CAREER: Bordered Floer homology and applications

低维拓扑学研究一维到四维空间的形状，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打结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，它可以被视为在空间中的束缚。这个项目将进一步发展和应用最新的剪切和粘贴工具在低维拓扑结构。该项目的一部分涉及的问题是什么样的几何结构，特别是什么样的“接触结构”，一个给定的三维空间可以支持。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开发了一套强大的不变量，用于纽结和3维和4维空间，通常称为Heegaard Floer同调。Heegaard Floer同调在低维拓扑学中占有重要的地位，并帮助研究人员获得了许多新的结果，解决了许多旧的问题。有界弗洛尔同调将Heegaard Floer同调推广到有边界的流形，并提供了计算闭流形的Heegaard Floer不变量的很好的技术，通过将流形切割成碎片（例如将结切割成缠结），并研究单个碎片及其粘合。该项目旨在进一步开发我们目前拥有的边界Heegaard Floer工具。PI计划继续从凸边界切触3-流形的有边Floer同调中发展一个不变量，并将其用于解决切触拓扑中的公开问题;将有边Floer同调和缠结Floer同调扩展到积分系数;理解和发展结弗洛尔同源性和量子代数之间的联系。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用评估支持基金会的学术价值和更广泛的影响审查标准。

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.