CAREER: Cluster Algebras in Representation Theory, Geometry, and Physics

职业：表示论、几何和物理学中的簇代数

• 批准号: 2143922
• 负责人: Harold Williams
• 金额: $ 45万
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-03-15 至 2027-02-28
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2143922&HistoricalAwards=false
• 关键词: CAREER; Cluster; Algebras; Representation; Theory

This project pursues fundamental advances connecting a range of mathematical disciplines, specifically representation theory (the study of symmetry), algebraic geometry (the study of polynomial equations), and symplectic geometry (the geometry of classical mechanics). The projects are connected by their use cluster algebras to enrich these subjects and develop new connections among them. The research will contribute to the fundamental goal of developing the mathematical foundations of quantum field theory by providing new, mathematically rigorous definitions of certain notions studied in modern theoretical physics. This project provides research training opportunities for graduate students.In more detail, the PI will develop the theory of a class of coherent sheaves inaugurated in his prior work. Key conjectures to be proved include that categories of such Kp-sheaves are monoidal cluster categorifications in wide generality, that in the adjoint case they obey a coherent variant of the geometric Satake equivalence, and that in the context of quiver gauge theories they admit a type of Schur-Weyl duality with affine Khovanov-Lauda-Rouquier algebras. The PI will continue his work using cluster combinatorics to develop new ideas in symplectic geometry. Finally, the PI will establish new cases of a conjecture describing acyclic cluster algebras in terms of the representation theory of Kac-Moody Lie groups.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.

该项目追求连接一系列数学学科的基本进步，特别是代表理论（对称性的研究），代数几何学（多项式方程的研究）和符号几何学（经典力学的几何学）。这些项目通过使用群集代数来丰富这些主题并在其中建立新的联系。这项研究将有助于通过提供对现代理论物理学中某些概念的新的，数学上严格的定义来开发量子场理论的数学基础的基本目标。该项目为研究生提供了研究培训机会。更详细的是，PI将开发出他先前的工作中启动的一类连贯的滑轮的理论。 Key conjectures to be proved include that categories of such Kp-sheaves are monoidal cluster categorifications in wide generality, that in the adjoint case they obey a coherent variant of the geometric Satake equivalence, and that in the context of quiver gauge theories they admit a type of Schur-Weyl duality with affine Khovanov-Lauda-Rouquier algebras. PI将使用集群组合学以形成符号几何形状来继续他的工作。最后，PI将根据Kac-Moody Lie Group的代表理论来建立描述无环代数的猜想的新案例。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的影响审查标准通过评估来进行评估的。