Higher Representation Theory and Heegaard Floer Homology

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、对称的深层概念以及材料科学和量子计算等迅速发展的领域之间惊人的联系。一组被称为“Heegaard flower同源”的结构汇集了许多这些观点，并在这个思想网络中形成了关键的联系。因此，人们对Heegaard flower同源性很感兴趣，因为它可以作为更广泛地理解这个网络的指路明灯，并推动最具数学和实际意义的应用。该项目旨在利用Heegaard flower同调中固有的局部性来理解基于更高对称性原理的长期追求的代数运算，并利用这种新的结构来推进邻近领域的研究。该项目将通过研究合作、指导活动和外联活动，包括组织会议和研讨会，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 flower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。这种张量积长期以来被认为是该领域进一步发展所需的关键元素，但仅在某些情况下才被定义。这个项目将扩展我们对Heegaard flower同调结构的知识，使用高张量积作为组织原理，同时也使用从广泛的Heegaard flower文献中获得的见解来了解更多关于高张量积的一般知识，并与从几何表示理论到物理振幅面等领域建立新的联系。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。

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.