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

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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.