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.

低维拓扑学是研究几何形状和四维空间的学科，它被证明比高维拓扑学更困难，这也许并不直观。在这门学科中存在结理论，缠结的绳子在它们的末端绑在一起，它与物理学（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，应用于药物设计）有各种联系。为了更好地理解结和其他几何物体，拓扑学家发明了一种叫做不变量的工具。一些现代不变量的灵感来自理论物理，如规范理论。一方面，这些不变量已经成功地应用于解决许多重要的和长期存在的拓扑问题；另一方面，它们的行为还远未完全被理解。本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，利用它们的力量探索低维拓扑和接触几何之间的联系，这是一个相关的数学领域，其根源在于牛顿力学，近年来已成为一个令人兴奋的研究领域。作为该项目的一部分，研究者将为本科生和研究生提供研究培训，使更广泛的受众能够访问现代不变量，并继续努力进行数学推广。花理论包含了瞬子、单极子和heeggaard花同调，它是三流形和结点及其协点的不变量的一个大集合，起源于规范理论和辛几何。近年来，Heegaard flower同调已被证明是算法上可计算的，使用组合图或有边不变量。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，它来自于理论和计算的结合。首先，花理论提供了在拓扑或几何约束下，三流形之间和结点之间是否存在共体的信息。它也被认为与接触几何密切相关，产生了证明接触三流形紧密性的不变量，并平滑地区分非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协调和接触几何。为此，研究者旨在建立自然性结果，将同构类不变量细化到具体的同构群元素。所涉及的组合图也将揭示弗洛尔理论和表示理论不变量之间的重要而神秘的联系，这种联系已经以光谱序列的形式建立起来。同样，该项目还旨在推进边界Floer不变量，这将激活更多的拓扑应用，并显着增加上述接触不变量的使用。该项目由拓扑学和促进竞争研究的既定计划（EPSCoR）共同资助。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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