Dima Arinkin
Derived category of the stack of Higgs bundles
The moduli of Higgs bundles is equipped with a Hitchin map; this lets us consider its smooth fibers are essentially abelian varieties, and singular fibers can be viewed as degenerate abelian varieties. By a results of Donagi and Pantev, the Hitchin fibrations for Langlands dual groups are generically dual to each other: the bases of the two Hitchin fibrations can be identified, and smooth fibers are dual abelian varieties. It is conjectured that the duality extends to singular fibers; such extension could be thought of as the 'classical limit' of the geometric Langlands correspondence.
In this talk, I will explore the derived category of coherent sheaves on the stack of Higgs bundles. Working with a stack, rather than the moduli space of semistable Higgs bundles, is natural from the point of view of the Langlands program. As we will see, this also leads to certain issues (once we leave the elliptic locus on the Hitchin base). My goal is to show that, even in the case of Higgs bundles over GL(n), the most 'naive' version of the statement is inconsistent, and to see what claims have a chance of being true.
Ekaterina Bogdanova
Local systems with restricted variation on the formal punctured disc via factorization
We will talk about the restricted local geometric Langlands conjecture, which is a local counterpart of the work of Arinkin, Gaitsgory, Kazhdan, Raskin, Rozenblyum and Varshavsky. Roughly speaking, it states that there is a canonical equivalence between the category of restricted categorical representations of the loop group of a reductive group $G$ and the category of sheaves of categories on the stack of $\check{G}$-local systems with restricted variation on the formal punctured disc. We will define this stack, and will explain how to describe the latter category in terms of the category of representations of $\check{G}$ via Beilinson-Drinfeld factorization.
Alexis Bouthier
Affine Grassmannians of pseudo-reductive groups.
When one studies affine smooth groups over general fields, one is naturally led to pseudo-reductive groups. The first systematic study of it have been made by Conrad-Gabber-Prasad. When we go to deeper applications such as a classification of G-torsors on P^1, Cartan or Birkohff decomposition, affine grassmannians of such naturally appear. We will thus explain several foundational results on these new affine grassmannians and discuss applications. Work in common with K. Cesnavicius and F. Scavia.
Dennis Gaitsgory
Restricted categorical representations of loop groups
Pengfei Huang
Filtered Stokes local systems and their moduli spaces
Analogous to the role of filtered local systems in Simpson’s tame nonabelian Hodge correspondence, filtered Stokes local systems are the appropriate topological objects in wild nonabelian Hodge correspondence. In this talk, we will introduce filtered Stokes local systems and demonstrate a purely algebro-geometric construction of their moduli spaces. As an application, we will derive a wild nonabelian Hodge correspondence. This approach is applicable to general reductive groups. Based on joint works with Hao Sun.
Xin Jin
Mirror symmetry for the affine Toda systems
I'll present recent work on mirror symmetry for the affine Toda systems, which can be viewed as a Betti Geometric Langlands correspondence in the wild setting. More explicitly, we realize the affine Toda system (associated to a complex semisimple group) as a moduli space of Higgs bundles on P^1 with certain automorphic data, and the dual side is the group version of the universal centralizer (associated to the dual group), which is a wild character variety. We show that the wrapped Fukaya category of the former is equivalent to the dg-category of coherent sheaves of the latter. This is joint work with Zhiwei Yun.
Qiongling Li
Harmonic metrics on Higgs bundles over non-compact surfaces
For a Higgs bundle over a compact Riemann surface of genus at least 2, the Hitchin-Kobayashi correspondence says the existence of a harmonic metric is equivalent to the polystability of the Higgs bundle. In this talk, we discuss some recent progress on the existence and uniqueness of harmonic metrics on Higgs bundles over general non-compact Riemann surfaces. This is joint work with Takuro Mochizuki.
Claude Sabbah
Geometric properties of the irregular Hodge filtration
I will give an overview of recent advances concerning the irregular Hodge filtration (introduced by Deligne 40 years ago) and I will focus on Kodaira vanishing theorems similar to those of Saito for mixed Hodge modules. If time permits, I will show how (still) conjectural specialization theorems with slopes lead to a good control of the smallest piece of the irregular Hodge filtration à la Kollár-Saito.
Annette Werner
On new developments in non-abelian p-adic Hodge Theory
This talk will introduce background, achievements and challenges in the quest to find p-adic analogs of the celebrated Corlette-Simpson correspondence, which on Kähler manifolds relates representations of the fundamental group to certain Higgs bundles. In the non-Archimedean world, vector bundles for the v-topology on Scholze’s diamonds have proven to be a useful framework to study analogous problems. This leads to some results and many questions even beyond the scope of the classical setting. In particular, I will explain results for abeloid varieties obtained jointly with Ben Heuer, Lucas Mann and Mingjia Zhang.