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

表示论是数学的一个分支，通过将抽象的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，其中群运算对应于矩阵乘法。表示论在整个数学中有着广泛的影响。 它在物理学、化学和其他科学中也起着重要的作用，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，这是现代表征理论的一个重要组成部分。 朗兰兹纲领是一个广泛而有影响力的数学网络，它将看似无关的数学领域联系起来。 在他们原来的制定的代数联系数论和代表性理论的代数群。 最近对朗兰兹纲领的重新表述已进入几何学领域。 最简单的情况断言一阶矩阵微分方程和一组可逆矩阵的表示与罗朗级数作为条目之间的关系。 PI发展了一种新的方法来研究约化群G的不规则奇异平坦G-丛：Moy和Prasad关于p进群的基本层（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--与一个形式的平坦G-丛相关联。直觉上，这一层扮演着平坦G丛的“主导项”的角色，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将在具有toral奇点的射影线上构造平坦G-丛的模空间--奇点与某种特殊类别的地层相关联，称为规则地层。 他还将调查几何的不规则monodromy地图在此设置。PI将研究具有toral奇点的平坦G-丛的Deligne-Simpson和刚性问题。 他将构建德拉姆类似云的广义Kloosterman层，并表明他们是刚性的。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方面的对象相关联，并将研究地层上的诱导朗兰兹对偶。 特别是，他的目的是证明当地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.