I shall discuss the problem of determining the topological entropy of a polynomial map of the affine space. To attack this difficult problem, it is often important to find suitable compactifications adapted to the dynamics. And the existence of such compactifications is tightly related to the dynamics induced by the polynomial map on the space of valuations at infinity.

This is joint work with Bernd Siebert aiming to construct mirror pairs in complete generality, starting with either log Calabi-Yau manifolds (X,D) with maximally degenerate boundary or maximally unipotent degenerations of Calabi-Yau manifolds X->S. We construct the (homogeneous) coordinate ring of the mirror using a new type of logarithmic Gromov-Witten theory of (X,D) called punctured invariants, developed jointly with Abramovich, Chen and Siebert.

Recently mixed Hodge modules have been key to some developments in birational geometry. Originally they have been built by Saito as a Hodge theoretic counterpart of l-adic mixed perverse sheaves over finite fields. In this lecture, I will explain how the notion of perverse sheaf of geometric origin over a field of characteristic zero can be refined using the theory of étale motives of Ayoub-Morel-Voevodsky. This refinement shows that many constructions done with perverse sheaves are motivic and provides a bridge between positive characteristic and characteristic zero that allows to prove results in Hodge theory using arithmetic. I will illustrate this phenomenon with intersection cohomology. (This is joint work with Sophie Morel).

In his 1990 thesis, Y. Zhu attached to any vertex algebra V a commutative ring R_V. The affine scheme Spec(R_V) is known as the associated scheme of V, and the corresponding reduced scheme is known as the associated variety. These geometric objects were introduced by T. Arakawa, and they carry important information about the structure of V. In this talk, I will discuss how these constructions interact with the orbifold construction in vertex algebra theory. Given a group G of automorphisms of V, the G-orbifold of V is just the invariant vertex subalgebra V^G. An old conjecture, which was recently established by M. Miyamoto, is that if V is lisse (a finiteness condition that implies that its associated variety consists of just a point), and G is a finite group, then V^G is also lisse. A conjectural generalization of this result is that if G is finite, the associated varieties of V and V^G have the same dimension. One way to approach this conjecture is to find a precise relationship between R_{V^G} and the invariant ring (R_V)^G. I will formulate such a conjecture and describe some interesting examples where it holds.

Groechenig, Wyss and Ziegler have recently proved a conjecture of Hausel and Thaddeus concerning an equality between stringy Hodge numbers of moduli spaces of Higgs bundles for $mathrm{SL}_n$ and $mathrm{PGL}_n$. A crucial ingredient in their approach is the use of $p$-adic integration in the fibres of the Hitchin fibration. We will present a motivic version of their result which is obtained by using motivic integration. This is joint work with Dimitri Wyss.

To a compactification of an open variety, we can associate the dual boundary complex, a topological space that encodes the combinatoric of the boundary and reflects the geometry of the open variety. The points of the dual boundary complex determine valuations on the function field of the variety, defining a so-called skeleton in the Berkovich space of the variety. Thanks to this interpretation, dual boundary complexes can be studied both from the point of view of the birational geometry of the variety, and using Berkovich techniques. In this talk I will explain how this approach applies to the study of character varieties, central objects in non-abelian Hodge theory. According to the geometric P=W conjecture, it is expected that the dual boundary complex of the compactification of character varieties has the homotopy type of a sphere. In a joint work with Mirko Mauri and Matthew Stevenson, we compute the first non-trivial examples of these dual boundary complexes in the compact case, providing new evidence for the conjecture.

Some problems of polarized endomorphisms are reduced to some problems of local endomorphisms. For example, let X be a complex normal projective admitting a polarized endomorphism f, then this endomorphism induces the local endomorphism of some affine cone on the vertex. Furthermore, if X is Q-Gorenstein, then every irreducible components of log canonical locus is totally invariant replacing f by some iterate. Localization at the generic point of non log canonical locus we have the local endomorphism. But Broustet and Horing proved that X is log canonical, so the above set is empty. First of my talk, we consider valuation log canonical singularities, where it is an anoulogous singularity to log canonical singularity in non Q-Gorenstein setting, and non valuation log canonical locus is closed and totally invariant replacing f by some iterate. Next, we consider the relationship between valuative singularities and existence of local endomorphism.