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# Résultats de recherche

**2804**

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## D. Semola - Boundary regularity and stability under lower Ricci bounds

The theory of non smooth spaces with lower Ricci Curvature bounds has undergone huge developments in the last thirty years. On the one hand the impetus came from Gromov’s precompactness theorem and the Cheeger-Colding theory of Ricci limit spaces. On the other hand “synthetic” theories of lower Ricci bounds have been developed, based on semigroup tools (the Bakry-Émery theory) and on Optimal Transport (the Lott-Sturm-Villani theory). The Cheeger-Colding theory did not consider manifolds with boundary, while in the synthetic framework even understanding what ... Voir la vidéole (1h3m35s)

## C. Li - Classifying sufficiently connected PSC manifolds in 4 and 5 dimensions

In this talk, I will discuss some recent developments on the topology of closed manifolds admitting Riemannian metrics of positive scalar curvature. In particular, we will prove if a closed PSC manifold of dimension 4 (resp. 5) has vanishing π2 (resp. vanishing π2 and π3), then a finite cover of it is homotopy equivalent to Snor connected sums of Sn-1 x S1. This extends a previous theorem on the non-existence of Riemannian metrics of positive scalar curvature on aspherical manifolds in 4 and 5 dimensions, due to ... Voir la vidéole (1h2m34s)

## Y. Lai - A family of 3d steady gradient Ricci solitons that are flying wings

We find a family of 3d steady gradient Ricci solitons that are flying wings. This verifies a conjecture by Hamilton. For a 3d flying wing, we show that the scalar curvature does not vanish at infinity. The 3d flying wings are collapsed. For dimension n ≥ 4, we find a family of Z2 × O(n − 1)-symmetric but non-rotationally symmetric n-dimensional steady gradient solitons with positive curvature operator. We show that these solitons are non-collapsed. 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 (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)