(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?

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.

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

Hadamard Lectures 2018

 

Retrouvez toutes les informations sur le site de la Fondation Mathématique Jacques Hadamard :

 

https://www.fondation-hadamard.fr/fr/financements-accueil-206-cours-avances/accueil-lecons-hadamard

Séminaire Laurent Schwartz — EDP et applications

We study the spectrum of local operators living on a defect in a generic conformal field theory, and their coupling to the local bulk operators.  We establish the existence of universal accumulation points in the spectrum at large s, s being the charge of the operators under rotations in the space transverse to the defect. Our tools include a formula that inverts the bulk to defect OPE and is analytic in s, analogous to the Caron-Huot formula for the four-point function. Some important assumptions are made in deriving this result: we comment on them.

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.

Hall algebras play an important role in representation theory, algebraic geometry and combinatorics. The Hall algebra of an exact or a triangulated category captures information about the extensions between objects. We consider twisted and extended Hall algebras of triangulated categories and note that in some cases they are well-defined even when their non-extended counterparts are not. We show that each exact category with weak equivalences with an appropriate extra structure naturally gives rise a twisted extended Hall algebra of its homotopy category. If time permits, we will discuss the relation of this construction to graded quiver varieties and to categorification of modified quantum groups.