Microlocal Geometry in Gauge Theory

这笔拨款支持的研究处于数学和物理的十字路口。它的目的是将起源于量子场论的思想应用于表象理论中的问题。一个主要的主题是研究傅立叶类型的对称性，这在传统观点中并不明显。另一个主题是用直观的组合术语描述复杂的同伦结构。开发和应用的主要工具属于微域几何，这是一门具有丰富历史的学科，可以追溯到经典力学。一个广泛的目标是在这个快速发展的方向的新前沿教育学生。还将有大量机会扩大到其他领域，并增加公众对数学的参与。具体项目包括用于循环群的特征线的椭圆理论、分类的Verlinde公式以及标记亏格零的计算，以及微局部线的组合模型。这些方法涉及到微分方程组和可构造层的传统微局部几何，以及在派生代数几何中发现的相干层的新的同伦微局部几何。应用包括新的亏格0和亏格1上的几何朗兰兹对应，以及微局部层的组合模型的计算。

The research supported by this grant lies at the crossroads of mathematics and physics. It aims to apply ideas originating in quantum field theory to problems in representation theory. A main theme is the study of Fourier-type symmetries not evident from traditional viewpoints. Another theme is the description of complicated homotopical constructions in intuitive combinatorial terms. The primary tools developed and applied belong to microlocal geometry, a subject with a rich history going back to classical mechanics. A broad goal is the education of students in the new frontiers of this rapidly developing direction. There will also be ample opportunities for outreach to other fields and for increased public engagement with mathematics. Specific projects include an elliptic theory of character sheaves for loop groups, a categorical Verlinde formula along with marked genus zero calculations, and a combinatorial model of microlocal sheaves. The methods involve the traditional microlocal geometry of differential equations and constructible sheaves, as well as the new homotopical microlocal geometry of coherent sheaves found within derived algebraic geometry. Applications include new geometric Langlands correspondences in genus zero and genus one, and calculations with combinatorial models of microlocal sheaves.