Floer homology and surfaces in 3- and 4-manifolds

该项目研究低维拓扑中的各种问题，研究3维和4维空间的整体形状，其应用范围从天体物理学（宇宙的形状）到生物化学（DNA分子的打结）到理论物理学。关于三维或四维空间的许多有趣的拓扑信息都被嵌入该空间的表面类型所捕获，本项目旨在获得对可以在各种空间中找到的此类表面类型的限制。PI的主要工具来自Heegaard Floer homology，这是一组不变量，自21世纪初由Ozsváth和Szabó引入以来，已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，并扩大这些领域研究人员之间的话语。该项目的具体目标是：（1）理解三维流形的Heegaard Floer同调与拓扑性质之间的关系，如不可压缩曲面的存在性，拉紧叶理和基本群上的左序;（2）为了解决四维流形中光滑嵌入曲面的具体拓扑问题，部分是通过使用新的计算技术，使有界Floer同调成为可能;（3）使用最近发展的生成树模型的结Floer同调和Khovanov同调，以阐明长期以来这两个理论之间的关系。

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.