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.

低维拓扑研究维度一到四个方面的空间的形状，并且具有从物理和宇宙学等应用，其中宇宙的形状是研究到生物化学，它试图了解打结的DNA的行为。与3维空间和4维空间的研究密切相关的是结的研究，可以将其视为绑在空间中。该项目将进一步开发并应用低维拓扑的最新剪切工具。该项目的一部分涉及哪种几何结构，特别是哪种“接触结构”，给定的三维空间可以支持哪种几何结构。除了直接应用数学应用外，接触结构还发现了许多在物理学中的应用，包括经典力学，热力学和控制理论。与研究部分并行，PI将进一步进一步进行教育和外展工作。例如，PI将监督本科和研究生研究，并建立一个高中富集计划。在2000年代初期，Ozsvath和Szabo开发了一套强大的结，用于结和3维空间的功能强大的不变性，通常称为Heegaard Floer同源性。此后，Heegaard Floer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并解决了许多旧的猜想。边界的Floer同源性将Heegaard Floer同源性推广到具有边界的歧管，并通过将歧管切成碎片（例如结成缠结的结）并研究单个碎片及其胶水，提供了计算封闭歧管的Heegaard Floer不变性的精美技术。该项目旨在进一步开发我们目前拥有的接壤的Heegaard浮动工具。 PI计划继续从边界的浮子同源性开发一个不变的凸边界3个manifolds，并使用它来解决联系拓扑中的空旷问题；将边界的浮子同源性和缠结的浮子同源性扩展到整体兼容性；理解并发展结浮情同源性和量子代数之间的联系。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响审查标准，被视为通过评估来获得珍贵的支持。