CAREER: Bordered Floer homology and applications

职业：Bordered Floer 同源性和应用

• 批准号: 2145090
• 负责人: Ina Petkova
• 金额: $ 49.98万
• 依托单位: Dartmouth College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2145090&HistoricalAwards=false
• 关键词: CAREER; Bordered; Floer; homology; applications

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.

低维拓扑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旨在了解打结 DNA 行为的生物化学，都与研究密切相关。 3 维和 4 维空间的研究是对结的研究，结可以被视为空间中的束缚。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。什么类型的几何结构，特别是给定的 3 维空间可以支持哪种“接触结构”，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。对于研究部分，PI 将进一步开展教育和推广工作，例如，PI 将监督本科生和研究生的研究，并建立高中强化计划。2000 年代初，Ozsvath 和 Szabo 开发了一套方案。结和 3 维和 4 维空间的强大不变量，通常称为 Heegaard Floer 同调，从此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结果并解决了许多旧的有界猜想。 Floer 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 不变量的好技术：将流形切割成碎片（例如，将结切割成缠结），并研究各个碎片及其粘合。该项目旨在进一步开发我们目前拥有的有边 Heegaard Floer 工具，PI 计划继续开发有边 Floer 同源性的不变量。对于具有凸边界的接触 3-流形，并用它来解决接触拓扑中的开放问题：将有界 Floer 同调延伸并将 Floer 同调缠结到积分；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。