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.

This is joint work with C. Florentino. The theory of motifs is an attempt to establish a universal cohomology theory for algebraic varieties. This has been completed in the case of smooth projective varieties, but the open case remains unsolved.
In this seminar, I will talk about our attempt to adapt our previous work on Hodge structures of character varieties to the simplest type of motifs, the so called naive motifs. I will also try to convey some recent ideas on equivariant Chow motifs to solve these problems.

There is already a generalized description of classes of moduli spaces of geometric structures on surfaces by Giansiracusa. That description includes moduli of spin surfaces, $r$-spin surfaces, surfaces with a principal G-bundle, surfaces with maps to a background space, among others. Here, we want to compute the homotopy groups of the moduli spaces of $k$-Higgs bundles, with the modular operads as main tool, and then, compare them with the results of Hausel and the results of Bradlow~et.al. Joint work with Jesús E. Sánchez Guevara.

Mirror symmetry is a prediction arising from theoretical physics which roughly conjectures that if we are given a pair of mirror Calabi-Yau varieties, called mirror partners, then the symplectic geometry of one of them (the Fukaya category) is somehow ``reflected'' on the holomorphic/algebraic geometry of the other (derived category of coherent sheaves). It is usually hard to find such mirror partners, but one important breakthrough was proposed by Strominger, Yau and Zaslow (SYZ) in the 90's, who established conditions to construct them.

Following the work of Miranda for triple covers, given a covering map $f\colon X\rightarrow Y$, I will show how to determine the structure as an algebra of the $\mathbb{O}_Y$ module $f_*\mathbb{O}_X$. When applied to Gorenstein covering maps this method brings the structure theorem of Casnati and Ekedahl to new light and can be used to describe a family of codimension $4$ Gorenstein ideals associated with covering maps of degree $6$.