*Speaker:* **Sunrose Shrestha** (Tufts University, USA)

*Title: ***The topology and geometry of random square-tiled surfaces**

*Abstract:* A square-tiled surface (STS) is a branched cover of the standard square torus with branching over exactly one point. They are concrete examples of translation surfaces which are an important class of singular flat metrics on 2-manifolds with applications in Teichmüller theory and polygonal billiards. In this talk we will consider a randomizing model for STSs based on permutation pairs and use it to compute the genus distribution. We also study holonomy vectors (Euclidean displacement vectors between cone points) on a random STS. Holonomy vectors of translation surfaces provide coordinates on the space of translation surfaces and their enumeration up to a fixed length has been studied by various authors such as Eskin and Masur. In this talk, we obtain finer information about the set of holonomy vectors, Hol(S), of a random STS. In particular, we will see how often Hol(S) contains the set of primitive integer vectors and find how often these sets are exactly equal.

*Speaker:* **Sandro Vaienti ** (Centre de Physique Théorique, Marseille, France)

*Title: ***Thermodynamic formalism for random weighted covering systems**

*Abstract:* We develop a quenched thermodynamic formalism for random dynamical systems generated by countably branched, piecewise-monotone mappings of the interval that satisfy a random covering condition.

Joint with J. Atnip, G. Froyland and C. Gonzalez-Tokman

*Speaker:* **Francesco Cellarosi** (Queen’s University, Canada)

*Title:* *Rational Horocycle lifts and the tails of Quadratic Weyl sums*

*Abstract:* Equidistribution of horocycles on hyperbolic surfaces has been used to dynamically answer several probabilistic questions about number-theoretical objects. In this talk we focus on horocycle lifts, i.e. curves on higher-dimensional manifolds whose projection to the hyperbolic surface is a classical horocycle, and their behaviour under the action of the geodesic flow. It is known that when such horocycle lifts are `generic’, then their push forward via the geodesic flow becomes equidistributed in the ambient manifold. We consider certain ‘non-generic’ (i.e. rational) horocycle lifts, in which case the equidistribution takes place on a sub-manifold. We then use this fact to study the tail distribution of quadratic Weyl sums when one of their arguments is random and the other is rational. In this case we obtain random variables with heavy tails, all of which only possess moments of order less than 4. Depending on the rational argument, we establish the exact tail decay, which can be described with the help of the Dedekind \psi-function.

Joint work with Tariq Osman.

**NOTE:** *The seminar will be streamed live on our YouTube channel then saved there. If you ask questions, with your video feed on or off, you agree to the use of your image/spoken words for said purpose.*

*Speaker:* **Martin Leguil** (Université Paris-Sud 11, France)

*Title: ***Some rigidity results for billiards and hyperbolic flows **

*Abstract:* In a project with P. Bálint, J. De Simoi and V. Kaloshin, we have been studying the inverse problem for a class of open dispersing billiards obtained by removing from the plane a finite number of smooth strictly convex scatterers satisfying a non-eclipse condition. The dynamics of such billiards is hyperbolic (Axiom A), and there is a natural labeling of periodic orbits. We show that it is generically possible, in the analytic category and for billiard tables with two (partial) axial symmetries, to determine completely the geometry of those billiards from the purely dynamical data encoded in their Marked Length Spectrum (lengths of periodic orbits + marking). An important step is the obtention of asymptotic estimates for the Lyapunov exponents of certain periodic points accumulating a reference periodic point, which turn out to be useful in the study of other rigidity problems. In particular, I will explain the results obtained in a joint work with J. De Simoi, K. Vinhage and Y. Yang on the question of entropy rigidity for 3-dimensional Anosov flows and dispersing billiards.

**NOTE:** *The seminar will be streamed live on our YouTube channel then saved there. If you ask questions, with your video feed on or off, you agree to the use of your image/spoken words for said purpose.*