For some classes of hyperbolic surfaces, all locally finite ergodic measures  invariant under the horocycle flow are described. See the works of Omri Sarig and those of Lindenstrauss-Landesberg in that sense. This talk focuses on the study of another class of hyperbolic surfaces. Precisely, we give an analytical construction of a family of surfaces of infinite type whose corresponding horocycle flow is conservative but not ergodic with respect to the Liouville measure.

Compact Hyperkaeler manifolds play a central role in complex algebraic geometry, as they arise as one of the building blocks of compact Kaehler manifolds with trivial canonical class. I will briefly recall the (very rich) theory of compact Hyperkaehler manifolds, hence I will pass to their analogue in the singular setting, where several different definitions are possible and convenient for different purposes.

In 1975 Narasimhan and Ramanan studied the action of a line bundle $L$ of finite order on the moduli space of vector bundles of fixed rank and degree via tensorization. They proved that fixed points are pushforwards of vector bundles of lower degree over an Étale cover of $X$ determined by $L$. We extend this study to $G$-Higgs bundles for $G$ reductive and consider the more general action of a finite group $\Gamma$. The action of an element in $\Gamma$ involves "tensorization by a line bundle", extension of structure group by an automorphism of $G$ and rescaling the Higgs field.

Abstract. In this talk we will focus on semicomplete Halphen vector fields.
It is shown that rational semicomplete Halphen vector fields are in ``one-to-one'' correspondence with singular uniformizable projective structures on compact Riemann surfaces.
In turn, many examples of the projective structures in question can be obtained by means of Teichmüller space techniques or, more precisely, from Bers simultaneous uniformization theorem.
The proof of this correspondence is the main topic for this discussion.

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.