Higher Representation Theory and Heegaard Floer Homology

更高表示理论和 Heegaard Floer 同调

• 批准号: 2101916
• 负责人: Andrew Manion
• 金额: $ 15.78万
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-09-01 至 2021-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2101916&HistoricalAwards=false
• 关键词: Higher; Representation; Theory; Heegaard; Floer

Modern research at the interface of mathematics and physics has revealed startling connections between geometric properties of shapes, theories in high-energy physics, deep concepts of symmetry, and rapidly developing areas like materials science and quantum computation. A set of constructions known as "Heegaard Floer homology" brings together many of these perspectives and forms a key nexus in this web of ideas. As a result, there is much interest in Heegaard Floer homology as a guiding light for understanding this web more broadly and pushing out toward the applications of most mathematical and practical significance. This project aims to exploit the locality inherent in Heegaard Floer homology to understand a long-sought-after algebraic operation based on principles of higher symmetry, and to use this new structure to advance adjacent fields of research. This project will support the professional development of students and other early-career mathematicians through research collaborations, mentoring activities, and outreach, including organization of conferences and seminars. In more detail, this project builds on recent work which uncovered a relationship between the lower-dimensional or more-local aspects of Heegaard Floer homology and tensor products for higher representations of categorified quantum groups. Such tensor products have long been recognized as a crucial element needed for further progress in this area, but only in certain cases have they been defined. This project will expand our knowledge of the structure of Heegaard Floer homology using higher tensor products as an organizing principle, while also using insights from the extensive Heegaard Floer literature to understand more about higher tensor products in general and to develop new links with areas ranging from geometric representation theory to amplituhedra in physics.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.

数学和物理交叉领域的现代研究揭示了形状的几何特性、高能物理理论、对称性的深层概念以及材料科学和量子计算等快速发展的领域之间的惊人联系。一组被称为“Heegaard Floer 同源性”的结构汇集了许多这些观点，并形成了这个思想网络中的关键联系。因此，人们对 Heegaard Floer 同源性很感兴趣，认为它可以作为更广泛地理解这个网络并推动最具数学和实际意义的应用的指路明灯。该项目旨在利用 Heegaard Floer 同调中固有的局部性来理解长期追求的基于更高对称性原理的代数运算，并利用这种新结构来推进相邻的研究领域。该项目将通过研究合作、指导活动和推广（包括组织会议和研讨会）来支持学生和其他早期职业数学家的专业发展。更详细地说，该项目建立在最近的工作基础上，该工作揭示了 Heegaard Floer 同调的低维或更局部方面与分类量子群的更高表示的张量积之间的关系。长期以来，此类张量积一直被认为是该领域取得进一步进展所需的关键要素，但仅在某些情况下才对它们进行了定义。该项目将使用更高张量积作为组织原则来扩展我们对 Heegaard Floer 同调结构的了解，同时还利用大量 Heegaard Floer 文献中的见解来更多地了解一般更高张量积，并与以下领域建立新的联系：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。