Microlocal Sheaves in Geometric Representation Theory

微分方程为研究随时间和空间变化的量提供了基本工具。线性微分方程组可以用称为d模的代数结构进行编码；这种编码使我们能够将方程组的代数性质与方程组解的特征问题分离开来。在空间对称存在的情况下，那些被对称保留的d模具有一种特殊的结构，反映了空间本身相应的分解成更简单的块。本研究的重点是细化这些结构特征，并将其应用于数学和物理。d模是几何表示理论的基本对象。它们和它们的更灵活的辛对应物，我们称之为“微局部束”，已经在这个主题中系统地挖掘了几十年。在最近的工作中，PI（与McGerty联合）已经建立了一个新的结构特征的存在性，即在各种X上的G等变d模的范畴莫尔斯分解，或者，在更一般的代数堆类上，用G泛化X的商。这种分类分解反映了一个分类水平更高的，期待已久的关于X的等变协切束（即商堆栈的协切束）几何的莫尔斯理论断言。这项研究将范畴莫尔斯分解的结构置于一个相互连接的方向和问题网络的中心，这些方向和问题是由表示理论、拓扑、几何和超对称场论的数学所通知的，并与它们的应用有关。范畴莫尔斯分解的“剥离顶层”导致了代数的表示理论的应用，这些代数是通过量子哈密顿约简实现的（例子包括许多辛反射代数）。脱范畴导致应用到经典拓扑不变量的超凯勒和代数辛商。将该框架应用于超对称规范理论中作为真空模出现的代数变体，将使我们深入了解这些物理定义空间的数学结构。

Differential equations provide basic tools for studying quantities that vary in space and time. A system of linear differential equations can be encoded in an algebraic structure known as a D-module; this encoding allows us to isolate algebraic properties of the system of equations from questions about features of its solutions. In the presence of spatial symmetry, those D-modules preserved by the symmetry possess a special structure reflecting a corresponding decomposition of space itself into simpler pieces. The focus of this research is to refine these structural features and apply them in mathematics and physics.D-modules provide fundamental objects in geometric representation theory. They and their more flexible symplectic counterparts, which we call "microlocal sheaves," have been systematically mined in the subject for several decades. In recent work, the PI (jointly with McGerty) has established the existence of a new structural feature, categorical Morse decomposition, for G-equivariant D-modules on a variety X, or, generalizing the quotient of X by G, on a more general class of algebraic stacks. This categorical decomposition mirrors, one categorical level higher, long-anticipated Morse-theoretic assertions for the geometry of the equivariant cotangent bundle of X (i.e., the cotangent bundle of the quotient stack). This research places the structure of categorical Morse decomposition at the center of a web of interconnected directions and problems informed by, and with applications to, representation theory, topology, geometry, and the mathematics of supersymmetric field theories. "Peeling off the top layer" of the categorical Morse decomposition leads to applications to the representation theory of algebras that are realized by quantum Hamiltonian reduction (examples include many symplectic reflection algebras). Decategorifying leads to applications to classical topological invariants of hyperkahler and algebraic symplectic quotients. Applying the framework to algebraic varieties that arise as moduli of vacua in supersymmetric gauge theories will yield insight into mathematical structures of these physically-defined spaces.