New Topologically Inspired Directions in Higher Representation Theory

几何和代数技术之间的相互作用和相互作用在数学和理论物理中推动了许多令人兴奋的发展。这些深层联系中固有的是对称的概念。当我们将数学的边界推向未知的边界时，越来越清楚的是，下一代理论需要更精确的对称概念。该项目利用几何学的前沿发展来揭示控制这些新的“更高”对称性的新代数结构。该项目还将为学生提供研究培训机会。利用通过低维拓扑和表示理论之间的相互作用获得的新见解，该项目详细介绍了一个程序，以实现仅通过一种专业化无法实现的结果。PI将从根本上引入高级表示理论的新工具，这些工具受到拓扑新兴领域进展的启发，包括边界heegard - flower理论、奇链路同调、非半简单TQFT和谱化（或链路同调的稳定同伦改进）。通过表征理论的透镜传递和重新解释这些最新的进展将阐明新的联系，并允许扩展，这不仅有助于推进表征理论领域，而且还可以反馈到低维拓扑和周围领域的研究中。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。

The interaction and interplay between geometric and algebraic techniques have fueled many exciting developments throughout mathematics and theoretical physics. Inherent within many of these deep connections is the notion of symmetry. As we push the boundaries of mathematics to unexplored frontiers, it is becoming increasingly clear that a more refined notion of symmetry is required for the next generation of theories. This project leverages cutting-edge developments in geometry to uncover new algebraic structures governing these new "higher" symmetries. The project will also provide research training opportunities for students.Leveraging new insights gained through the interplay between low-dimensional topology and representation theory, this project details a program to achieve results not apparent through one specialization alone. The PI will introduce fundamentally new tools in higher representation theory inspired by advances in emerging areas of topology, including bordered Heegaard-Floer theory, odd link homologies, non-semisimple TQFT, and spectrafication (or stable homotopy refinements of link homologies). Transporting and reinterpreting these recent advances through a representation-theoretic lens will illuminate new connections and allow for extensions that are not only useful in advancing the field of representation theory but can also be back-fed into the study of low-dimensional topology and surrounding fields.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.