Floer Invariants, Cobordisms, and Contact Geometry

Floer 不变量、配边和接触几何

• 批准号: 2010863
• 负责人: Chuen Ming Mike Wong
• 金额: $ 12.28万
• 依托单位: Louisiana State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2020-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2010863&HistoricalAwards=false
• 关键词: Floer; Invariants; Cobordisms; Contact; Geometry

Low-dimensional topology is the study of geometric shapes and spaces in dimensions up to four, which has, perhaps unintuitively, proved to be more difficult than high-dimensional topology. Within this subject lies the theory of knots, loops of tangled string that are tied together at their ends, which has various connections to physics (via quantum theory and string theory), chemistry (via molecular knots), and biology (via DNA structure, with applications to drug design). To better understand knots and other geometric objects, topologists have invented tools called invariants. Some modern invariants are inspired by theoretical physics, such as gauge theory. On the one hand, these invariants have been successfully applied to solve many important and long-standing questions in topology; on the other hand, their behaviors are far from entirely understood. The goal of this research project is to further the development of modern invariants in terms of both theory and computation, harnessing their power to explore the link between low-dimensional topology and contact geometry, a related area of mathematics that has its roots in Newtonian mechanics and that has emerged as an exciting area of research in recent years. As part of this project, the investigator will provide research training to undergraduate and graduate students, make modern invariants accessible to a wide audience, and continue efforts in mathematical outreach. Floer theory, which encompasses instanton, monopole, and Heegaard Floer homologies, is a large package of invariants for three-manifolds and knots, as well as their cobordisms, that originate from gauge theory and symplectic geometry. In recent years, Heegaard Floer homology has been shown to be algorithmically computable, using combinatorial diagrams or bordered invariants. This project aims to harness the power of Floer invariants that comes from combining theory and computation, in several related directions. First, Floer theory provides information on the existence or non-existence of cobordisms between three-manifolds and between knots, with topological or geometric constraints. It is also known to be closely related to contact geometry, giving rise to invariants that certify tightness of a contact three-manifold, and distinguish smoothly isotopic knots that are not Legendrian isotopic. One goal of the project is to further extend these applications to cobordisms and contact geometry. To do so, the investigator aims to establish naturality results that will refine isomorphism class invariants to concrete homology group elements. The combinatorial diagrams involved will also shed light on the significant yet mysterious link between Floer theory and representation-theoretic invariants, which has been established in the form of spectral sequences. Similarly, the project also aims to advance bordered Floer invariants, which will activate more topological applications and significantly augment the use of the contact invariants above. This project is jointly funded by Topology and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

低维拓扑学是研究四维以下的几何形状和空间的学科，它被证明比高维拓扑学更难，这可能并不直观。在这门学科中有结的理论，即在末端绑在一起的缠绕的绳子的环，它与物理(通过量子理论和弦理论)、化学(通过分子结)和生物学(通过DNA结构，以及在药物设计中的应用)有各种联系。为了更好地理解结和其他几何对象，拓扑学家发明了称为不变量的工具。一些现代不变量受到理论物理的启发，例如规范理论。一方面，这些不变量已经被成功地应用于解决许多重要的和长期存在的拓扑学问题；另一方面，它们的行为还远未完全被理解。这个研究项目的目标是在理论和计算方面进一步发展现代不变量，利用它们的力量来探索低维拓扑和接触几何之间的联系，接触几何是一个相关的数学领域，起源于牛顿力学，近年来成为一个令人兴奋的研究领域。作为该项目的一部分，研究人员将为本科生和研究生提供研究培训，使广大受众能够接触到现代不变量，并继续在数学推广方面做出努力。Floer理论包括瞬子、单极和Heegaard Floer同调，是一大套关于三维流形和纽结的不变量，以及它们的余边线，起源于规范理论和辛几何。近年来，Heegaard Floer同调已被证明是可用组合图或边界不变量进行算法计算的。这个项目的目的是在几个相关的方向上利用Floer不变量的力量，这些不变量来自于理论和计算的结合。首先，Floer理论提供了关于具有拓扑或几何约束的三维流形之间和纽结之间存在或不存在上界的信息。众所周知，它与接触几何密切相关，产生了证明接触三维流形的紧密性的不变量，并顺利地区分了不是Legendrian同位素的同位素结。该项目的一个目标是进一步将这些应用扩展到COBORDISM和接触几何。为了做到这一点，研究人员的目标是建立自然性结果，将同构类不变量提炼为具体的同构群元。所涉及的组合图还将阐明弗洛尔理论和表示论不变量之间重要而神秘的联系，这种联系已经以谱序列的形式建立起来。同样，该项目还旨在推进有边界Floer不变量，这将激活更多的拓扑应用，并显著增加上述接触不变量的使用。该项目由拓扑学和已建立的激励竞争性研究计划(EPSCoR)共同资助。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。