Frenkel--Gross introduced a notable family of $G$-connections on the projective line minus two points with irregular singularities. We get hypergeometric connections from this construction for classical groups. In this talk, I will talk about an ongoing project with Andreas Hohl, Peter Spacek, and Christian Sevenheck, focusing on the exploration of the irregular Hodge filtration on Frenkel--Gross connections. Time permitting, I will also mention a new method for calculating that on hypergeometric connections, developed in collaboration with Daxin Xu.
To a projective homogeneous space embedded via a power of its anticanonical bundle one associates a D-module on the space of sections of the line bundle, called the tautological system. As shown recently in a joint work with Reichelt, Sevenheck, Steiner, Walther, this D-module underlies the structure of a complex mixed Hodge module that describes the (twisted) cohomology of hyperplane section complements. In this talk, I will discuss progress on toric degenerations of such tautological systems, relating them inside the category of complex mixed Hodge modules via a degenerating family of the homogeneous variety to a certain quotient of a GKZ system, a tautological system of the toric fiber and its automorphism group. Particular emphasis is put on Grassmannians in their Plücker embedding, especially the case of Gr(2,n).
Tensor categories of perverse sheaves play an important role in the geometry and arithmetic of irregular varieties. The arising Tannaka groups are powerful invariants but usually hard to compute. For perverse sheaves on elliptic curves, Collas-Dettweiler-Reiter-Sawin have obtained new examples of exceptional groups starting from a topological description of the convolution product via braid group actions; in the talk I will discuss recent progress that seems to indicate a surprisingly simple interpretation of the arising Lie algebras in terms of vanishing cycles.
In recent years there has been much interest in the study of the canonical Hodge and weight filtrations associated to the sheaf of meromorphic functions on a complex manifold with poles along some fixed divisor, instigated mainly by Mustață and Popa. We are interested in the computation of weighted Hodge ideals with an aim towards understanding these filtrations and thus in turn the (mixed) Hodge theory of the complement of the divisor in question. We have particular interest in the generation level of the Hodge filtration on the respective steps of this weight filtration, and in trying to determine how many steps the weight filtration has, and how this is related to multiplicities of roots of the Bernstein-Sato polynomial. Work in progress.
We use the l-adic Fourier transform and Laumon's theory of epsilon constants in order to determine Frobenius determinants of the arithmetic middle convolution. This leads to new Galois realizations of special linear groups.