## Ricci flows with Rough Initial Data

Peter Topping

University of Warwick

March 8, 2019

Alexander Nabutovsky

University of Toronto; Member, School of Mathematics

March 8, 2019

Abstract: Uryson k-width of a metric space X measures how close X is to being k-dimensional. Several years ago Larry Guth proved that if M is a closed n-dimensional manifold, and the volume of each ball of radius 1 in M does not exceed a certain small

constant e(n), then the Uryson (n-1)-width of M is less than 1. This result is a significant generalization of the famous Gromov's inequality relating the volume and the filling radius that plays a central role in systolic geometry.

Stéphane Sabourau

Université Paris-Est Créteil

March 7, 2019

Ernst Kuwert

University of Freiburg

March 7, 2019

Abstract: For surfaces immersed into a compact Riemannian manifold, we consider the curvature functional given by the $L^2$ integral of the second fundamental form. We discuss an an area bound in terms of that functional, with application to the existence of minimizers (joint work with V. Bangert).

Gerard Besson

Université de Grenoble

March 7, 2019

Abstract : It is a joint work with G. Courtois, S. Gallot and A.Sambusetti. We prove a compactness theorem for metric spaces with anupper bound on the entropy and other conditions that will be discussed.Several finiteness results will be drawn. It is a work in progress.

Michael Struwe

ETH Zürich

March 6, 2019

Abstract: Finding non-constant harmonic 3-spheres for a closed target manifold N is a prototype of a super-critical variational problem. In fact, the

direct method fails, as the infimum of the Dirichlet energy in any homotopy class of maps from the 3-sphere to any closed N is zero; moreover, the

harmonic map heat flow may blow up in finite time, and even the identity map from the 3-sphere to itself is not stable under this flow.

Nancy Hingston

The College of New Jersey

March 6, 2019

Neshan Wickramasekera

University of Cambridge; Member, School of Mathematics

March 6, 2019

Yanyan Li

Rutgers University

March 6, 2019

Abstract: For a given finite subset S of a compact Riemannian manifold (M; g) whose Schouten curvature tensor belongs to a given cone, we establish a necessary and

sufficient condition for the existence and uniqueness of a conformal metric on $M \setminus S$ such that each point of S corresponds to an asymptotically flat end and

that the Schouten tensor of the new conformal metric belongs to the boundary of the given cone. This is a joint work with Luc Nguyen.

Tristan Rivière

ETH Zürich; Member, School of Mathematics

March 5, 2019

Abstract : What are the possible limits of smooth curvatures with uniformly bounded $L^p$ norms ?

We shall see that the attempts to give a satisfying answer to this natural question from the calculus of variation of gauge theory brings us to numerous analysis challenges.