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。它围绕着辛流形的结构和对称性的问题——辛流形是一个偶维空间，局部看起来像包含运动粒子的位置和动量的相空间。现代物理学已经导致了辛流形许多重要而复杂的不变量的发现，并预测了许多与数学不同领域的深刻联系。这些不变量的当前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为其中一些不变量开发一种替代方法，这种方法更容易获得，并将形成有效计算的基础。该方法基于微局部束理论，微局部束理论是一种研究微分方程的代数和拓扑方法。该项目将在表征理论领域有许多应用，表征理论是一个专注于研究数学和物理中出现的对称性的丰富学科。PI还将继续举办相关主题的研讨会，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，PI将使用微局部束理论来量化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束范畴的定义，该定义预计将等同于重要的辛不变量，称为Fukaya范畴。该方法是纯拓扑的，它允许系数环是环谱，这为稳定同伦理论开辟了有趣的联系。PI将在精确全纯辛流形的全纯拉格朗日的复杂背景下发展一个平行的故事，它将展示新的和更丰富的结构，并将在几何表示理论中具有重要的应用。PI将用它来量化辛分辨率下的全纯拉格朗日量，这是现代表示理论中心的一类全纯辛流形，目的是理解辛对偶的神秘现象。表征理论的其他应用包括使用辫群作用的束量化作为辛形态在Hecke范畴中的计算，以及使用微局部反常束实现非abel Hodge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.