Floer homology and surfaces in 3- and 4-manifolds

这个项目研究低维拓扑学中的各种问题，研究三维和四维空间的全球形状，其应用范围广泛，从天体物理(宇宙的形状)到生物化学(DNA分子的打结)再到理论物理。有关三维或四维空间的许多有趣的拓扑信息都被嵌入到该空间中的曲面类型中捕获，该项目旨在获得对可以在各种类型的空间中找到的此类曲面类型的限制。PI的主要工具来自Heegaard Floer同调，这是一套不变量，自从Ozsváth和Szabó在21世纪初引入以来，它已经成为低维拓扑研究的主导领域之一。这些技术汇集了几个不同的数学领域，包括表示论、同调代数、分析和数学物理，PI希望阐明这些不同领域之间的联系，并扩大这些领域的研究人员之间的讨论。这个项目的具体目标是(1)了解3-流形的Heegaard Floer同调与拓扑性质之间的关系，例如基本群上存在不可压缩曲面、张叶和左序；(2)解决关于4-流形中光滑嵌入曲面的具体拓扑问题，部分是通过使用由有界Floer同调实现的新的计算技术；以及(3)使用最近发展的纽结Floer同调和Khovanov同调的生成树模型来阐明这两个理论之间长期猜测的关系。

This project investigates a variety of questions in low-dimensional topology, the study of the global shapes of 3- and 4-dimensional spaces, which has a wide variety of applications ranging from astrophysics (the shape of the universe) to biochemistry (the knotting of DNA molecules) to theoretical physics. Much of the interesting topological information about a 3- or 4-dimensional space is captured in the types of surfaces that are embedded in that space, and this project aims to obtain restrictions on the types of such surfaces that can be found in various classes of spaces. The PI's primary tools come from Heegaard Floer homology, a package of invariants that has become one of the dominant areas of research in low-dimensional topology since its introduction in the early 2000s by Ozsváth and Szabó. These techniques bring together several different fields of mathematics, including representation theory, homological algebra, analysis, and mathematical physics, and the PI hopes to elucidate the connections between these different areas and expand the discourse among researchers in these fields. The specific goals of the project are (1) to understand the relationship between the Heegaard Floer homology of a 3-manifold and topological properties such as the existence of incompressible surfaces, taut foliations, and left-orderings on the fundamental group; (2) to solve concrete topological problems regarding smoothly embedded surfaces in 4-manifolds, in part by using new computational techniques made possible by bordered Floer homology; and (3) to use the recently-developed spanning tree models for knot Floer homology and Khovanov homology to shed light on the long-conjectured relationship between the two theories.