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.

In this talk we discuss deformations of Lie groupoids and introduce the cohomology which controls them. We then study some of its properties, and give a geometric interpretation for it in low degrees. We will also look at relations with other cohomologies: deformation cohomologies for classical examples, such as Lie group actions and foliations, and deformation cohomology of Lie algebroids.

For a semisimple real Lie group $G$, parabolic $G$-Higgs bundles over a punctured Riemann surface emerge as the appropriate holomorphic objects corresponding to fundamental group representations of the surface with fixed holonomy around these punctures. In the absense of punctures, the study of the moduli spaces of (non-parabolic) $G$-Higgs bundles is a highly non-trivial and particularly fruitful subject, that has come to be the rapidly growing area known nowadays as Higher Teichmüller Theory.

Hitchin moduli spaces are non-compact hyperkähler manifolds parameterising solutions of gauge-theoretic equations on a Riemann surface.  There are two distinguished complex structures for which this gauge-theoretic moduli space can be realised as a moduli space of Higgs bundles and as a moduli space of local systems respectively on the Riemann surface.  In this talk I will focus on describing ten Hitchin moduli spaces on the Riemann sphere of hyperkähler dimension one (i.e.

During the first half of the talk I will give a survey about the Hitchin connection on the sheaf of non-abelian theta functions and its analogues in the theory of Conformal Blocks and TQFT. During the second part I will sketch the construction of the Hitchin connection (starting from the heat equation for abelian theta functions) from an algebro-geometric perspective. This is joint work with T. Baier, M. Bolognesi and J. Martens.

There exists a unique (smooth) cubic hypersurface of dimension 7 in $\mathbb{P}^8$ which is invariant under the order 3420 simple group $\mathrm{PSL}(2,19)$.
We study the intermediate Jacobian of that hypersurface and we prove that its subjacent 85-dimensional torus has a structure of an Abelian variety.
We also study a natural 9-dimensional abelian variety with $\mathrm{PSL}(2,19)$ action, which is related to the intermediate Jacobian.

A classical result due to Seidenberg states that every singular holomorphic foliation on a
complex surface can be turned into a foliation possessing only elementary singular points
(i.e. singular points possessing at least one eigenvalue different from zero) by means of a
finite sequence of (one-point) blow-ups. However, in dimension 3, the natural analogue
of Seidenberg theorem no longer holds as shown by Sancho and Sanz.