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.

拓扑是一个数学领域，旨在理解形状的内在特性。这为许多科学领域提供了工具，包括研究酶如何打结DNA或在大型数据集中找到趋势。该项目的目的是研究打结的字符串的特性。主要目标之一是在数学上了解结，如果打开打开，并且末端的连接方式也有所不同，就像酶对DNA一样。预测是，这始终在打结上具有可衡量的变化，PI试图通过数学验证这一验证，增强对结基本结构的理解。 PI在该项目中的活动，包括指导和发展虚拟数学社区，将为本科和研究生创造新的教育机会，特别关注增加数学的多样性和访问。该项目将通过Lagrangian Submanifolds构建的各种不变性来研究低维拓扑的关键问题和结构。这包括使用沉浸在四个函数领域中的沉浸式拉格朗日人来研究结节的缠结者的描述，以研究卵形交叉猜想，在基本的Conway Sphere的存在下获得结节同源性理论等级的下限，并证明了缠结的缠结类别的纠缠型缠结的结果。 PI将使用来自边界的Heegaard Floer同源性的沉浸式Lagrangian不变式，以对卫星结的无打结。 PI还将在SU（2）（2）的枕套中使用Lagrangians，朝向三夏猜想的结（2） - 特征的特征品种，这预测，在三个球体中的一个结中Dehn手术不能由两个以上的素数组成。 PI还将指导研究生，以开发符号Instanton同源性的新结构特性，并研究隧道第一结的结浮情同源性。除了组织各种研讨会外，PI还将有助于开发Floer同源性开放问题清单，并继续指导和其他活动，以促进数学中代表性不足的群体。该奖项反映了NSF的法定任务，并通过基金会的知识分子优点和广泛的影响来评估NSF的法定任务。