Alessandro Cucinotta (University of Oxford)
Mattia Fogagnolo (University of Padova)
Erik Hupp (University of Bonn)
Peter Topping (University of Warwick)
Masoumeh Zarei (University of Hamburg)
Alessandro Cucinotta - On manifolds with almost non-negative Ricci curvature and positive kᵗʰ scalar curvature
Abstract: In this talk, we consider n-manifolds with almost non-negative Ricci curvature, where the sum of the lowest k eigenvalues of the Ricci tensor is bounded below by a strictly positive constant. We show that, at large scales, these n-manifolds behave like (k-1)-dimensional objects. If k=n, meaning that the strictly positive lower bound is on the scalar curvature, assuming additional hypotheses, the dimensional behavior at large scales can be improved to n-2. Based on joint work with Andrea Mondino.
Mattia Fogagnolo - Some topics in the analysis of scalar curvature bounds in low regularity regimes
Abstract: The talk concerns some recent advances in the understanding of basic properties of 3-manifolds with nonnegative scalar curvature when classical regularity assumptions are dropped. Namely, I will discuss Penrose-type inequalities in nonsmooth asymptotic regimes and related local properties of continuous metrics that are uniform limits of smooth metrics with scalar curvature bounds. The results mentioned are obtained, or in progress, jointly with Antonelli, Benatti, Gatti, Mazzieri, Pluda and Pozzetta.
Erik Hupp - Non-manifold structure and collapsed Ricci limit spaces
Abstract: This talk will describe some topologically unpleasant features of general Ricci limit spaces. These are Gromov-Hausdorff limits of Riemannian manifolds of a fixed dimension with a uniform lower bound on the Ricci curvature. One of the (many) achievements emerging from the work of Cheeger-Colding in the late 90’s was the improved structure of the limit space when a uniform volume-noncollapsing assumption is added. For instance, in this setting they conclude that on an open dense set whose complement has codimension 2, the limit is homeomorphic to a manifold. This fails if volume is allowed to collapse, as shown in joint work of the speaker with Aaron Naber and Kai-Hsiang Wang, who produced examples of 6-dimensional manifolds collapsing to 4 (rectifiable) dimensional limit spaces for which every open set has nontrivial homology. We will review this construction, survey some related results that have appeared since then, and highlight some possible future directions of investigation.
Peter Topping - Delayed parabolic regularity for curve shortening flow
Abstract: Linear parabolic PDEs like the heat equation have well-known smoothing properties. In reasonable situations we can control the Cᵏ norm of solutions at time t in terms of t and a weak norm of the initial data. This idea often carries over to nonlinear parabolic PDEs such as geometric flows. In this talk I will discuss a totally different phenomenon that can occur in some natural situations, in which there is an explicit magic positive time before which we have no regularity estimates at all, but after which parabolic regularity is switched on and we obtain full regularity. I plan to focus on the case of curve shortening flow, which will mean that almost no prerequisites will be assumed. Joint work with Arjun Sobnack.
Masoumeh Zarei - Ricci flow emerging from singular spaces with bounded curvature
Abstract: A metric flow (M,g(t)) on a compact manifold M satisfying the Ricci flow equation is said to have a metric space (X,d) as its initial condition if (X,d) is the Gromov-Hausdorff limit of (M,g(t)) as t → 0. A fundamental problem is to understand the assumptions on (X,d) that ensure the existence of a Ricci flow (M,g(t)) with (X,d) as its initial condition, as well as the additional regularity conditions on (X,d) and (M,g(t)) that improve the convergence. In this talk, I will present results on the existence and uniqueness of solutions to the Ricci flow, where the initial condition is a compact length space with bounded curvature, specifically a space that is both Alexandrov and locally CAT, and I will discuss how these conditions improve the convergence. This work is joint with Diego Corro and Adam Moreno.