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 $6$.

A group $G$ is Jordan if there exists a constant $C$ such that
any finite subgroup of $G$ has an abelian subgroup of index at most $C$.
For example, $\mathrm{GL}(n,\mathbb{R})$ is Jordan for every $n$. Some 30 years ago E. Ghys
asked whether diffeomorphism groups of closed manifolds are Jordan.
A number of papers have been written on this question in the past few
years. It is known that there are lots of manifolds whose
diffeomorphism group is Jordan, and also lots of manifolds for which

A generalised Kummer surface $X=Km(T,G)$ is the minimal resolution of the quotient $T/G$ of a complex torus $T$ by a finite group $G$ of symplectic automorphisms.
In this talk we will give an account on the previous work on the subject, which was launched by Nikulin, then followed by Bertin and Garbagnati.
We will then finish the classification of these surfaces, obtaining results analogous to the well-known work of Nikulin for the group $G=\mathbb{Z}/2\mathbb{Z}$ and the (classical) Kummer surfaces.

In this talk we will discuss a construction of scalar-flat Kähler toric metrics on non-compact 4-manifolds. The construction actually gives a new perspective on Gibbons-Hawking’s so called gravitational instantons. Perhaps more interestingly, it allows us to write down some new examples of complete scalar-flat Kähler metrics on some important non-compact manifolds (namely non-compact toric surfaces). We will discuss the energy of such metrics and point to some open directions regarding the construction. Some of this is joint work with Miguel Abreu.

The Bethe Ansatz equations were initially conceived as a
method to solve some particular Quantum Integrable Models (IM), but
are nowadays a central tool of investigation in a variety physical and
mathematical theory such as string theory, super-symmetric Gauge
theories, and Donaldson-Thomas invariants. Surprisingly, it has been
observed, in several examples, that the solutions of the same Bethe
Ansatz equations are provided by the monodromy data of some ordinary
differential operators with an irregular singularity (ODE/IM
correspondence).

I’ll explain how the absolute Galois group of the rationals acts on 
a space which is closely related to the space of all knots. The path components
of this space form a finitely generated abelian group which is, conjecturally, a
universal receptacle for (integral) finite-type knot invariants. The added Galois 
symmetry allows us to extract new information about its homotopy and 
homology. This is joint work with Geoffroy Horel.

We  explain  the construction  of an explicit regulator map  at the level of complexes:
\[
\operatorname{Reg} \colon {CH_\Delta^{p}(X, n)}   \longrightarrow H^{2p-n}_{\mathscr{D}}(X;\mathbb{Z}(p)),
\]
from the higher Chow groups of a smooth complex algebraic variety \(  X \),  in their simplicial formulation with \(  \mathbb{Z} \) coefficients, into integral Deligne-Beilinson cohomology.