CAREER: Low dimensional topology via Floer theory

这个项目的重点是研究低维拓扑中的各种问题，这是对维数最多为4的空间形状的研究。这门学科的中心是研究三维空间中的圆的打结和连接，称为纽结理论。例如，结和链接可以用来构建三维和四维空间。低维拓扑学和纽结理论与宇宙学和物理学（宇宙的形状和弦理论）以及生物化学（DNA分子的打结行为）有着各种各样的联系。非直观地，拓扑学中的许多分类问题在三维和四维中更难，在过去的三十年里，拓扑学家已经开发出了研究这些问题的现代工具（源于物理学）。这种工具的两个例子是Heegaard Floer不变量，它产生于规范理论，和Khovanov同调，它起源于表示论。本项目将进一步发展这些不变量，并研究它们的相似性和关系。此外，它将利用他们的力量来研究曲面（二维空间）的对称性，与双曲几何和动力学有关，并将研究低维拓扑学中的几个基本问题，例如找到一个结必须穿过自己才能解开的最小次数。与此同时，该项目旨在通过各级教育活动，使广大受众能够接触数学，特别是拓扑学，并强调多样性和包容性。这些活动包括，在雅典-克拉克县公共图书馆建立数学圈计划，为高中和中学生组织暑期数学夏令营，为本科生举办暑期研究体验项目，为研究生和博士后举办拓扑学暑期学校。该研究计划的主要目标围绕四个领域组织。首先，发展新的不变量用于研究空间图和图协调。主要的工具将是一个推广的（减）Heegaard弗洛尔同调称为，缠结弗洛尔同调。研究空间图的一致性在研究节点之间的等变一致性和链接之间的强一致性方面有应用。第二，利用Heegaard Floer同调的另一种推广--加边Heegaard Floer同调，研究了空间体上的映射类群和曲面同调的扩张。第三，通过推广和改进PI及其合著者定义的接触不变量，进一步开发有边界的Heegaard Floer同调工具，并使用它来解决接触拓扑中的开放问题。最后，PI侧重于Khovanov同调和Heegaard Floer不变量之间的联系和相似性，其三个主要目标是定义新的协调不变量来研究光滑的4D Poincare猜想，为解结数找到新的下限，并为横向结开发新的不变量。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进行评估，被认为值得支持和更广泛的影响审查标准。

The focus of this project is investigating a variety of questions in low dimensional topology, which is the study of shapes of spaces with dimension at most four. Central in this subject is the study of knotting and linking of circles in three-dimensional spaces, called knot theory. For instance, knots and links can be used to construct three- and four-dimensional spaces. Low dimensional topology and knot theory have various connections to cosmology and physics (the shape of the universe and string theory), and biochemistry (the knotting behavior of DNA molecules). Unintuitively, many classification questions in topology are harder in dimensions three and four, and over the past three decades, topologists have developed modern tools (with roots in physics) for studying these questions. Two examples of such tools are Heegaard Floer invariants, which grew out of gauge theory, and Khovanov homology, which has roots in representation theory. This project will further develop these invariants and investigate their similarities and relations. Moreover, it will harness their power to study symmetries of surfaces (two-dimensional spaces), in connection with hyperbolic geometry and dynamics, and will investigate several fundamental questions in low-dimensional topology such as finding the minimum number of times a knot must cross itself to become unknotted. In parallel, this project aims to make mathematics and in particular topology accessible to a broad audience, through educational activities at all levels with an emphasis on diversity and inclusion. These activities include, establishing a Math Circle program in the Athens-Clarke County public library and organizing summer Math Camps for high school and middle school students, running a summer research experience project for undergraduate students and a topology summer school for graduate students and postdocs. The major goals of this research program are organized around four areas. First, developing new invariants for studying spatial graphs and graph concordance. The main tool will be a generalization of (minus) Heegaard Floer homology called, tangle Floer homology. Studying spatial graphs up to concordance has applications in studying equivariant concordance between knots, and strong concordance between links. Second, studying mapping class group and extensions of surface diffeomorphisms over handlebodies using another generalization of Heegaard Floer homology, called bordered Heegaard Floer homology. Third, further developing the bordered Heegaard Floer homology tools by generalizing and refining the contact invariant defined by the PI and her co-authors and use it to address open questions in contact topology. Lastly, the PI focuses on connections and similarities between Khovanov homology and Heegaard Floer invariants, with the three major goals of defining new concordance invariants to study smooth 4D Poincare conjecture, finding new lower bounds for the unknotting number and developing new invariants for transverse knots.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.