## Séminaire - Juin

Le séminaire aura lieu en deux mercredis de ce mois (exceptionnellement) de 14h à 15h à l'Institut Henri Poincaré à Paris. Pour télécharger les affiches du mois: juin1.pdf et juin2.pdf.

- 05/06/2019 - Simion Filip (IAS) - salle 05
- 19/06/2019 - Luke Jeffreys (University of Glasgow) - salle 05

Titre:
Dynamics on K3 surfaces: homogeneous and inhomogeneous

Resumé: K3
surfaces are defined by algebraic equations and have moduli spaces that
are homogeneous for appropriate Lie groups. At the same time, K3s admit
dynamically interesting automorphisms and geometric structures which
are far from homogeneous. I will discuss the interplay between the
homogeneous structures on the moduli spaces and the geometry and
dynamics of K3 automorphisms.
Joint work with Valentino Tosatti.

Titre:
Single-cylinder square-tiled surfaces

Resumé: Square-tiled
surfaces are Abelian differentials given by branched covers of the
square torus. They can be thought of as the integer points of a stratum
of the moduli space of Abelian differentials. As such, their counting
asymptotic is important in computing volumes of strata. An important
piece of combinatorial data is the number of horizontal and vertical
cylinders in a square-tiled surface. By recent work of
Delecroix-Goujard-Zograf-Zorich, square-tiled surfaces with one
horizontal and one vertical cylinder, which we shall call
1,1-square-tiled surfaces have a definite proportion among all
square-tiled surfaces and in fact, equidistribute as the number of
squares goes to infinity.

It is natural to ask what is the minimum number of squares required for a 1,1-square-tiled surface in a given stratum. In this talk, we construct such 1,1-square-tiled surfaces in every connected component of every stratum of Abelian differentials. With the exception of the hyperelliptic components, this can be done with the minimum number of squares required for a square-tiled surface in the ambient stratum. As an application, time permitting, we will use the main result to demonstrate the ubiquity of "ratio-optimising" pseudo-Anosovs.

It is natural to ask what is the minimum number of squares required for a 1,1-square-tiled surface in a given stratum. In this talk, we construct such 1,1-square-tiled surfaces in every connected component of every stratum of Abelian differentials. With the exception of the hyperelliptic components, this can be done with the minimum number of squares required for a square-tiled surface in the ambient stratum. As an application, time permitting, we will use the main result to demonstrate the ubiquity of "ratio-optimising" pseudo-Anosovs.