Mumford introduced in the 1960ies an algebraic approach to the construction of (almost) canonical bases of sections of ample line bundles on abelian varieties that permitted him to construct quasi-projective moduli spaces. His construction was later re-interpreted by Welters as a flat projective connection before being generalized by Hitchin to the non-abelian setting.

# Seminars

One of the key problems that has been considered on a pseudovariety V of finite monoids consists in deciding whether a given finite system of equations with rational constraints has a solution in every monoid from V. An instance of this problem is the separation problem, which has received considerable attention lately due to its role in investigations on the Straubing-Thérien hierarchy of star-free languages.

Let *S* be a smooth projective surface over **C** and* B *a smooth projective curve. A fibration *f* : *S* → *B* is a surjective morphism such that the general fibre is a smooth connected curve.

This talk will focus on some properties of fibrations with general fibre of genus ≥ 2, discussing in particular the existence and number of singular fibres on a fibration.

In 2008, Gehrke, Grigorieff, and Pin proposed Stone duality as a means for studying Boolean algebras of (non-necessarily regular) languages. They showed how that tool could be understood as a generalization of some key concepts found in the study of varieties of regular languages, such as the syntactic monoid of a language. In this talk, I will present the main points of this approach.

Let FIM(X) be the free inverse monoid on a finite set X. The word problem of FIM(X) is easily seen to be recognisable in linear space (e.g. using Munn trees), and this has been improved to log space (Lohrey and Ondrusch 2007).

Let Λ be a D-algebra in the sense of Bernstein and Beilinson. Higgs bundles, vector bundles with flat connections, co-Higgs bundles... are examples of Λ-modules for particular choices of Λ. Simpson studied the moduli problem for the classification of Λ-modules over Kähler varieties, proving the existence of a moduli space of Λ-modules. Using the Polishchuck-Rothstein transform for modules of D-algebras over abelian varieties, we obtain a description of the moduli spaces of Λ-modules of rank 1. We also proof that polystable Λ-modules decompose as a direct sum of rank 1 Λ-modules.

Takens' last problem. Whether are there persistent classes of smooth dynamical systems such that the set of initial states which give rise to orbits with

historic behavior has positive Lebesgue measure?

Colli-Vargas' conjecture. Every two-dimensional diffeomorphism with homoclinic tangency can be approximated in the Cr-topology by diffeomorphisms

having non-trivial wandering domains.

Non-abelian gerbes are a generalization of principal G-bundles, involving the replacement of the Lie group G by a Lie 2-group, or crossed module of groups, not necessarily Abelian. Apart from providing a nice example of categorification in geometry, they have found a number of applications in physics, e.g. in higher gauge theory and topological states of matter.

We generalize the classical MacDonald formula for smooth curves to reduced curves with planar singularities. More precisely, we show that the cohomologies of the Hilbert schemes of points on a such a curve are encoded in the cohomologies of the fine compactified Jacobians of its connected subcurves, via the perverse Leray filtration. A crucial step in the proof is the case of nodal curves, where the weight polynomials of the spaces involved can be computed in terms of the underlying dual graph. This is a joint work with Luca Migliorini and Vivek Schende.