(Joint work with Yi Huang) Goncharov and Shen introduced a Landau-Ginzberg potential on the Fock-Goncharov $A_{G,S}$ moduli space, where $G$ is a semisimple Lie group and $S$ is a ciliated surface. They used the potential to formulate a mirror symmetry via Geometric Satake Correspondence. This potential is the markoff equation for $A_{ PSL(2,R), S_{1,1} }$. When $S=S_{g,m}$, such potential can be written as a sum of rank $G*m$ partial potentials. We obtain a family of generalized Mcshane's identities by splitting these partial potentials for $A_{PSL(n,R),S_{g,m}}$ by certain pattern of cluster transformations with geometric meaning. We also find some interesting new phenomena in higher rank case, like triple ratio is bounded in mapping class group orbit. As applications, we find a generalized collar lemma which involves $lambda 1 / lambda 2$ length spectral, discreteness of that spectral etc. In further research, we would like to ask how can we integrate to obtain the generalized Mirzakhani's topological recursion with $mathcal{W}_n$ constraint?

Degenerations of maximal representations of a surface group may be seen as maximal representations in Sp(2n,F) for some non-Archimedean real closed field F. We associate to every such maximal representation a geodesic current whose intersection number is the length function of the representation for the L1 norm. When  the current is a measured lamination, we reconstruct an equivariant isometric embedding of the dual real tree in the Bruhat-Tits  building of Sp(2n,F).  This involves a general construction of an intersection current associated to a non necessarily continuous positive cross-ratio.

This is joint work with Marc Burger, Alessandra Iozzi, and Beatrice Pozzetti.

Nous organisons un petit groupe de travail pour essayer de mieux comprendre les liens entre la méthode de BKW complexe, la correspondance de Hodge nonabélienne sauvage et la récursion toplogique d’Eynard-Orantin.

The basic aim is to try to better understand the relation between exact WKB, wild nonabelian Hodge theory and the topological recursion of Eynard-Orantin, as well as links to the (nonlinear) Stokes phenomenon.

Nous organisons un petit groupe de travail pour essayer de mieux comprendre les liens entre la méthode de BKW complexe, la correspondance de Hodge nonabélienne sauvage et la récursion toplogique d'Eynard-Orantin.

The basic aim is to try to better understand the relation between exact WKB, wild nonabelian Hodge theory and the topological recursion of Eynard-Orantin, as well as links to the (nonlinear) Stokes phenomenon.

Séminaire Laurent Schwartz — EDP et applications

Séminaire Laurent Schwartz — EDP et applications

Among representations of surface groups into Lie groups, the Anosov representations are the ones with the nicest dynamical properties.

Guichard-Wienhard and Kapovich-Leeb-Porti have shown that their actions on generalized flag manifolds often admit co-compact domains of discontinuity, whose quotients are closed manifolds carrying interesting geometric structures.

Dumas and Sanders studied the topology and the geometry of the quotient in the case of quasi-Hitchin representations (Anosov representations which are deformations of Hitchin representations). In a conjecture they ask whether these manifolds are homeomorphic to fiber bundles over the surface.

In joint work with Qiongling Li, we can prove that the conjecture is true for (quasi-)Hitchin representations in SL(n,R) and SL(n,C), acting on projective spaces and partial flag manifolds parametrizing points and hyperplanes.

An important application of bounded cohomology is the theory of maximal representations: a class of homomorphisms of fundamental groups of Kähler manifolds (most notably fundamental groups of surfaces and finite volume manifolds covered by complex hyperbolic spaces) in Hermitian Lie groups (such as Sp(2n,R) or SU(p,q)). These representations have striking geometric properties and, in some cases, are even superrigid. In my talk I will discuss a joint work with Bruno Duchesne and Jean Lécureux in which we study generalizations to actions on infinite dimensional Hermitian symmetric spaces.

We will first explain the concepts of relatively hyperbolic group and the Bowditch boundary. We will then give some interesting examples of groups whose boundaries embed in the two-sphere. The most prominent family of this type is the class of geometrically finite Kleinian groups. However, we show that there are lots of relatively hyperbolic groups with planar boundaries that are not virtually Kleinian. We formulate a conjecture about which groups with planar boundary are virtually Kleinian, and prove this in a certain case. This is joint work in progress with Chris Hruska.

Séminaire de Probabilités et de Physique Théorique

 

We present results on the critical behaviour of long-range models of multi-component ferromagnetic spins and weakly self-avoiding walk in dimensions 1, 2, and 3. The range of the interaction is adjusted so that the models are below their upper critical dimension.  Critical exponents are computed for the susceptibility, specific heat, and critical two-point function, using a renormalisation group method to perturb around a non-Gaussian fixed point.  This provides a mathematically rigorous version of the epsilon expansion.

I will discuss the extension of Geroch's solution-generating method to the case of Einstein spaces. This will include the reduction to a three-dimensional coset space, the description of the dynamics in terms of a sigma-model and its transformation properties under the SL(2,R) group, and the reconstruction of new four-dimensional Einstein spaces. The detailed analysis of the space of solutions will be presented in a minisuperspace reduction, and will be performed using the Hamilton-Jacobi method. The cosmological constant will appear in this framework as a constant of motion.