Lagrangians and Low-Dimensional Topology

拉格朗日和低维拓扑

• 批准号: 2105469
• 负责人: Tye Lidman
• 金额: $ 40.88万
• 依托单位: North Carolina State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-15 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2105469&HistoricalAwards=false
• 关键词: Lagrangians; Low; Dimensional; Topology

Topology is a mathematical field that seeks to understand the intrinsic properties of shape. This provides tools for many areas of science, including studying how enzymes knot up DNA or finding trends in large data sets. The aim of this project is to study the properties of knotted strings. One of the major goals is to understand mathematically how a knot changes if it is cut open and the ends are connected together differently, just as enzymes do to DNA. The prediction is that this always has a measurable change in the knottedness, and the PI seeks to verify this mathematically, enhancing understanding of the fundamental structure of knots. The PI's activities in this project, including mentorship and developing virtual mathematics communities, will create new educational opportunities for undergraduate and graduate students, with special focus on increasing diversity in and access to mathematics. The project will study key questions and structures in low-dimensional topology through a variety of invariants built from Lagrangian submanifolds. This includes using the description of knot Floer and Khovanov tangle invariants in terms of immersed Lagrangians in the four-punctured sphere to study the nugatory crossing conjecture, obtain lower bounds on the rank of knot homology theories in the presence of an essential Conway sphere, and prove detection results for the tangle invariants for certain classes of tangles. The PI will use the immersed Lagrangian invariants from bordered Heegaard Floer homology to give new bounds on the unknotting numbers of satellite knots. The PI will also use Lagrangians in the pillowcase arising from SU(2)-character varieties of knots towards the three-summands conjecture, which predicts that Dehn surgery on a knot in the three-sphere cannot consist of more than two prime summands. The PI will also mentor graduate students to develop new structural properties of symplectic instanton homology and to study the knot Floer homology of tunnel number one knots. In addition to organizing various seminars, the PI will assist in the development of the Floer homology open problem list and continue mentorship and other activities with the aim of promotion of underrepresented groups in mathematics.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.

拓扑学是一个旨在理解形状内在属性的数学领域。这为许多科学领域提供了工具，包括研究酶如何连接 DNA 或发现大数据集中的趋势。该项目的目的是研究打结绳的特性。主要目标之一是从数学角度理解如果将结切开并且末端以不同方式连接在一起，结会如何变化，就像酶对 DNA 所做的那样。据预测，结的数量总是会发生可测量的变化，PI 试图从数学上验证这一点，从而增强对结基本结构的理解。 PI 在该项目中的活动，包括指导和开发虚拟数学社区，将为本科生和研究生创造新的教育机会，特别注重增加数学的多样性和获取数学的机会。该项目将通过拉格朗日子流形构建的各种不变量来研究低维拓扑中的关键问题和结构。这包括使用四穿刺球中的浸没拉格朗日对结Floer和Khovanov缠结不变量的描述来研究无用交叉猜想，在存在本质康威球的情况下获得结同调理论的等级下界，并证明某些类缠结的缠结不变量的检测结果。 PI 将使用来自有界 Heegaard Floer 同调的浸没拉格朗日不变量来给出卫星结解开数量的新界限。 PI 还将在枕套中使用由 SU(2) 字符结变体产生的拉格朗日量来实现三被加数猜想，该猜想预测对三球体中的结进行 Dehn 手术不能包含两个以上素数被加数。 PI 还将指导研究生开发辛瞬子同源性的新结构特性，并研究隧道一号结的结弗洛尔同源性。除了组织各种研讨会之外，PI 将协助制定 Floer 同源开放问题清单，并继续开展指导和其他活动，旨在促进数学领域代表性不足的群体。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。