Alternatively have a look at the program.

## Lyapunov exponents of the Hodge bundle and diffusion in periodic billiards

Asymptotic behavior of leaves of a measured foliation on a Riemann surface is governed by the mean monodromy of the Hodge bundle along the associated trajectory of the Teichmüller geodesic flow in the moduli space. As a consequence, recent progress in the study of the Teichmüller flow (inspired by the fundamental work of A. Eskin and M. Mirzakhani) and in the study of the Lyapunov exponents of the Hodge bundle along this flow leads to new results on measured foliations on surfaces.

## Presentation and discussion of some problems in ergodic theory.

These include "moving" version of Poincaré recurrence theorem (discussed in a joint paper with E. Glasner) and combinatorial questions related to Interval Exchange Transformations.

## Dynamical systems of non-algebraic origins: fixed points, orbit lengths and distribution

We give a short survey of several theoretic and heuristic results, about the fixed points and orbit lengths of several dynamical system associated with iterations of functions of number theoretic nature. These include (in the historical order of study):

## Large deviations upper bound for the Teichmueller flow and applications

Based on the work of A. Eskin, M. Mirzakhani and A. Mohammadi on invariant measures of the $SL(2,R)$ action on translation surfaces, we provide an upper bound for deviations of Birkhoff integrals of the Teichmueller flow. It is a refinement of previous work of J. Chaika and A. Eskin. As a consequence, we obtain an upper bound for the Hausdorff dimension of angles in the periodic windtree model for which the associated flow is transient. This strengthen a previous result of A. Avila and P. Hubert who prove that this set has zero measure. Joint work with Artur Avila.

## Mahler measure, Fuglede-Kadison determinants and entropy of algebraic systems

Lind-Schmidt-Ward proved the logarithmic Mahler measure of an integral polynomial equals the entropy of a corresponding dynamical system (building on pioneering work of Yuzvniskii). After a breakthrough due to Deninger, this equality has been generalized greatly over recent years by several authors to an equality between the Fuglede-Kadison determinant of an element of the integral group ring of a group G and the sofic entropy of a corresponding action of G by automorphisms on a compact abelian group.

## Amorphic complexity of zero entropy systems

We introduce amorphic complexity as a new topological invariant which measures the complexity of dynamical systems in the regime of zero entropy. After stating some basic properties, we discuss its application to regular Toeplitz flows, cut and project quasicrystals and general almost automorphic minimal systems. Joint work with Maik Gröger and Gabriel Fuhrmann.

## Diophantine approximation for group actions

The classical theory of Diophantine approximation quantifies the density of rational number in the real line. In a joint work with A. Ghosh and A. Nevo we consider an analogous problem of approximating by dense orbits for actions on homogeneous spaces. We explain a general approach which allows to establish quantitative density and gives the best possible exponents of approximation in a number of cases.

## On the enumeration of surface coverings

We discuss how one can count branched covers of surfaces of prescribed degree and ramification type. Coverings of the sphere with simple ramification and Belyi maps serve us as main examples. We also compare these examples with the problem of counting square-tiled and pillowcase surfaces that is relevant in dynamics.

## Billiard flows, Laplace eigenfunctions, and Selberg zeta functions

Eigenfunctions of Laplace operators on Riemannian manifolds and orbifolds are objects of common interest in various fields, e.g. number theory, spectral theory, harmonic analysis, and mathematical physics. In particular by results from the latter field it is long known that these eigenfunctions are intimately related to geometric properties of the orbifold. However, the full extent of this relation and its consequences is still an active field of research.

## Lyapunov exponents of non-arithmetic complex hyperbolic lattices

To a flat vector bundle over a Riemannian manifold, one can associate its Lyapunov exponents, the logarithmic growth rates of sections when parallel transported along the geodesic flow. Flat bundles occurring in nature are the relative cohomology bundles associated to families of curves, or more generally Kaehler manifolds. In the case of a family of curves over a hyperbolic curve, there is a beautiful formula, first discovered by Kontsevich, that relates the sum of Lyapunov exponents to the degrees of certain line bundles.