New Invariants of Knots and 3-Manifolds

该奖项为拓扑学项目提供资金，拓扑学是现代数学的中心领域。主要目标是利用代数和理论粒子物理（规范理论）的思想来开发研究几何形状的新工具。PI计划应用这些工具来研究高维形状可以被三角化的问题，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。该项目将支持研究生教育，并帮助向公众传播数学思想。用更专业的术语来说，这个项目涉及到与结点和三流形相关的一些同调不变量。PI将研究Khovanov同调在除三球以外的流形中的结的扩展，并研究它们在各种四流形中关于协点和曲面的性质。在另一个方向上，PI将使用Pin(2)-等变Seiberg-Witten Floer稳定同伦和对合Heegaard Floer同伦来更多地了解三维同调配群的结构，从而揭示高维流形三角剖分的分类。此外，PI的目的是构造结花稳定同伦类型和加花稳定同伦类型。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识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 award provides funding for a project in topology, a central field in modern mathematics. The main goal is to use ideas from both algebra and theoretical particle physics (gauge theory) to develop new tools for studying geometric shapes. The PI 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 will support graduate education and help disseminate mathematical ideas to the general public.In more technical terms, the project concerns some of the homological invariants that are associated to knots and three-manifolds. The PI will investigate extensions of Khovanov homology to knots in manifolds other than the three-sphere, and study their properties with regard to cobordisms and surfaces in various four-manifolds. In a different direction, the PI will use Pin(2)-equivariant Seiberg-Witten Floer stable homotopy and involutive Heegaard Floer homology to understand more about the structure of the three-dimensional homology cobordism group, and thus shed light on the classification of triangulations of high-dimensional manifolds. Additionally, the PI aims to construct knot Floer stable homotopy types and Heegaard Floer stable homotopy types.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.