June 5, 2018
TITLE: Ricci Flow, non-negative curvatures & beyond
ABSTRACT: In order to use the Ricci flow to prove classification results in geometry and control the behaviour of solutions as times goes by, it is crucial to look for properties of the manifold that are preserved under the flow. During the talk we will see that this is typically the case for a large family of non-negative curvature conditions.
In contrast, the condition of almost non-negative curvature operator (e.g. the condition that its smallest eigenvalue is larger than -1) is not preserved under Ricci flow. In this second talk we will present a work (joint with Richard Bamler and Burkhard Wilking) in which we generalize most of the known Ricci flow invariant non-negative curvature conditions to less restrictive negative bounds that remain sufficiently controlled for a short time. As an application of our almost preservation results we deduce a variety of gap and smoothing results of independent interest, including a classification for non-collapsed manifolds with almost non-negative curvature operator and a smoothing result for singular spaces coming from sequences of manifolds with lower curvature bounds. We also obtain a short-time existence result for the Ricci flow on open manifolds with almost non-negative curvature (without requiring upper curvature bounds).
June 4, 2018
TITLE: The ABC of Ricci Flow
ABSTRACT: Geometric flows have been used to address successfully key questions in Differential Geometry like isoperimetric inequalities, the Poincaré conjecture, Thurston’s geometrization conjecture, or the differentiable sphere theorem. During this talk we will give an intuitive introduction to Ricci flow, which is sort of a non-linear version of the heat equation for the Riemannian metric. The equation should be understood as a tool to canonically deform a manifold into a manifold with nicer properties, for instance, some kind of constant curvature. We will emphasize the features that convert evolution equations into a powerful tool in geometry.