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

表示理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抽象代数结构。例如，群的表示是群中元素作为可逆矩阵的具体实现，群运算对应于矩阵乘法。表征理论在数学中有着广泛的影响。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，是现代表征理论的主要组成部分。朗兰兹计划是一个影响深远的猜想网络，将看似无关的数学领域联系在一起。在他们最初的表述中，他们的猜想将数论和代数群的表示论联系起来。最近对朗兰兹纲领的重新表述进入了几何领域。最简单的例子证明了一阶矩阵微分方程与一组以洛朗级数为项的可逆矩阵的表示之间的关系。PI开发了一种新的方法来研究可约群G的不规则奇异平面G束：这是Moy和Prasad关于p进群的基本地层（或最小k型）理论的几何版本。在几何理论中，人们将一个基本层——涉及环路代数上适当过滤的数据——与一个正式的平面g束联系起来。直观地说，这个地层在平坦的g束中起着“主导项”的作用，并允许人们定义其斜率。PI将使用这种方法来回答与不规则平坦g束和几何朗兰兹对应有关的各种问题。PI将在具有全部奇异点的射影线上构造平面g束的模空间——奇异点与某一类特殊的称为规则地层的地层有关。他还将在这种情况下研究不规则单形地图的几何形状。PI将研究具有总奇点的平面g束的delign - simpson问题和刚性问题。他将构建de Rham类似于Yun的广义Kloosterman轴，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方面的对象相关联，并将研究地层上的诱发朗兰兹对偶性。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.