Floer Homology and Low-Dimensional Topology

低维拓扑是研究三维和四维空间及其定性几何性质的学科。这些空间的分类在今天仍然是一个基本问题。这个项目的主要主题是研究空间的拓扑性质使用某些代数不变量，称为Floer同调和Khovanov同调。这些不变量已经成为现代拓扑学的核心工具，并与辛几何、量子物理和生物学等领域有联系。除了研究部分，该项目还包括指导和推广工作计划，重点是增加数学科学中代表性不足的群体的数学可及性。本项目致力于通过进一步发展Floer同调和Khovanov同调技术来研究三维和四维流形。项目的第一部分是研究同调配合和结的协调，包括构造同调球中结的新的协调同态。第二部分研究了开卷分解的单一性和接触三流形的Stein填充性。第三部分研究了辛sl(n)同调的连杆不变量的性质及其与Khovanov-Rozansky同调的联系。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。

Low-dimensional topology is the study of spaces of dimensions three and four and their qualitative geometric properties. The classification of these spaces remains a fundamental problem today. The main theme of this project is to study topological properties of spaces using certain algebraic invariants, called Floer homology and Khovanov homology. These invariants have become central tools in modern topology and have connections to fields ranging from symplectic geometry to quantum physics to biology. In addition to its research component, the project includes plans for mentoring and outreach efforts, with a focus on increasing the accessibility of mathematics to groups underrepresented in the mathematical sciences. The project is devoted to studying three- and four-dimensional manifolds by further developing techniques in Floer homology and Khovanov homology. The first part of the project is to study homology cobordism and knot concordance, including constructing new concordance homomorphisms for knots in homology spheres. The second part of the project concerns properties of the monodromy of open book decompositions and Stein fillability of contact three-manifolds. The third part is to study properties of a link invariant called symplectic sl(n) homology and its connection to Khovanov-Rozansky homology.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.