# Trimester Seminar

**Venue:** HIM lecture hall, Poppelsdorfer Allee 45

Organizers: Martin Huesmann, Christian Ketterer, Michael Munn, André Schlichting

## Monday, January 12

15:00 - 16:00 Welcome Meeting

Abstract: Welcome meeting of the participants of the Junior Trimester Program.

## Tuesday, January 13

14:00 - 15:00 Sajjad Lakzian: The curvature dimension condition CD(K,N) of Sturm-Lott-Villani

Abstract: Sajjad Lakzian will present some of the basic ideas behind the curvature dimension condition CD(K,N) of Sturm-Lott-Villani. This will be a very informal meeting and everyone is welcome to attend.

## Wednesday, March 18

11:00 - 12:00 Paul Gassiat: On Root's solution to the Skorokhod embedding problem

Abstract: Given a target probability measure μ, the classical Skorokhod Embedding Problem consists in finding a stopping time τ such that the stopped Brownian motion B_{τ} has distribution μ. In 1968, Root showed that there exists a subset of time-space such that its hitting time by Brownian motion gives a solution to this problem. Root's proof was nonconstructive, leaving open the question of how this barrier can be computed in practical cases. We will report on recent progress in this direction, applications to numerical simulations, as well as extensions to general Markov processes. Based on joint works with A. Mijatovic, H. Oberhauser, O. Reichmann and G. dos Reis.

## Tuesday, March 31

15:00 - 16:00 Stefano Lisini: A gradient flow approach to fractional porous medium equations

Abstract: In this talk I will show a construction of weak global solutions for a family of fractional porous medium equation. The proof, alternative to the one given by Caffarelli and Vazquez, is based on the gradient flow interpretation and works for a general class of initial data. An energy dissipation inequality and the decay of the L^{p} norms along the solutions will be illustrated. Finally I will show the convergence of the solutions of the s-fractional porous medium equation to the unique solution of the classical porous medium equation as the parameter s goes to 0. Joint work with E. Mainini and A. Segatti.

## Wednesday, April 8

14:00 - 15:00 Fernando Galaz-Garcia: Three-dimensional Alexandrov spaces

Abstract: I will talk about the topology and geometry of three-dimensional Alexandrov spaces (with a lower curvature bound). I will discuss the classification of closed three-dimensional Alexandrov spaces of positive and nonnegative curvature, the Poincaré Conjecture in dimension three (in the context of Alexandrov spaces) and an analogue of the geometrization conjecture for closed three-dimensional Alexandrov spaces. This is joint work with Luis Guijarro.

## Friday, April 10

14:30 - 15:30 Alex Amenta: An extended theory of tent spaces on metric measure spaces

Abstract: Tent spaces associated to the Euclidean space have proven to be quite useful in harmonic analysis, particularly in the study of Hardy spaces and related function spaces. The extension of this theory to more general metric measure spaces, particularly doubling spaces, is relatively straightforward. Recently we have extended this theory further to include what we call 'weighted tent spaces'; these have an additional parameter, analogous to a regularity parameter, which is needed when studying boundary value problems with data in fractional smoothness spaces (e.g. Besov and Hardy-Sobolev spaces).

From the viewpoint of abstract tent space theory, this additional parameter reflects some internal structure of the tent space scale which was not previously known (even in the Euclidean case). In particular, there are embeddings between weighted tent spaces from which we can recover various useful inequalities. We also identify some real interpolation spaces between weighted tent spaces (with different weights), recovering function spaces which appear in the study of elliptic BVPs with Besov space data.