Mini-courses

• François Labourie - Θ-positive representations of surface groups.

Θ-positive representations of surface groups are a certain type of representations of the fundamental group of a closed surface in some semi-simple groups, which seem to play a significant role amongst representations of surface groups. In this lecture, we will tour the subject and try to get a feeling that these representations form connected components of character varieties. The prototype of these connected components is Teichmüller space.

1) We will first describe and give a dynamical definition of generalised flag manifolds. We will then explain that these manifolds can be listed using an algebraic definition.

2) We then will explain what is a Θ-positive structure on a generalised flag manifold again using a dynamical approach, and its consequences in defining positive quadruples, positive curves, positive representations, and give examples.

3) We will then move to give a definition of Anosov representations, and explain that positive representations are a special case of Anosov representations. We will then show that positive representations form an open set in the set of Anosov representations and give a criterion (*) showing that a limit of positive representations is itself positive.

4) We finally show, thanks to a collar lemma akin to the celebrated collar lemma in hyperbolic geometry, that criterion (*) is actually always satisfied. We will explain the idea behind the proof of the collar lemma. 

These lectures are based on the work of Guichard—Wienhard, Guichard—Labourie—Wienhard, Beyrer—Guichard—Labourie—Pozzetti—Wienhard. This will be a Higgs-bundle free series of lectures and I plan to adapt its level to the culture of the audience. 

• Peter Smillie - Harmonic maps and Hitchin's equations at high energy 

The topic of these lectures will be the asymptotic behavior as t goes to infinity of the harmonic metrics on the family of stable Higgs bundle over a compact Riemann surface as the Higgs field is multiplied by t. The main goal will be to present most of the proof of the theorem of Mochizuki-Szabo that when the spectral curve is smooth, these metrics converge to a certain singular limit.

A key element in the proof if the asymptotic decoupling phenomenon: at any point of the Riemann surface where the eigenvalues of the Higgs field are distinct, the eigenlines of the Higgs field become asympotically orthogonal. This phenomenon is fundamental for many applications of Higgs bundles, and I intend to give a complete proof of it.

One application of this theorem is towards the conjecture of Gaiotto-Moore-Neitzke on the asymptotics of the Hitchin metric. Another application, at least of the asymptotic decoupling estimate, is to the study of equivariant minimal surfaces. I hope to say a bit about each of these.

Abstracts

• Colin Davalo - Constructing geometric structures from cyclic Higgs bundles

Since Baraglia's thesis, several methods have been introduced to geometrize the holonomy of cyclic Higgs bundles by constructing associated geometric structures modelled on flag manifolds. In this talk we will discuss a unified construction, recovering in particular previously known constructions, and see how it can be applied to study representations of surface groups in the exceptional group $G_2'$. This construction relies on the geometry of the associated non -positively curved symmetric space. This is a joint work with Parker Evans.

• Alexander Fruh - The Higgs centraliser decomposition and orbit method multiplicities

We consider a decomposition of the moduli stack of G-Higgs bundles using the centraliser dimension of the Higgs field. The largest piece in the decomposition is the familiar open dense locus of generically regular G-Higgs bundles, but much less is known about the remaining pieces, 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 loci and illustrate these results for low rank examples of G. We will also discuss how the geometry of the decomposition 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.