It is interesting to study the topology of the space of smoothly embedded $n$-spheres in $\mathbb R^{n+1}$. By Smale’s theorem, this space is contractible for $n=1$ and by Hatcher’s proof of the Smale conjecture, it is also contractible for $n=2$. These results are of great importance, generalising in particular the Schoenflies theorem and Cerf’s theorem. In this talk, I will explain how mean curvature flow with surgery can be used to study a higher-dimensional variant of these results, proving in particular that the space of two-convex embedded spheres is path-connected in every dimension $n$. We then also look at the space of two-convex embedded tori where the question is more intriguing and the result in particular depends on the dimension $n$. This is all joint work with Robert Haslhofer and Or Hershkovits.

The talk will focus on the following results established with Cyril Lecuire: a finitely generated group quasi-isometric to the fundamental group of a compact $3$–manifold or to a finitely generated Kleinian group contains a finite index subgroup isomorphic to the fundamental group of a compact $3$–manifold or to a finitely generated Kleinian group.

The problem of finding (complete) metrics with constant Q-curvature in a prescribed conformal class is a famous fourth-order cousin of the Yamabe problem. In this talk, I will provide some background on Q-curvature and discuss how several non-uniqueness results for the Yamabe problem can be transplanted to this context. However, special emphasis will be given to multiplicity phenomena for constant Q-curvature that have no analogues for the Yamabe problem, confirming expectations raised by the lack of a maximum principle.

Nous verrons que toute application quasi-isométrique entre variétés de Hadamard pincées est à distance bornée d'une unique application harmonique.

In this talk I will discuss how to construct surface groups acting by diffeomorphisms on the closed interval. This is a joint work with Ludovic Marquis.

It is interesting to study the topology of the space of smoothly embedded n-spheres in $\mathbb R^{n+1}$. By Smale’s theorem, this space is contractible for $n=1$ and by Hatcher’s proof of the Smale conjecture, it is also contractible for $n=2$. These results are of great importance, generalising in particular the Schoenflies theorem and Cerf’s theorem. In this talk, I will explain how mean curvature flow with surgery can be used to study a higher-dimensional variant of these results, proving in particular that the space of two-convex embedded spheres is path-connected in every dimension $n$. We then also look at the space of two-convex embedded tori where the question is more intriguing and the result in particular depends on the dimension $n$. This is all joint work with Robert Haslhofer and Or Hershkovits.

In this talk we present an optimal transport characterization of lower sectional curvature bounds for smooth Riemannian manifolds. More generally, we characterize lower bounds for the $p$-Ricci tensor in terms of convexity of the relative Reny entropy on Wasserstein space with respect to the $p$-dimensional Hausdorff measure. The $p$-Ricci tensor corresponds to taking the trace of the Riemannian curvature tensor on $p$-dimensional planes.

This is a joint work with Andrea Mondino.

The famous 1939 Myers–Steenrod theorem states that a distance preserving map between (smooth) Riemannian manifolds is actually a smooth isometry. In this talk I will give a similar statement for Finsler manifolds under minimal regularity (i.e. we do not assume the metric to be smooth). I shall explain the concepts from Finsler geometry that are needed as well as some history (Riemannian and Finslerian) of the subject. I will give a sketch of the proof and discuss some related results.

Dans cet exposé, je parlerai d'espaces métriques à courbure minorée au sens d'Alexandrov. D'un point de vue analytique, on peut considérer ces espaces comme des variétés riemanniennes ouvertes, à un lieu singulier près. Je m'intéresserai à la régularité de ces métriques riemanniennes et donnerai un résultat dans le cas des surfaces. Si le temps le permet, j'aborderai aussi des résultats partiels valables en dimension supérieure.

Une partie des résultats a été obtenue en collaboration avec L. Ambrosio (SNS Pisa).