O. Paris-Romaskevich - Triangle tiling billiards

In  the  first two lectures I will try to tell (or, rather, to  give  an  idea)  of  how  Maryam Mirzakhani has counted simple  closed  geodesics on hyperbolic surfaces. I plan to briefly  mention her

Multicurves have played a fundamental role in the study of mapping class groups of surfaces since the work of Dehn. A beautiful method of describing such systems on the n-punctured disk is given by

In  the  first two lectures I will try to tell (or, rather, to  give  an  idea)  of  how  Maryam Mirzakhani has counted simple  closed  geodesics on hyperbolic surfaces. I plan to briefly  mention her

In these lectures we summarized results on the cohomological equation for translation flows on translation surfaces (myself, Marmi, Moussa and Yoccoz, Marmi and Yoccoz) and apply these results to the

We will give the proof of Mirzakhani's theorem that the earthquake flow and Teichmuller unipotent flow are measurably isomorphic. We will assume some familiarity with quadratic differentials, but no

We discuss the problem of finding the minimal dilatation among pseudo-Anosov mapping classes on a fixed closed surface. In particular, for every nonorientable closed surface of even genus, we consider

Mathematicians have long understood periodic trajectories on the square billiard table. In the present work, we describe periodic trajectories on the regular pentagon – their geometry, symbolic

1. Symbolic dynamics: Arnoux - Rauzy words and Rauzy gasket 2. Topology: Arnoux - Yoccoz example and its generalization 3. Novikov’s problem: how dynamics meets topology and together they help to

Effective convergence of ergodic averages and cusp excursions of geodesics on moduli spaces We survey some applications of effective convergence of ergodic averages to the analysis of cusp ex

In the principal stratum in genus two, McMullen observed that something odd happens - there is only one nonarithmetic Teichmuller curve - the one generated by the decagon.  This strange phenomenon

a) basic definitions and examples b) strata and genus c) reduced and primitive origamis, SL(2,R) action, Veech groups d) automorphisms and affine homeomorphisms e) homology of origamis f) Kontsevich