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，包括研究酶如何使DNA打结或在大型数据集中发现趋势。本计画的目的是研究打结弦的性质。其中一个主要目标是从数学上理解，如果一个结被切开，末端以不同的方式连接在一起，它是如何变化的，就像酶对DNA所做的那样。预测是，这总是有一个可测量的变化，在打结，和PI试图验证这一数学，加强了解的基本结构的结。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，特别关注增加数学的多样性和获得数学的机会。该项目将通过从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。这包括使用的描述结Floer和Khovanov缠结不变量的沉浸拉格朗日在四穿孔领域的研究无效穿越猜想，获得下界的秩的结同源理论的存在下，一个本质的康威球，并证明检测结果的缠结不变量为某些类的缠结。PI将使用浸入拉格朗日不变量从边界Heegaard Floer同源给出新的界限卫星结的unknotting数量。PI还将使用枕头套中的拉格朗日量，这些拉格朗日量来自于SU（2）-字符的各种纽结，用于三个被加数猜想，该猜想预测了在三个球体中的纽结上的Dehn手术不能由两个以上的素被加数组成。PI还将指导研究生开发辛瞬子同源的新结构特性，并研究隧道一号结的结Floer同源性。除了组织各种研讨会，PI还将协助Floer同源性开放问题列表的开发，并继续指导和其他活动，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。

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.