witten Floer homotopy type of Seifert spaces. In addition, the principal investigator proposes to develop techniques for calculating involutive Floer homology, in order to find new phenomena in the homology cobordism group of 3-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.