CAREER: Homotopical representation theory and TQFTs

这个项目将在现代数学和量子场论之间建立新的联系。量子场论是现代物理学的有效工具。该项目将通过利用抽象代数和几何的最新创新，以数学术语明确地描述量子场论，发展严格描述和分析这些理论所需的数学。除了科学进步之外，该项目还将为数学家开发基于文本的资源，这将提高研究人员获取和部署代数和几何前沿方法的能力，并为洛杉矶地区的社区教育项目提供支持。更准确地说，PI将在表示理论中使用同调方法来提供真空相关模是非紧致的拓扑量子场论的严格数学实现。这种类型的拓扑场论作为超对称量子场论的扭曲自然出现。在产生上述实现的过程中，PI将发展量子群表示与施普林格分辨率几何之间的新关系。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。

This project will develop novel connections between modern mathematics and quantum field theory. Quantum field theories are an effective tool in modern physics. The project will develop the mathematics needed to rigorously describe and analyze such theories by leveraging recent innovations in abstract algebra and geometry to explicitly describe quantum field theories in mathematical terms. In addition to scientific advances, the project will both develop text-based resources for mathematicians that will advance researchers' abilities to access and deploy cutting edge methods in algebra and geometry and provide support to community-based education programs in the Los Angeles area. More precisely, the PI will use homotopical approaches in representation theory to provide rigorous mathematical realizations of topological quantum field theories whose associated moduli of vacua are non-compact. Topological field theories of this type arise naturally as twists of supersymmetric quantum field theories. In producing the aforementioned realizations, the PI will develop new relationships between representations of quantum groups and the geometry of Springer resolutions.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.