Low-Dimensional Topology via Bordered Floer Theory

这个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打结曲线和曲面。由于我们居住的空间是三维的，这个领域有广泛的应用，从理解宇宙的可能形状到描述聚合物和DNA分子的打结。理解三维空间的特性需要一系列复杂的数学工具，包括代数拓扑、几何、分析和表示理论。研究三维空间的每个工具都带有一个概念，即哪些空间是“简单的”，哪些是“复杂的”。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概念之间的关系。最终研究者希望巩固和加深我们对这些空间的认识，并在各种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有边界的Heegaard flower同调，这是具有边界的3流形的Heegaard flower同调的一个版本。特别是，该项目集中于开发一种新的几何解释的边界Heegaard花不变量；这是边界heegard花同源性和某些深谷类别之间深层联系的具体实现。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没曲线。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 -空间的分类（相对于Heegaard花同调是“简单”的3-流形）。作为一个应用，研究者试图证实一个猜想等价于3流形的三个简单性度量：是l空间，具有非左序基群，不承认共取向紧叶。该项目的其他目标包括限制l空间结的可能分解，并将Heegaard flower同源性与jaco - shalen - johnson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.