Floer Invariants, Cobordisms, and Contact Geometry

Floer 不变量、配边和接触几何

• 批准号: 2238131
• 负责人: Angela Wu
• 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: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2238131&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.

低维拓扑是对多达四个尺寸的几何形状和空间的研究，这也许不直觉，这比高维拓扑更加困难。在这个主题中，是结的理论，即在其末端绑在一起的纠结弦的循环，与物理学（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，在药物设计中应用）具有各种联系。为了更好地了解结和其他几何对象，拓扑师发明了称为不变的工具。一些现代不变的人灵感来自理论物理学，例如仪表理论。一方面，这些不变的人已成功地用于解决拓扑中许多重要和长期存在的问题。另一方面，他们的行为远非完全理解。该研究项目的目的是在理论和计算方面进一步发展现代不变的人，并利用其探索低维拓扑与接触几何形状之间的联系，这是数学的相关领域，它扎根于牛顿力学，并且近年来已经成为一个令人兴奋的研究领域。作为该项目的一部分，研究人员将向大学生和研究生提供研究培训，使广大受众群体可访问现代不变性，并继续在数学宣传方面进行努力。浮子理论涵盖了Instanton，Monopole和Heegaard Floer同源性，是三个manifolds和nots的大量不变式，以及它们的共同体，源自仪表理论和符号几何形状。近年来，Heegaard Floer同源性使用组合图或边界不变性词被证明是可计算的算法。该项目的目的是利用来自多个相关方向将理论和计算结合起来的浮动不变性的力量。首先，浮子理论提供了有关三个manifolds之间以及结之间以及拓扑或几何约束之间的共同存在的存在或不存在的信息。众所周知，它与触点几何形状密切相关，这会导致证明触点三个manifold的紧密度的不变性，并区分非legendrian同位素的平滑同位素结。该项目的一个目标是将这些应用程序进一步扩展到同居和接触几何形状。为此，研究者旨在建立自然结果，以完善异构级不变性，以使同源组元素不变。所涉及的组合图还将阐明浮子理论与表示不变性之间的显着但神秘的联系，这些联系已以光谱序列的形式建立。同样，该项目还旨在推进边界的浮动不变性，这将激活更多的拓扑应用，并大大增加上面的触点不变性的使用。该项目由拓扑和启发竞争研究的既定计划共同资助。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的影响评估标准通过评估来支持的。

项目成果

Ribbon homology cobordisms

带状同源配边

• DOI: 10.1016/j.aim.2022.108580
• 发表时间: 2022
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Daemi, Aliakbar;Lidman, Tye;Vela-Vick, David Shea;Wong, C.-M. Michael
• 通讯作者: Wong, C.-M. Michael