Representation Theory and Symplectic Geometry Inspired by Topological Field Theory

几何表示论和辛几何是当前数学中的两个重要课题。他们从数学物理学中汲取原始灵感，通常以量子场论的形式，特别是对其对称性的研究。这是一个历史上富有成果的方向，由推广傅立叶理论的对偶指导。该项目的研究涉及多种追求，包括开发新工具和解决开放问题。一个贯穿始终的共同主题是找到用初等组合术语思考复杂几何系统的方法。这项研究还为进入这些学科的学生提供了机会，使他们能够通过应用最新的工具和探索新的方法做出重大贡献。其他活动包括相关主题的教育和学术写作，数学和物理研究人员之间的新互动，以及对数学公众参与的持续投资。具体项目承担超对称规范理论的核心挑战，特别是关于规范场的相空间，二维sigma模型，以及来自四维场论的膜上的更高结构。主要的主题是仿射Hecke范畴的余中心和椭圆特征层，局部Langlands等价和相对Langlands对偶，以及Weinstein流形的Lagrangian拓扑。该项目的主要目标包括识别仿射Hecke范畴与椭圆特征层的共中心作为自守胶合的实例，Langlands参数空间的循环对称性应用于Langlands分类的范畴形式，以及偏振Weinstein流形与树空间的比较。该奖项反映了NSF的法定使命，并通过评估被认为值得支持使用基金会的知识价值和更广泛的影响审查标准。

Geometric representation theory and symplectic geometry are two subjects of central interest in current mathematics. They draw original inspiration from mathematical physics, often in the form of quantum field theory and specifically the study of its symmetries. This has been an historically fruitful direction guided by dualities that generalize Fourier theory. The research in this project involves a mix of pursuits, including the development of new tools and the solution of open problems. A common theme throughout is finding ways to think about intricate geometric systems in elementary combinatorial terms. The research also offers opportunities for students entering these subjects to make significant contributions by applying recent tools and exploring new approaches. Additional activities include educational and expository writing on related topics, new interactions between researchers in mathematics and physics, and continued investment in public engagement with mathematics.The specific projects take on central challenges in supersymmetric gauge theory, specifically about phase spaces of gauge fields, their two-dimensional sigma-models, and higher structures on their branes coming from four-dimensional field theory. The main themes are the cocenter of the affine Hecke category and elliptic character sheaves, local Langlands equivalences and relative Langlands duality, and the topology of Lagrangian skeleta of Weinstein manifolds. The primary goals of the project include an identification of the cocenter of the affine Hecke category with elliptic character sheaves as an instance of automorphic gluing, the application of cyclic symmetries of Langlands parameter spaces to categorical forms of the Langlands classification, and a comparison of polarized Weinstein manifolds with arboreal spaces.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.