Robert Haslhofer - The moduli space of 2-convex embedded spheres

Description : We investigate the topology of the space of smoothly embedded n-spheres in R^{n+1}, i.e. the quotient space M_n:=Emb(S^n,R^{n+1})/Diff(S^n). By Hatcher’s proof of the Smale conjecture, M_2 is contractible. This is a highly nontrivial theorem generalizing in particular the Schoenflies theorem and Cerf’s theorem. In this talk, I will explain how geometric analysis can be used to study the topology of M_n respectively some of its variants.I will start by sketching a proof of Smale’s theorem that M_1 is contractible. By a beautiful theorem of Grayson, the curve shortening flow deforms any closed embedded curve in the plane to a round circle, and thus gives a geometric analytic proof of the fact that M_1 is path-connected. By flowing, roughly speaking, all curves simultaneously, one can improve path-connectedness to contractibility.In the second half of my talk, I’ll describe recent work on space of smoothly embedded spheres in the 2-convex case, i.e. when the sum of the two smallest principal curvatures is positive. Our main theorem (joint with Buzano and Hershkovits) proves that this space is path-connected, for every n. The proof uses mean curvature flow with surgery. Mots-clés libres : Grenoble, CNRS, institut fourier, summer school, geometric analysis, metric geometry, topology, UGA, moduli space Classification générale : Mathématiques Accès à la ressource : http://www.canal-u.tv/video/institut_fourier/rober... rtmpt://fms2.cerimes.fr:80/vod/institut_fourier/ro... Conditions d'utilisation : Droits réservés à l'éditeur et aux auteurs. CC BY-NC-ND 4.0

DONNEES PEDAGOGIQUES Type pédagogique : cours / présentation Niveau : doctorat DONNEES TECHNIQUES Format : video/x-flv Taille : 1.78 Go Durée d'exécution : 50 minutes 2 secondes

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