Hitchin's equations are a system of gauge theoretic equations on a Riemann surface that are of interest in many areas including representation theory, Teichmu ̈ller theory, and the geometric Langlands correspondence. In this talk, I'll describe what solutions of SL(n, C)-Hitchin's equations “near the ends” of the moduli space look like, and the resulting compactification of the Hitchin moduli space. Wild Hitchin moduli spaces are an important ingredient in this construction. This construction generalizes Mazzeo-Swoboda-Weiss-Witt's construction of SL(2, C)-solutions of Hitchin's equations where the Higgs field is “simple.”[-]

In this talk I will provide a brief and gentle introduction to Witten's conjecture, which predicts that the generating series of certain intersection numbers on the moduli space of curves is a tau function of the KdV integrable hierarchy, as a motivation for r-spin Witten's conjecture that concerns much more complicated geometric objects and specialises to the original conjecture for r=2. The r=2 conjecture was proved for the first time by Kontsevich making use of maps arising from a cubic hermitian matrix model with an external field. Together with R. Belliard, S. Charbonnier and B. Eynard, we studied the combinatorial model that generalises Kontsevich maps to higher r. Making use of some auxiliary models we manage to find a Tutte-like recursion for these maps and to massage it into a topological recursion. We also show a relation between a particular case of our maps and the r-spin intersection numbers, which allows us to prove that these satisfy topological recursion. Finally, I will explain how, in joint work with G. Borot and S. Charbonnier, we relate another specialisation of our models to fully simple maps, and how this identification helps us prove that fully simple maps satisfy topological recursion for the spectral curve in which one exchanges x and y from the spectral curve for ordinary maps. This solved a conjecture from G. Borot and myself from '17.[-]

Mirzakhani's recursion for Weil-Petersson volumes was shown by Eynard and Orantin to be equivalent to Topological Recursion with a specific choice of spectral curve. However, such a recursion is known to produce formal power series with factorially growing coefficient which, according to the theory of Resurgence, should be upgraded to “transseries” via the computation of non-perturbative contributions (i.e. instantons). In this talk I will show how a non-perturbative formulation of Topological Recursion allows for the computation of such contributions which, through simple resurgent relations, allow to obtain large genus asymptotics of Weil-Petersson volumes.[-]

Après avoir expliqué la notion de Z-invariance pour les modèles de mécanique statistique, nous introduisons une famille à un paramètre (dépendant du module elliptique) de Laplaciens massiques Z-invariants définis sur les graphes isoradiaux. Nous démontrons une formule explicite pour son inverse, la fonction de Green massique, qui a la propriété remarquable de ne dépendre que de la géométrie locale du graphe. Nous expliquerons les conséquences de ce résultat pour le modèle des forêts couvrantes, en particulier la preuve d'une transition de phase d'ordre 2 avec le modèle des arbre couvrants critiques sur les graphes isoradiaux, introduit par Kenyon. Finalement, nous considérons la courbe spectrale de ce Laplacien massique et montrons qu'il s'agit d'une courbe de Harnack de genre 1.
Il s'agit d'un travail en collaboration avec Cédric Boutillier et Kilian Raschel.[-]

In the past several decades, it has been established that numerous fundamental invariants in physics and geometry can be expressed in terms of the so-called Witten-Kontsevich intersection numbers. In this talk, I will present a novel approach for calculating their large genus asymptotics. Our technique is based on a resurgent analysis of the n-point functions of such intersection numbers, which are computed using determinantal formulae and depend significantly on the presence of an underlying ODE. I will show how, with this approach, we are able to extend the recent results of Aggarwal with the computation of all subleading corrections. If time permits, I will also explain how the same technique can be applied to address other enumerative problems.
Based on a joint work with B. Eynard, E. Garcia-Failde, P. Gregori, D. Lewanski.[-]

In 2007, Fan, Jarvis, and Ruan constructed an analogue of the Gromov-Witten (GW) theory of hypersurfaces in weighted projective spaces. The new theory is attached to quasi-homogeneous polynomial singularities and is usually called Fan-Jarvis-Ruan-Witten theory (FJRW). It is part of the general picture of Witten, where GW and FJRW theories arise as two distinct GIT quotients of the same model. I will first explain this idea under the light of mirror symmetry. Then I will present FJRW theory and the geometric problem it illustrates. In particular, I will highlight a geometric property called concavity. For now, it is a necessary condition for explicit results on GW theory of hypersurfaces. But on the FJRW side, the situation has recently changed and I will describe my method based on Koszul cohomology to overcome this difficulty. As a consequence, I obtain a mirror symmetry theorem without concavity.[-]