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。一个线性微分方程组可以被编码在一个称为D-模的代数结构中;这种编码使我们能够将方程组的代数性质与关于其解的特征的问题隔离开来。在空间对称性存在的情况下，那些由对称性保持的D-模具有一种特殊的结构，反映了空间本身相应地分解成更简单的部分。本研究的重点是提炼这些结构特征并将其应用于数学和物理中。D-模为几何表示理论提供了基本对象。 他们和他们的更灵活的辛对应物，我们称之为“微局部层”，已经系统地挖掘了几十年的主题。在最近的工作中，PI（与McGerty一起）已经建立了一个新的结构特征，范畴莫尔斯分解的存在性，G-等变D-模在一个簇X上，或者，推广X的商G，在一个更一般的代数栈类。这种范畴分解反映了一个更高的范畴层次，长期预期的Morse理论断言，即X的等变余切丛的几何（即，商栈的余切丛）。本研究将范畴莫尔斯分解的结构置于互联方向和问题网络的中心，并应用于表示论、拓扑学、几何学和超对称场论的数学。“剥离顶层”的范畴莫尔斯分解导致应用到表示理论的代数，实现了量子哈密顿约化（例子包括许多辛反射代数）。Decategorifying导致应用程序的经典拓扑不变量的hyperkahler和代数辛代数。将这个框架应用于超对称规范理论中作为真空模出现的代数簇，将使我们深入了解这些物理定义的空间的数学结构。

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.