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推动了整个数学和理论物理学的许多令人兴奋的发展。 在许多这些深层联系中，固有的是对称的概念。当我们将数学的边界推向未探索的前沿时，越来越清楚的是，下一代理论需要更精确的对称性概念。 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，以实现仅通过一个专业化不明显的结果。PI将在更高的表示理论中引入全新的工具，这些工具受到拓扑学新兴领域进步的启发，包括边界Heegaard-Floer理论，奇链同调，非半单TQFT和光谱化（或稳定同伦改进的链接同调）。 通过表征理论的透镜传输和重新解释这些最新进展将照亮新的连接，并允许扩展，不仅在推进表征理论领域有用，而且还可以反馈到低-该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的评估支持影响审查标准。

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.