Low-Dimensional Topology via Bordered Floer Theory

这项低维拓扑学的研究项目研究了各种三维空间的整体形状以及其中的纽结曲线和曲面。由于我们居住的空间是三维的，这一领域有着广泛的应用，从理解宇宙的可能形状到描述聚合物和DNA分子的打结。理解三维空间的属性需要一系列复杂的工具，这些工具来自数学中的一系列学科，包括代数拓扑学、几何学、分析和表示理论。每个研究三维空间的工具都带有一个概念，即哪些空间“简单”，哪些“复杂”。这个项目的主要目标是探索用于描述三维空间的不同工具和相应的简单性概念之间的关系。最终，研究人员希望巩固和加深我们对这些空间的知识，并在不同的数学领域之间建立桥梁。在这项研究中使用的一个关键技术工具是边界Heegaard Floer同调，它是具有边界的三维流形的Heegaard Floer同调的一个版本。特别是，该项目的中心是建立一个新的几何解释的边界Heegaard Floer不变量；这是一个具体的实现之间的深层联系的边界Heegaard Floer同调和某些Fukaya范畴。在环面边界的情况下，不变量可以解释为边界环面中的装饰浸入曲线。这个框架极大地简化了计算，并导致了有趣的粘合结果的证明。这将应用于L空间(相对于Heegaard Floer同调“单”的3-流形)的分类。作为一个应用，研究者试图证实一个猜想，该猜想等同于三维流形的三个简单性度量：是L空间，具有不可左序的基本群，以及不允许有可共向的紧分层。该项目的其他目标包括限制L空间节点的可能分解，并将Heegaard Floer同调与Jaco-Shalen-Johannson分解的复杂性联系起来。

This research project in low-dimensional topology investigates the global shape of various three-dimensional spaces and of knotted curves and surfaces within them. Since the space we inhabit is three-dimensional, this field has wide-ranging applications, from understanding the possible shapes of the universe to describing the knotting of polymers and DNA molecules. Understanding the properties of three-dimensional spaces requires an array of sophisticated tools from a range of disciplines within mathematics, including algebraic topology, geometry, analysis, and representation theory. Each tool for studying three-dimensional spaces carries with it a notion of which spaces are "simple" and which are "complicated." The primary goal of this project is to explore the relationship between different tools used to describe three-dimensional spaces and the corresponding notions of simplicity. Ultimately the investigator hopes to consolidate and deepen our knowledge of these spaces and to build bridges between a variety of mathematical fields.A key technical tool to be used in this investigation is bordered Heegaard Floer homology, a version of Heegaard Floer homology for 3-manifolds with boundary. In particular, the project centers on developing a new geometric interpretation of bordered Heegaard Floer invariants; this is a concrete realization of a deep connection between bordered Heegaard Floer homology and certain Fukaya categories. In the case of torus boundary, the invariants may be interpreted as decorated immersed curves in the boundary torus. This framework greatly simplifies computations and leads to proofs of interesting gluing results. This will be applied to the classification of L-spaces (3-manifolds that are "simple" with respect to Heegaard Floer homology). As one application, the investigator seeks to confirm a conjecture equating three measures of simplicity for 3-manifolds: being an L-space, having non-left-orderable fundamental group, and not admitting a co-orientable taut foliation. Other goals of the project include restricting the possible decompositions of L-space knots and relating Heegaard Floer homology to the complexity of the Jaco-Shalen-Johannson decomposition.