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 consider the asymptotic behavior of eigenfunctions of the Laplacian on a compact locally symmetric manifold M « in the level aspect », that is as the volume of M tends to infinity. I will formulate a precise conjecture of « Berry type », and describe partial results obtained in a joint work with Miklos Abert and Étienne Le Masson.

A classification will be given of all separable minimal hypersurfaces in ${mathbb R}^{n geq 3}$.

An important conjecture in quantum mechanics is that non-interacting, disordered 3d quantum systems should exhibit a localization-delocalization transition as a function of the disorder strength. From a mathematical viewpoint, a lot is known about the localized phase for strong disorder, much less about the delocalized phase at weak disorder. In this talk I will present results for a hierarchical supersymmetric model for a class of 3d quantum systems, called Weyl semimetals. We use rigorous renormalization group methods to compute the correlation functions of the system. In particular, I will report a result about the algebraic decay of the averaged two-point correlation function, compatible with delocalization. Our method is based on a rigorous implementation of RG, reminiscent of the Gawedzki-Kupiainen block spin transformations; the main technical novelty is the multiscale analysis of Gaussian measures with purely imaginary covariances.

Joint work with Luca Fresta and Marcello Porta.

In 1996 Rinat Kashaev introduced a new way to write the star-triangle move (or Yang-Baxter equations) of the Ising model as a single polynomial relation. In 2013 Richard Kenyon and Robin Pemantle understood that this relation could be seen as a kind of spatial recurrence. I will show how the iteration of Kashaev’s recurrence can be related the combinatorics of a loop model with two colors, that was introduced for different reasons by Ole Warnaar and Bernard Nienhuis in 1993. It is also possible to couple this loop model with known relatives of the Ising model: a dimer model and a six-vertex model. Finally I will show a few limit shape phenomena.

I will show that for any integer N, there are only finitely many cuspidal algebraic automorphic representations of GL_m over Q whose Artin conductor is N and whose « weights » are in the interval {0,…,23} (with m varying). Via the conjectural yoga between geometric Galois representations (or motives) and algebraic automorphic forms, this statement may be viewed as a generalization of the classical Hermite-Minkowski theorem in algebraic number theory. I will also discuss variants of these results when the base field Q is replaced by an arbitrary number field.

We describe dynamical systems arising from the classification of locally homogeneous geometric structures on manifolds. Their classification mimics the classification of Riemann surfaces by the Riemann moduli space – the quotient of Teichmüller space by the properly discontinuous action of the mapping class group. However, this action is misleading: mapping class groups generally act chaotically on character varieties. For fundamental examples, these varieties appear as affine cubics, and we relate the projective geometry of cubic surfaces to dynamical properties of the action.

Let Γ be a group acting by isometries on a proper metric space (X,d). The critical exponent δΓ(X) is a number which measures the complexity of this action. The critical exponent of a subgroup Γ'<Γ is hence smaller than the critical exponent of Γ. When does equality occur? It was shown in the 1980s by Brooks that if X is the real hyperbolic space, Γ’ is a normal subgroup of Γ and Γ is convex-cocompact, then equality occurs if and only if Γ/Γ’ is amenable. At the same time, Cohen and Grigorchuk proved an analogous result when Γ is a free group acting on its Cayley graph.
When the action of Γ on X is not cocompact, showing that the equality of critical exponents is equivalent to the amenability of Γ/Γ’ requires an additional assumption: a « growth gap at infinity ». I will explain how, under this (optimal) assumption, we can generalize the result of Brooks to all groups Γ with a proper action on a Gromov hyperbolic space.
Joint work with R. Coulon, R. Dougall and B. Schapira.