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

Mapping class groups of infinite type surfaces, also called « big » mapping class groups, arise naturally in several dynamical contexts, such as two-dimensional dynamics, one-dimensional complex dynamics, « Artinization » of Thompson groups, etc.

In this talk, I will present recent objects and phenomena related to big mapping class groups. In particular, I will define two faithful actions of some big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. I will describe some properties of the objects involved, and give some fruitful relations between the dynamics of the two actions. For example, we will see that loxodromic elements (for the first action) necessarily have rational rotation number (for the second action). If time allows, I will explain how to use these simultaneous actions to construct nontrivial quasimorphisms on subgroups of big mapping class groups.
This is joint work with Alden Walker.

Séminaire Laurent Schwartz — EDP et applications

Séminaire Laurent Schwartz — EDP et applications

In this talk I will argue that after certain timescale (which scales with the system size as L^{d+2}) dynamics of a local observable becomes universal and it can be described by a random matrix.

 

This talks is based on https://arxiv.org/abs/1804.08626 and other recent works.

Mathematicians have long understood periodic trajectories on the square billiard table, which occur when the slope of the trajectory is rational. In this talk, I will explain my joint work with Samuel Lelièvre on periodic trajectories on the regular pentagon, describing their geometry, symbolic dynamics, and group structure. The periodic trajectories are very beautiful, and some of them exhibit a surprising « dense but not equidistributed » behavior.

In my talk, I will report on my ongoing collaborating project together with Yifeng Liu, Liang Xiao, Wei Zhang, and Xinwen Zhu, which concerns the rank 0 case of the Beilinson-Bloch-Kato conjecture on the relation between L-functions and Selmer groups for certain Rankin–Selberg motives for GL(n) x GL(n+1). I will state the main results with some examples coming from elliptic curves, sketch the strategy of the proof, and then focus on the key geometric ingredients, namely the semi-stable reduction of unitary Shimura varieties of type U(1,n) at non-quasi-split places.