Mini-courses

Abstracts

• Alexander Fruh - Higgs centraliser strata and orbit method multiplicities

We consider a stratification of the moduli stack of G-Higgs bundles by the centraliser dimension of the Higgs field. The open stratum is the familiar locus of generically regular G-Higgs bundles, but much less is known about the remaining strata, which in particular lie over the locus of non-reduced cameral curves in the Hitchin base. We will give some general statements on the structure of the Hitchin map over these strata and illustrate these results for low rank examples of G. We will also discuss how the geometry of the strata is related to a representation-theoretic notion of multiplicity via Losev's version of the orbit method.

• Guillermo Gallego - Multiplicative Hitchin fibrations and Langlands duality

Hitchin fibrations for Langlands dual groups, when restricted to a dense open locus of the Hitchin base, are known to be dual (relative) Beilinson 1-motives; that is, the neutral connected components of their coarse moduli spaces are dual abelian schemes and their inertia and component groups are exchanged under Cartier duality. Following the recent preprint (arxiv:2509.14364), we explore a similar situation in the context of multiplicative Hitchin fibrations. These are group-valued analogues of the Hitchin fibration, modelled over the Steinberg map from a reductive group to the GIT quotient of it by its own adjoint action. The corresponding pairs matched under duality can be classified in terms of the duality for affine Dynkin diagrams. In this talk, we will provide a self-contained introduction to multiplicative Hitchin fibrations (untwisted and twisted) as well as an explanation of this duality.

• Miguel González - Very stable parabolic Higgs bundles and affine flag varieties

Very stable and wobbly Higgs bundles were introduced by Hausel and Hitchin, motivated by the study of mirror symmetry phenomena in moduli spaces of Higgs bundles over smooth projective complex curves. We will recall these notions and, motivated by similar mirror symmetry aspects that appear in moduli spaces of strongly parabolic G-Higgs bundles, we will explain how very stable points with generically regular Higgs field can be studied by performing Hecke transformations. As a result, we will provide a classification in terms of the combinatorics of the affine flag variety for G and the Bruhat order on its extended affine Weyl group. From the point of view of mirror symmetry, the resulting very stable points determine BAA-branes and we will analyse what is the corresponding mirror BBB-brane.   

• Robert Hanson - Higgs bundles and Langlands duality

We survey old and new ideas on the Dolbeault geometric Langlands conjecture of Donagi—Pantev. I will explain (1) how Fourier—Mukai transforms solve an irreducible subset of the conjecture; (2) a trick for computing the transforms by normalising singular spectral curves; (3) methods for spreading out Fourier—Mukai duality using parabolic induction. (2) is based on joint work with Franco, Horn, and Oliveira (arXiv:2405.11860), while (3) is based on arXiv:2512.24239 and recent work of Padurariu—Toda.

• Enya Hsiao - Odd magical triples and maximal Higgs bundles

Higher Teichmüller theory is the study of connected components in the G^R-character variety consisting entirely of discrete and faithful surface group representations. Under the nonabelian Hodge correspondence, various problems in higher Teichmüller theory can be approached by using topological methods on the G^R-Higgs bundle moduli space. 

Following recent developments of Theta-positivity on the character variety side, it has been proposed by Bradlow, Collier, Garcia-Prada, Gothen and Oliveira that the corresponding components on the Higgs bundle moduli space are characterized by Slodowy slices of magical sl2-triples. In this talk, I will explain how the above framework can be extended to include the case of maximal components of a nontube type Hermitian Lie group by introducing the notion of odd magical triples, further supporting the expectation that all higher Teichmüller components arise via a Cayley correspondence. 

• Elsa Maneval - Topological mirror symmetry for SLn/PGLn Higgs bundles

I will first introduce the moduli spaces of Higgs bundles that appear in the Hausel-Thaddeus topological mirror symmetry conjecture, present its different proofs and generalisations. In particular I will explain the p-adic integration approach of Groechenig, Wyss and Ziegler. Finally, I will present my result, which is a generalisation beyond the original coprime case of the key intermediate step of this approach, which we call a non-archimedean topological mirror symmetry.