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# Résultats de recherche

**5343**

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## A. Eskin - Counting closed geodesics on translation surfaces

We show that the number of closed geodesics in the flat metric on a translation surface of length at most R is asymptotic to e hR / (hR). This is joint work with Kasra Rafi Voir la vidéole (57m36s)

## C. Judge - Systoles in translation surfaces

I will discuss joint work with Hugo Parlier concerning the shortest noncontractible loops—’systoles’—in a translation surface. In particular, we provide estimates (some sharp) on the number of systoles (up to homotopy) in the strata H(2g-2) and the stratum H(1,1). We also determine the maximum systolic ratio (length squared/area) in H(2g-2), and we give a conjectural value for maximum systolic ratio in H(1,1). Voir la vidéole (57m17s)

## R. Santharoubane - Quantum representations of surface groups

I will show how we can produce exotic representations of surface groups from the Witten-Reshetikhin-Turaev TQFT. These representations have infinite images and give points on character varieties that are fixed by the action of the mapping. Moreover we can approximate these representations by representations into finite groups in order to build exotic regular finite covers of surfaces. These covers have the following property: the integral homology is not generated by pullbacks of simple closed curves on the base. This is joint work with Thomas Koberda. Voir la vidéole (59m45s)

## O. Paris-Romaskevich - Triangle tiling billiards

Tiling billiards is a dynamical system in which a billiard ball moves through the tiles of some fixed tiling in a way that its trajectory is a broken line, with breaks admitted only at the boundaries of the tiles. One can think about this system as a movement of the refracted light. Voir la vidéole (1h3m54s)

## A. Wright - Nearly Fuchsian surface subgroups of finite covolume Kleinian groups

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 the Dynnikov coordinate system. In this talk we describe polynomial time algorithms for calculating the number of connected components of a multi curve, and the geometric intersection number of two multicurves on the n-punctured disk, taking as input their Dynnikov coordinates. This is joint work Voir la vidéole (1h4m44s)

## J. Aramayona - MCG and infinite MCG (Part 2)

The first part of the course will be devoted to some of the classicalresults about mapping class groups of finite-type surfaces. Topics may include: generation by twists, Nielsen-Thurston classification,abelianization, isomorphic rigidity, geometry of combinatorial models.In the second part we will explore some aspects of "big" mapping class groups, highlighting the analogies and differences with their finite-type counterparts, notably around isomorphic rigidity, abelianization, and geometry of combinatorial models. Voir la vidéole (1h28m28s)

## J. Aramayona - MCG and infinite MCG (Part 3)

The first part of the course will be devoted to some of the classical results about mapping class groups of finite-type surfaces. Topics may include: generation by twists, Nielsen-Thurston classification, abelianization, isomorphic rigidity, geometry of combinatorial models. In the second part we will explore some aspects of "big" mapping class groups, highlighting the analogies and differences with their finite-type counterparts, notably around isomorphic rigidity, abelianization, and geometry of combinatorial models. Voir la vidéole (53m9s)

## D. Davis - Periodic paths on the pentagon

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 dynamics, and group structure. Some of the periodic trajectories exhibit a surprising "dense but not equidistributed" behavior. I will show pictures of periodic trajectories, which are very beautiful. This is joint work with Samuel Lelièvre and Barak Weiss. Voir la vidéole (1h31m15s)