Lagrangians and Low-Dimensional Topology

拓扑学是一个数学领域，旨在理解形状的内在属性。这为许多科学领域提供了工具，包括研究酶如何结合DNA或在大型数据集中发现趋势。这个项目的目的是研究打结弦的性质。其中一个主要目标是用数学方法理解一个结是如何改变的，如果它被切开，末端的连接方式不同，就像酶对DNA所做的那样。预测是，结度总是有一个可测量的变化，PI试图在数学上验证这一点，增强对结的基本结构的理解。PI在这个项目中的活动，包括指导和发展虚拟数学社区，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得数学的机会。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。这包括在四刺球中使用浸没拉格朗日量对结Floer和Khovanov纠缠不变量的描述来研究核交叉猜想，在本质Conway球存在下获得结同调理论秩的下界，并证明某些类纠缠不变量的检测结果。PI将使用来自有边Heegaard flower同调的浸入式拉格朗日不变量来给出卫星结解结数的新界限。PI还将在枕套中使用拉格朗日量，该拉格朗日量由SU(2)-三和数猜想的结的特征变体引起，该猜想预测三球中结的Dehn手术不能由两个以上的素数和组成。PI还将指导研究生开发辛瞬子同调的新结构性质，并研究隧道1号结的结花同调。除了组织各种研讨会外，PI还将协助开发弗洛尔同调开放问题清单，并继续指导和其他活动，以促进数学领域代表性不足的群体。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。

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.