I will give a formula for the Masur-Veech volume of the moduli space of quadratic differentials in terms of psi-classes (in the spirit of Mirzakhani’s formula for the Weil-Peterson volume of the moduli space of hyperbolic surfaces). I will also show that Mirzakhani’s frequencies of simple closed hyperbolic geodesics of different combinatorial types coincide with the frequencies of the corresponding square-tiled surfaces. I will conclude with a (mostly conjectural) description of the geometry of a « random » square-tiled surface of large genus and of a « random » multicurve on a topological surface of large genus.

The talk is based on joint work in progress with V. Delecroix, E. Goujard and P. Zograf. It is aimed at a broad audience, so I will try to include all necessary background.

Quasi-circles in the complex plane are fundamental objects in complex analysis; they were used by Bers to define an infinite-dimensional analogue of the usual Teichmüller space. After introducing the notion of quasi-circles in the boundary of the pseudo-hyperbolic space $H^{2,n}$, I will explain how to construct a unique complete maximal surface in $H^{2,n}$ bounded by a given quasi-circle. This construction relies on Gromov’s theory of pseudo-holomorphic curves and provides a generalization of maximal representations of surface groups into rank 2 Lie groups. This joint work with François Labourie and Mike Wolf.

We consider discrete groups admitting proper cocompact topological actions by homeomorphisms on $R^4$. We will say that such a group Γ is geometrized if we can build an action of Γ by projective transformations on a properly convex open subset of the real projective 4-space, or a convex cocompact action of Γ on the real hyperbolic 5-space or on its Lorentzian counterpart, the anti-de Sitter 5-space.

Certain uniform lattices of the isometry group of hyperbolic 4-space are geometrizable by the three geometries mentioned above. We will discuss the existence of groups which are not uniform lattices in hyperbolic 4-space, and which yet admit several of these three geometries. If time allows, we will also discuss the corresponding deformation spaces.

This is joint work with Gye-Seon Lee (Heidelberg).

In the first introductory part of the talk (that may be skipped by the request of the audience), we remind the classical Witten’s result that the de Rham complex is isomorphic to the Hilbert space of wave functions in a certain supersymmetric quantum mechanical system. There is an « industrial » way to construct supersymmetric systems which is based on the superspace formalism, which we also  describe. We then discuss Kahler manifolds and show how the classical result that any Kahler metric can be derived from the Kahler potential, $h_{m bar n} = partial_m bar partial_n , K(z^p, bar z^p)$ can be easily derived in the {it extended} superspace formalism.

The hyper-Kahler models enjoy ${cal N} = 8$ supersymmetry. We do not know how to fully implement the latter in the conventional superspace approach, it is only possible in the harmonic superspace formalism that we  briefly outline. Any hyper-Kahler metric can be derived from a certain
function (prepotential) depending on the harmonic superfields and harmonics. In contrast to the Kahler case, the relation of the metric to the prepotential is not so simple, one has to solve a system of certain differential equations.

 Finally we go over   to the so-called HKT manifolds. These are the manifolds admitting three quaternionic complex structures that are covariantly constant with respect to a certain torsionful connection. (HK manifolds is a subclass of HKT manifolds where this connection is torsionless). We describe them in the harmonic superspace framework. In constrast to the HK metric, a general  HKT metric needs  {it two} different functions for its description. We show that the space of all HKT metrics is divided in families such that the Obata curvatures of all members of one family coincide.  

Consider a simplicial complex that allows for an embedding into ${mathbb R}^d$. How many faces of dimension $frac{d}{2}$ or higher can it have? How dense can they be? This basic question goes back to Descartes’ « Lost Theorem » and Euler’s work on polyhedra. Using it and other fundamental combinatorial problems, we introduce a version of the Kähler package beyond positivity, allowing us to prove the hard Lefschetz theorem for toric varieties (and beyond) even when the ample cone is empty. A particular focus lies on replacing the Hodge-Riemann relations by a non-degeneracy relation at torus-invariant subspaces, allowing us to state and prove a generalization of theorems of Hall and Laman in the setting of toric varieties and, more generally, the face rings of Hochster, Reisner and Stanley. This has several applications including full characterization of the possible face numbers of simplicial rational homology spheres, a generalization of the crossing lemma.)

Dans cet exposé, nous nous intéressons à des modèles de percolation de lignes de niveau de champs gaussiens planaires continus. Plus précisément, nous présentons des résultats qui suggèrent une parenté proche entre de tels modèles et la percolation de Bernoulli planaire (par exemple la percolation de Bernoulli sur $Z^2$). La percolation de Bernoulli peut être vue comme une loi de probabilité sur un espace produit discret ; deux étapes importantes seront donc de discrétiser le modèle continu considéré et de trouver une définition de celui-ci reposant sur un espace produit sous-jacent naturel. Travaux en commun avec Alejandro Rivera ainsi que Stephen Muirhead.

Soit $f:R^2 to R$ un champ Gaussien centré lisse stationnaire. On s’intéresse à la probabilité que l’ensemble ${ f geq -ell }$ contienne chemin continu qui traverse le rectangle $[0,3R]times [0,R]$ de gauche à droite. Ici, $Rto+infty$ et $ell$ est un paramètre réel fixé. Ce type d’événement est appelé événement croisement. Nous présenterons deux instances où l’étude de la probabilité de croisement fait naturellement intervenir le bord de cet événement.

En premier lieu, l’étude de certaines ‘influences’, qui sont  des fonctionnelles sur le bord de l’événement de croisement, permet de montrer un résultat de transition de phase des probabilités de croisement au paramètre auto-dual $ell=0$. Dans un second temps, nous donnerons une formule exacte pour la covariance entre deux événements de croisement en termes de probabilités de pivot. La démonstration passe par une étude précise de la géométrie du bord des événements pivots. Ce dernier résultat se généralise en fait aux événements ‘topologiques’ sur les lignes de niveau de champs Gaussiens sur des variétés lisses.

Les résultats que je présenterai ont été réalisés en collaboration avec Hugo Vanneuville ainsi que Stephen Muirhead et Dmitry Beliaev.

The conformal bootstrap aims to systematically constraint CFTs based on crossing symmetry and unitarity.

In this talk I will introduce a new approach to extract information from the crossing symmetry sum rules, based on the construction of linear functionals with certain positivity properties. I show these functionals allow us to derive optimal bounds on CFT data. Furthemore I will argue that special extremal solutions to crossing form a basis for the crossing equation, with the functionals living in the dual space. As an application we reconstruct physics of QFTs in AdS2 from the properties of 1d CFTs.

Let $G$ be a Lie group, $H$ a closed subgroup of $G$ and $Gamma$ a discontinuous group  for the homogeneous space $mathscr{X}=G/H$, which means that $Gamma$ is a discrete subgroup of $G$ acting properly discontinuously and fixed point freely on $mathscr{X}$. The subject of the talk is to to deal with some questions related to the geometry of the  parameter and the deformation spaces of the action of $Gamma$ on $mathscr{X}$, when the group $G$ is solvable. The local rigidity conjecture in the nilpotent case and  the analogue of the Selberg-Weil-Kobayashi rigidity Theorem in such non-Riemannian setting is also discussed.

I will present a new method to block-diagonalize some Hamiltonians describing quantum chains. The method is applied to study perturbations of the so called Kitaev Hamiltonian. This is a joint work with J. Froehlich.

Following the method outlined b Emmy Noether in her famous 1918 paper, we propose a version of the momentum-energy tensor in general relativity which is geometric and suitably covariant, giving conservation laws .

This conssuction appears to be novel in a field well explored.

Séminaire Laurent Schwartz — EDP et applications