Microlocal Sheaves, Symplectic Geometry and Applications in Representation Theory

辛几何是研究经典力学和量子力学的数学框架。它的中心是关于辛流形的结构和对称性的问题-一个偶数维空间，局部看起来像包含运动粒子的位置和动量的相空间。 现代物理学已经发现了辛流形的许多重要和复杂的不变量，并预测了许多与不同数学领域的深刻联系。目前对这些不变量的定义和方法是基于对曲面到辛流形的一种特殊映射的分析，称为全纯曲线理论。该项目的主要目标是开发一种替代方法来解决其中一些不变量，这种方法更容易获得，并将成为有效计算的基础。该方法是基于微局部层理论，这是发明作为一种代数和拓扑方法来研究微分方程。该项目将在表征理论领域有许多应用，这是一个丰富的主题，专注于研究数学和物理中出现的对称性。该研究所还将继续组织相关主题的研讨会，通过学术活动传播她的成果，并为本科生提供研究机会。更具体地说，该研究所将使用微局部层理论来构造精确辛流形中的拉格朗日子流形，并将给出辛流形的微局部层范畴的定义，期望它等价于重要的辛不变量，被称为福谷类。 这种方法是纯拓扑的，它允许系数环是一个环谱，这开辟了有趣的连接稳定同伦理论。PI将开发一个平行的故事，在复杂的设置全纯拉格朗日在一个精确的全纯辛流形，表现出新的和更丰富的结构，这将有重要的应用在几何表示理论。PI将使用它来计算辛解析中的全纯拉格朗日，这是现代表示论中心的一类全纯辛流形，目的是理解辛对偶的神秘现象。表示论的其他应用包括在Hecke范畴中使用辫子群作用的层量子化作为辛同胚的计算，以及使用微局部反常层实现非交换霍奇理论的建议。

Symplectic geometry is a mathematical framework for the study of classical and quantum mechanics. It centers around questions about the structures and symmetries of a symplectic manifold--an even-dimensional space that locally looks like the phase space containing the position and momentum of a moving particle. Modern physics has led to the discoveries of many important and sophisticated invariants of symplectic manifolds, and has predicted a number of deep connections to different fields of mathematics. Current definitions and approaches to these invariants are based on the analysis of a special kind of mapping of a surface to a symplectic manifold, known as the theory of holomorphic curves. The primary goal of the project is to develop an alternative approach to some of these invariants, which is more accessible and which will form the foundations of effective calculations. The approach is based on the microlocal sheaf theory, which was invented as an algebraic and topological method to study differential equations. The project will have many applications in the field of representation theory, a rich subject focusing on the study of symmetries appearing in mathematics and physics. The PI will also continue to organize seminars on related topics, disseminate her results through academic events, and provide research opportunities for undergraduate students.More specifically, the PI will use microlocal sheaf theory to quantize Lagrangian submanifolds in an exact symplectic manifold, and will give the definition of a microlocal sheaf category of a symplectic manifold, which is expected to be equivalent to the important symplectic invariant, known as the Fukaya category.  The approach is purely topological, and it allows the coefficient ring to be a ring spectrum, which opens up interesting connections to stable homotopy theory. The PI will develop a parallel story in the complex setting of holomorphic Lagrangians in an exact holomorphic symplectic manifold, which exhibits new and richer structures and which will have important applications in geometric representation theory. The PI will use this to quantize holomorphic Lagrangians in symplectic resolutions, a class of holomorphic symplectic manifolds in the center of modern representation theory, with the goal of understanding the mysterious phenomena of symplectic duality. Other applications to representation theory involve calculations in the Hecke category using sheaf quantizations of the braid group action as symplectomorphisms and a proposal to realize the nonabelian Hodge theory using microlocal perverse sheaves.