Flat G-Bundles, Isomonodromy, and the Geometric Langlands Program

表示论是数学的一个分支，它通过将抽象代数结构的元素表示为矩阵来研究它们。例如，群的表示是群的元素作为可逆矩阵的具体实现，群运算对应于矩阵乘法。表示论在整个数学中影响无处不在。它在物理、化学和其他科学中也扮演着重要的角色，因为它为研究物理系统中对称性的影响提供了正确的语言。这个项目是几何朗兰兹计划的一部分，几何朗兰兹计划是现代表现理论的一个主要组成部分。朗兰兹计划是一个影响深远的猜想网络，将看似互不相关的数学领域联系在一起。在它们最初的公式中，猜想将数论和代数群的表示理论联系在一起。朗兰兹计划最近的重新设计已经进入了几何学领域。最简单的情况是一阶矩阵微分方程与一组以Laurent级数为项的可逆矩阵组的表示之间的关系。PI发展了一种新的方法来研究约化群G的非规则奇异平坦G丛：它是Moy和Prasad的基本层次(或极小K型)理论的几何版本。在几何理论中，人们将一个基本层--涉及循环代数上的适当过滤的数据--与形式的平坦G-丛联系起来。直观地说，这一层起到了扁平G丛的“主导项”的作用，并允许人们定义它的斜率。PI将使用这种方法来回答与不规则平坦G丛和几何朗兰兹对应有关的各种问题。PI将在投影线上构造平坦G-丛的模空间，其中包含奇点--奇点与某一特殊类别的地层相关联，称为规则地层。他还将在这种情况下研究不规则单色地图的几何形状。PI将研究具有对角奇点的平坦G-丛的Deligne-Simpson和刚性问题。他将构建De Rham类似于Yun的广义Krousterman Sheet，并证明它们是刚性的。此外，PI将表明，基本地层可能与几何朗兰兹对应的表示理论侧的对象相关联，并将研究地层上的诱导朗兰兹对偶。特别是，他的目标是证明局部几何朗兰兹保持深度。

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as matrices. For example, a representation of a group is a concrete realization of the elements of the group as invertible matrices, with the group operation corresponding to matrix multiplication. Representation theory has a pervasive influence throughout mathematics. It also plays an important role in physics, chemistry, and other sciences as it provides the correct language to study the effects of symmetry in a physical system. This project is part of the the geometric Langlands program, a major component of modern representation theory. The Langlands program is a network of far-reaching and influential conjectures connecting seemingly unrelated areas of mathematics. In their original formulation the conjectures linked number theory and the representation theory of algebraic groups. More recent reformulations of the Langlands program have moved into the realm of geometry. The simplest case asserts a relationship between first-order matrix differential equations and representations of a group of invertible matrices with Laurent series as entries. The PI has developed a new approach to the study of irregular singular flat G-bundles for reductive groups G: a geometric version of Moy and Prasad's theory of fundamental strata (or minimal K-types) for p-adic groups. In the geometric theory, one associates a fundamental stratum -- data involving an appropriate filtration on the loop algebra -- to a formal flat G-bundle. Intuitively, this stratum plays the role of the "leading term" of the flat G-bundle and allows one to define its slope. The PI will use this approach to answer various questions related to irregular flat G-bundles and the geometric Langlands correspondence. The PI will construct moduli spaces of flat G-bundles on the projective line with toral singularities -- singularities associated to a certain special class of strata called regular strata. He will also investigate the geometry of the irregular monodromy map in this setting. The PI will study the Deligne-Simpson and rigidity problems for flat G-bundles with toral singularities. He will construct de Rham analogues of Yun's generalized Kloosterman sheaves and show that they are rigid. Furthermore, the PI will show that fundamental strata may be associated to objects on the representation-theoretic side of the geometric Langlands correspondence and will study the induced Langlands duality on strata. In particular, he aims to prove that local geometric Langlands preserves depth.