We consider the moduli space of G-Higgs bundles over a compact Riemann surface X, where G is a real semisimple Lie group. By the non-abelian Hodge correspondence this is homeomorphic to the moduli space of representations of the fundamental group of X in G. We are interested in the fixed point subvarieties under the action of C*, obtained by rescaling the Higgs field. The fixed points correspond to variations of Hodge structure. They also correspond to critical subvarieties of a Morse function on the moduli space of Higgs bundles, known as the Hitchin functional. We show that one can define in this context an invariant that generalizes the Toledo invariant in the case where G is of Hermitian type. Moreover, there are bounds on this invariant similar to the Milnor–Wood inequalities of the Hermitian case. These bounds also generalize the Arakelov inequalities of classical Hodge bundles arising from families of varieties over a compact Riemann surface. We study the case where this invariant is maximal, and show that there is a rigidity phenomenon, relating to Fuchsian representations and higher Teichmüller spaces (Joint work with O. Biquard, B. Collier and D. Toledo).

I will discuss a famous problem posed by S. P. Novikov in 1982 in connection with the conductivity theory of monocrystals: what one can say about different types of asymptotic behavior of plane sections of a triply periodic surface? Apparently the most efficient approach to this problem is closely related to the techniques used in Teichmuller dynamics. I will try to explain historical perspectives and recent developments as well as challenging questions that still remain open.

We shall review the geometric recursion and its relation to topological recursion. In particular, we shall consider the target theory of continuous functions on Teichmüller spaces and we shall exhibit a number of classes of mapping class group invariant functions, which satisfies the geometric recursion. Many of these classes of functions are integrable over moduli spaces and we prove that these averages over moduli spaces satisfies topological recursion. The talk will end with some future perspectives of applications of geometric recursion. The construction of geometric recursion and the results relating it to topological recursion is joint work with Borot and Orantin.

A half-translation surface is given by a quadratic differential on a closed Riemann surface. It can be visualized by polygons in the plane whose edges are identified via translations and reflection-translations. The geometric properties of a half-translation surface can be captured in the saddle connection complex. The vertices of this complex are saddle connections (i.e. geodesic segments that connect a zero or pole to a zero or pole) and the simplices are formed by saddle connections that are non-intersecting. I will explain properties of the saddle connection complex, in particular the following rigidity result: Every simplicial isomorphism between the saddle connection complexes of two half-translation surfaces is induced by an affine diffeomorphism between the underlying surfaces. The talk is based on joint work with Valentina Disarlo and Robert Tang.

The Mathematical Sciences Research Institute (MSRI) and George Csicsery have started production of a one-hour documentary film about a remarkable mathematician whose contributions were recognized with a Fields Medal just a few years before her untimely death.

The biographical film is about Maryam Mirzakhani, a brilliant woman, and Muslim immigrant to the United States who became a superstar in her field. The story of her life will be complemented with sections about Mirzakhani’s mathematical contributions, as explained by colleagues and illustrated with animated sequences. Throughout, we will look for clues about the sources of Mirzakhani’s insights and creativity.

A meander is a (homotopy class of) a pair of curves on the sphere with transverse intersections. The meandric number M_n is the count of "rooted" meanders with 2n intersections. One has M_1=1, M_2=2, M_3=8 and M_4=42. Understanding the asymptotic growth rate of M_n remains an open problem in mathematics/statistical physics. In this talk we will consider the refinement M_{n,k} that counts meanders with 2n intersections and whose complement has exactly k bigons. In this settings, we can describe the asymptotics as n tends to infinity for each fixed k.

Our solution relies on associating a quadratic differential to a meander (a so called square tiled surface) and relates the counting of M_{n,k} to the Masur-Veech volumes of quadratic differentials on the sphere with k punctures.

This is a joint work with E. Goujard, P. Zograf and A. Zorich.

Hitchin components, positive representations in the sense of Fock and Goncharov, and maximal representations are families of higher Teichmüller spaces and important examples of Anosov representations, which provide a large class of discrete subgroups in higher rank Lie groups. In this talk I will discuss some recent results and open open questions.

Let M (2, L) denote the moduli space of stable vector bundles of rank 2 and determinant L of odd degree, on a smooth curve of genus g ≥ 2. Owing to the work of Mumford, Newstead, Thaddeus, King, Kirwan and several others, questions such as the generators and relations, higher rank Torelli-type theorems as well as the Hodge conjecture for the cohomology ring of M (2, L) are well understood. In this talk I will survey some of these aspects for the smooth case and discuss analogous results for the case when the underlying curve is irreducible, nodal. This is joint work with S. Basu and A. Dan.

In joint work with David Aulicino, we show there are no Shimura-Teichmuller curves in genus 5. Together with Moeller's results this completes the classification of Teichmuller curves with completely degenerate Kontsevich-Zorich spectrum. In this talk I hope to highlight the main new technique - a parametric jump problem - with the hope it can be useful to answer other questions.

In this joint work with Jérémy Toulisse and Mike Wolf, we prove the existence (and discuss the uniqueness)  of maximal surfaces in the pseudo-hyperbolic space $H_{2,n}$ 

of signature (2,n) with a prescribed quasi-symmetric boundary at infinity. 

No knowledge of Lorentzian geometry will be assumed. We will mainly discuss the notion of quasi-symmetric curves in the boundary at infinity of $H_{2,n}$, boundary known as the EInstein universe