CAREER: Bordered Floer homology and applications

低维拓扑学研究1到4维空间的形状，其应用范围从研究宇宙形状的物理学和宇宙学，到试图理解结DNA行为的生物化学。与三维和四维空间的研究密切相关的是对结的研究，它可以被看作是在空间中捆绑的。该项目将进一步开发和应用低维拓扑中的最新剪切和粘贴工具。项目的一部分涉及到什么样的几何结构，特别是什么样的“接触结构”，一个给定的三维空间可以支持的问题。除了直接应用于数学之外，接触结构在物理学中也有许多应用，包括经典力学、热力学和控制理论。在开展研究工作的同时，PI将进一步开展教育和外联工作。例如，PI将监督本科生和研究生的研究，并建立一个高中充实计划。在21世纪初，Ozsvath和Szabo为结点和3维和4维空间开发了一套强大的不变量，通常被称为Heegaard flower同调。此后，Heegaard flower同源性在低维拓扑中占据了重要地位，并帮助研究人员获得了许多新的结果，解决了许多旧的猜想。有边花同调将Heegaard花同调推广到有边界的流形，并提供了计算封闭流形Heegaard花不变量的好技术，通过将流形切成块（例如一个结变成缠结），并研究单个块及其粘合。该项目旨在进一步开发我们目前拥有的边界Heegaard flower工具。PI计划继续开发具有凸边界的接触3流形的不变量，并使用它来解决接触拓扑中的开放性问题；将有边花同调和缠结花同调扩展到积分系数；理解和发展结花同调和量子代数之间的联系。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。

Low-dimensional topology studies the shapes of spaces in dimensions one through four, and has applications ranging from physics and cosmology in which the shape of the universe is studies to biochemistry, which seeks to understand the behavior of knotted DNA. Closely related to the study of 3- and 4-dimensional spaces is the study of knots, which can be viewed as tied in space. This project will further develop and apply recent cut-and-paste tools in low-dimensional topology. Part of the project concerns the question of what kinds of geometric structures, specifically what kind of “contact structures”, a given 3-dimensional space can support. In addition to direct applications to mathematics, contact structures have found numerous applications in physics, including classical mechanics, thermodynamics, and control theory. In parallel to the research component, the PI will further their educational and outreach efforts. For example, the PI will supervise undergraduate and graduate research, and establish a high school enrichment program.In the early 2000s, Ozsvath and Szabo developed a package of powerful invariants for knots and 3- and 4-dimensional spaces, generally known as Heegaard Floer homology. Heegaard Floer homology has since taken a major place in low-dimensional topology, and has helped researchers obtain many new results and settle numerous old conjectures. Bordered Floer homology generalizes Heegaard Floer homology to manifolds with boundary, and provides nice techniques for computing the Heegaard Floer invariants of closed manifolds, by cutting a manifold into pieces (e.g. a knot into tangles), and studying the individual pieces and their gluing. This project seeks to develop further the bordered Heegaard Floer tools we currently have. The PI plans to continue to develop an invariant from bordered Floer homology for contact 3-manifolds with convex boundary, and use it to address open questions in contact topology; extend bordered Floer homology and tangle Floer homology to integral coefficients; understand and develop the connections between knot Floer homology and quantum algebra.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.