Abstract:

The complexity of the singular fibers of the Hitchin system stems from the diversity of singularities of spectral curves. Already for G=GL(2,C) all singularities of type A appear. In light of the Deligne-Mumford compactification of the moduli space of smooth projective curves, it is a natural idea to compactify the family of smooth spectral curves over the regular locus of the Hitchin base to a family of nodal curves over a modified Hitchin base. In the talk, I will compare several approaches to achieve this goal in the GL(2,C)-case.

Hitchin systems are algebraically completely integrable systems foliating a dense subset of Higgs bundle moduli spaces by abelian torsors. This structure played a crucial role in recent developments in the field, e. g. Langlands duality and the GMN-conjecture. In this talk, we take a closer look at degenerate fibres of the symplectic and odd orthogonal Hitchin system. We give a new approach identifying a certain class of these fibres with SL(2,C)-Hitchin fibres of twisted Higgs bundles.

Hitchin's connection, originally constructed using techniques of Kähler geometry, is a flat projective connection on bundles of non-abelian theta functions. In this talk we will discuss its construction in algebraic geometry, and notably also for ground fields of (almost any) positive characteristic. Time permitting we will also discuss applications to higher rank Prym varieties.

This is joint work with M.Bolognesi, J.Martens and C.Pauly.

Consider the moduli space $\mathcal{M}(G)$ of $G$-Higgs bundles on a compact Riemann surface $X$, for a real semisimple Lie group $G$. Hitchin components in the split real form case and maximal components in the Hermitian case were, for several years, the only known source of examples of higher Teichmüller components of $\mathcal{M}(G)$.

A discrete statistical model is a subset of a probability simplex. Its maximum likelihood estimator (MLE) is a retraction from that simplex onto the model. We characterize all models for which this retraction is a rational function. This is a contribution via real algebraic geometry which rests on results on Horn uniformization due to Huh and Kapranov. We present an algorithm for constructing models with rational MLE, and we demonstrate it on a range of instances. Everything will be explained from basic principles and no knowledge of statistics will be assumed.

While it is well known that the moduli space of $G$-bundles over a smooth projective curve is compact, it is not the case for an arbitrary base variety. This motivated the definition of $G$-sheaves by Gomez and Sols who proved that their moduli space is a compactification of the moduli space of $G$-bundles.
In this talk I will study the deformation and obstruction theory of these objects when $G$ is either the symplectic or the orthogonal group. This is joint work with T. Gomez.

With $G=GL(n,\mathbb{C})$, let $\mathcal{X}_{\Gamma}G$ be the $G$-character variety of a given finitely presented group $\Gamma$, and let $\mathcal{X}_{\Gamma}^{irr}G\subset\mathcal{X}_{\Gamma}G$ be the locus of irreducible representation conjugacy classes. We provide a concrete relation, in terms of plethystic functions, between the generating series for Hodge-Euler (also called $E$-) polynomials of $\mathcal{X}_{\Gamma}G$ and the one for $\mathcal{X}_{\Gamma}^{irr}G$, generalizing a formula of Mozgovoy-Reineke.

Hyperpolygons spaces are a family of (finite dimensional, non-compact) hyperkähler spaces, that can be obtained from coadjoint orbits by hyperkähler reduction. Jointly with L. Godinho, we show that these space are diffeomorphic (in fact, symplectomorphic) to certain families of parabolic Higgs bundles. In this talk I will describe this relation and use it to analyse the fixed points locus of a natural involution on the moduli space of parabolic Higgs bundles.

In Donagi-Pantev's programme to deduce geometric Langlands from its abelianised version, wobbly and shaky bundles play a key role. A conjecture by both authors establishes the equality of these loci.

In this talk, I will explain a joint result with Christian Pauly giving a geometric criterion for wobbliness, and report on ongoing work about the applicability of the ideas therein towards the proof of the conjectural equality.