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# Résultats de recherche

**5509**

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## C. Sormani - Intrinsic Flat and Gromov-Hausdorff Convergence 4

We introduce various notions of convergence of Riemannian manifolds and metric spaces. We then survey results and open questions concerning the limits of sequences of Riemannian manifolds with uniform lower bounds on their scalar curvature. We close the course by presenting methods and theorems that may be applied to prove these open questions including older techniques developed with Lakzian, with Huang and Lee, and with Portegies. I will also present key new results of Allen and Perales. Students and postdocs interested in ... Voir la vidéole (52m2s)

## A. Lytchak - Convex subsets in generic manifolds

In the talk I would like to discuss some statements and questions about convex subsets and convex hulls in generic Riemannian manifolds of dimension at least 3. The statements, obtained jointly with Anton Petrunin, are elementary but somewhat surprising for the Euclidean intuition. For instance, the convex hull of any finite non-collinearset turns out to be either the whole manifold or non-closed. Voir la vidéole (56m54s)

## A. Song - On the essential minimal volume of Einstein 4-manifolds

Given a positive epsilon, a closed Einstein 4-manifold admits a natural thick-thin decomposition. I will explain how, for any delta, one can modify the Einstein metric to a bounded sectional curvature metric so that the thick part has volume linearly bounded by the Euler characteristic and the thin part has injectivity radius less than delta. I will also discuss relations to conjectural obstructions to collapsing with bounded sectional curvature or to the existence of Einstein metrics. Voir la vidéole (1h1m56s)

## F. Schulze - Mean curvature flow with generic initial data

Mean curvature flow is the gradient flow of the area functional and constitutes a natural geometric heat equation on the space of hypersurfaces in an ambient Riemannian manifold. It is believed, similar to Ricci Flow in the intrinsic setting, to have the potential to serve as a tool to approach several fundamental conjectures in geometry. The obstacle for these applications is that the flow develops singularities, which one in general might not be able to classify completely. Nevertheless, a well-known conjecture of Huisken states ... Voir la vidéole (1h3m55s)

## T. Ozuch - Noncollapsed degeneration and desingularization of Einstein 4-manifolds

We study the noncollapsed singularity formation of Einstein 4-manifolds. We prove that any smooth Einstein 4-manifold close to a singular one in a mere Gromov-Hausdorff (GH) sense is the result of a gluing-perturbation procedure that we develop. This sheds light on the structure of the moduli space of Einstein 4-manifolds near its boundary and lets us show that spherical and hyperbolic orbifolds (which are synthetic Einstein spaces) cannot be GH-approximated by smooth Einstein metrics. New obstructions specific to the compact situation moreover raise the ... Voir la vidéole (54m16s)

## R. Perales - Recent Intrinsic Flat Convergence Theorems

Given a closed and oriented manifold M and Riemannian tensors g0, g1, ... on M that satisfy g0 < gj, vol(M, gj)→vol (M, g0) and diam(M, gj)≤D we will see that (M, gj) converges to (M, g0) in the intrinsic flat sense. We also generalize this to the non-empty bundary setting. We remark that under the onditions we do not nexessarily obtain smooth, C0 or even Gromov-Hausdorff convergence. furthermore, these results can be applied to show stability of a class of tori and a class of complete and asymptotically flat manifolds. That is, any sequence ... Voir la vidéole (1h13m19s)

## R. Bamler - Compactness and partial regularity theory of Ricci flows in higher dimensions

We present a new compactness theory of Ricci flows. This theory states that any sequence of Ricci flows that is pointed in an appropriate sense, subsequentially converges to a synthetic flow. Under a natural non-collapsing condition, this limiting flow is smooth on the complement of a singular set of parabolic codimension at least 4. We furthermore obtain a stratification of the singular set with optimal dimensional bounds depending on the symmetries of the tangent flows. Our methods also imply the corresponding quantitative stratification result ... Voir la vidéole (1h14m45s)

## A. Mondino - Time-like Ricci curvature bounds via optimal transport

The goal of the talk is to present a recent work in collaboration with Cavalletti (SISSA) on optimal transport in Lorentzian synthetic spaces. The aim is to set up a “Lorentzian analog” of the celebrated Lott-Sturm-Villani theory of CD(K,N) metric measure spaces. The key idea being to analyse convexity properties of Entropy functionals along future directed timelike geodesics (with respect to a suitable Lorentzian Wasserstein distance) of probability measures. The smooth Lorentzian setting was previously investigated by McCann and Mondino-Suhr.After recalling the general setting ... Voir la vidéole (1h2m14s)