La rencontre du 16 mai 2019 a été co-financée par l'Institut de Mathématiques de Jussieu et le projet ANR Contraintes de courbures et espace des métriques.

**Exposés de synthèse :**

Jason Lotay (University of Oxford)

**Lagrangian mean flow**

It is well-known that in Kähler-Einstein manifolds, the mean curvature flow preserves the Lagrangian condition. This leads to important potential applications relevant to symplectic topology, Riemannian geometry and theoretical physics, particularly in the setting of CalabiYau manifolds. I will describe some of the key aspects of Lagrangian mean curvature flow, and provide a survey of progress and open problems in the field.

Samuel Tapie (Université de Nantes)

**Geodesic dynamics and Laplace spectrum in negative curvature.**

The geodesic flow on a Riemannian manifold is a dynamical system whose long time behaviour provides many topological and geometrical informations. The Laplace-Beltrami is an elliptic operator whose spectral properties also encapture many topological and geometrical datas. In many aspects, the dynamics of the geodesic flow and the specrum of the Laplacian are known to be related. We will present in this talk some precise relationships between the bottom of the spectrum of the Laplacian (first eigenvalue, associated eigenfunction, bottom of the essential spectrum. . .) and asymptotical dynamical properties (counting of geodesics, mixing of the flow. . .) which hold for hyperbolic non-compact manifolds. This talk will gather some classical results by Patterson and Sullivan together with recent works.

**Exposé de recherche: **

Heather Macbeth (ENS)

**The Kähler analogue of a Riemannian submersion**