Representation Theory and Symplectic Geometry Inspired by Topological Field Theory

几何表示理论和辛几何是当前数学研究的两个中心课题。他们从数学物理中获得原始灵感，通常以量子场论的形式，特别是对其对称性的研究。这在历史上是一个富有成果的方向，它由推广傅里叶理论的对偶性指导。这个项目的研究涉及多种追求，包括新工具的开发和开放性问题的解决。贯穿全书的一个共同主题是找到用基本组合术语来思考复杂几何系统的方法。该研究还为进入这些学科的学生提供了应用最新工具和探索新方法做出重大贡献的机会。其他活动包括相关主题的教育和说明文写作，数学和物理研究人员之间的新互动，以及对公众参与数学的持续投资。具体的项目承担了超对称规范理论的核心挑战，特别是规范场的相空间，它们的二维西格玛模型，以及来自四维场理论的膜上的更高结构。主要的主题是仿射Hecke范畴的中心和椭圆字符束，局部朗兰等价和相对朗兰对偶，以及Weinstein流形的拉格朗日骨架的拓扑。该项目的主要目标包括识别仿射Hecke范畴的中心与椭圆字符束作为自同构胶合的一个实例，朗兰兹参数空间的循环对称性在朗兰兹分类的范畴形式中的应用，以及极化Weinstein流形与树状空间的比较。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。

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.