Low-Dimensional Topology via Bordered Floer Theory

通过有边弗洛尔理论的低维拓扑

• 批准号: 1711926
• 负责人: Jonathan Hanselman
• 金额: $ 10.26万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-15 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1711926&HistoricalAwards=false
• 关键词: Low; Dimensional; Topology; via; Bordered

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.

低维拓扑的该研究项目研究了各种三维空间以及内部打结的曲线和表面的全球形状。由于我们居住的空间是三维的，因此该领域具有广泛的应用，从了解宇宙的可能形状到描述聚合物和DNA分子的打结。了解三维空间的特性需要来自数学中一系列学科的复杂工具，包括代数拓扑，几何，分析和表示理论。每个用于研究三维空间的工具都带有一个概念，其中哪个空间“简单”并且“复杂”。该项目的主要目标是探索用于描述三维空间的不同工具与简单性的相应概念之间的关系。最终，调查人员希望巩固和加深我们对这些空间的了解，并在各种数学领域之间建造桥梁。在此调查中使用的关键技术工具与Heegaard Floer同源性接壤，Heegaard Floer同源性是Heegaard Floer同源性的，用于具有边界的3个manifolds。特别是，该项目集中于对接壤的Heegaard Floer不变性的新几何解释；这是对边界的Heegaard浮子同源性与某些福卡亚类别之间的深厚联系的具体实现。在圆环边界的情况下，不变性可以解释为边界圆环中装饰的沉浸式曲线。该框架大大简化了计算，并导致了有趣的粘合结果证明。这将应用于L空间的分类（相对于Heegaard Floer同源性“简单”的3个manifolds）。作为一种应用，研究人员试图确认一个等同于三个序曲的猜想：是L空间，具有不可左级的基本组，并且不承认可共定位的绷紧叶面。该项目的其他目标包括限制L空间结的可能分解，并将Heegaard Floer同源性与Jaco-Shalen-Johannson分解的复杂性联系起来。