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

该项目追求连接一系列数学学科的基本进步，特别是表示论（对称性研究），代数几何（多项式方程研究）和辛几何（经典力学的几何）。这些项目通过使用集群代数来丰富这些主题并在它们之间建立新的联系。这项研究将有助于发展量子场论的数学基础，为现代理论物理学中研究的某些概念提供新的、数学上严格的定义。该项目为研究生提供了研究培训的机会。更详细地说，PI将发展他先前工作中开创的一类相干层的理论。要证明的关键命题包括：这种KP层的范畴是广义的么半群簇的等价，在伴随的情况下它们服从几何佐竹等价的一个相干变体，在规范理论的背景下，它们承认与仿射Khovanov-Lauda-Rouquier代数的一种Schur-Weyl对偶。PI将继续他的工作，使用集群组合，以发展辛几何的新思想。最后，PI将建立一个猜想描述的非循环集群代数在卡茨-穆迪李群的表示理论方面的新案例。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.