243176 Geometry and Analysis in non-smooth spaces (S) (WiSe 2023/2024)

Non-smooth spaces naturally arise in various branches of mathematics and its applications, for instance as limits of smooth Riemannian manifolds, through singularity formation in geometric flows or as discrete structure in data.

A natural framework for non-smooth spaces are metric measure spaces, i.e. spaces equipped only with the structure of a metric and a reference measure. In this seminar we will study analytic and geometric properties of such spaces and in particular how to develop a notion of curvature (lower bounds) for these objects using the theory of optimal transport.
Metric measure spaces with curvature bounds provide a robust framework that allows for the development of far reaching analytic and geometric results akin to classical results for smooth manifolds such a diameter and volume growth estimates or isoperimetric inequalities. The investigation of this class of spaces has been a highly active research field in the last decade and continues to spawn fascinating new developments.

We will work on familiarising ourselves with the foundations of the theory, developing central results and highlighting some recent developments according to the preferences of the participants.

Possible topics include:

• Introduction to metric geometry
• Basic theory of optimal transport and the Wasserstein distance
• Characterisation of curvature-dimension bounds on manifolds through optimal transport
• Synthetic curvature-dimension bounds for metric measure spaces
• Convergence of metric measure spaces and stability of curvature-dimension bounds
• Bakry-Emery criterion and characterisation of curvature-dimension bounds via the heat flow and Brownian motion
• Heat flow on metric measure spaces
• Equivalence of the Bakry-Emery and Optimal transport approaches
• Functional inequalities on metric measure spaces via curvature
• Synthetic curvature bounds for discrete spaces
• time-dependent metric measure spaces and (non-smooth) super Ricci flows

Preliminary meeting: Tuesday October 17 16:00

If you are interested in the seminar but not able to attend the meeting, please contact me by email.

The seminar will take place starting from December 2023. The precise schedule will be arranged according to the participants.

Requirements for participation, required level

Background in measure theory and functional analysis is desired. Depending on the topic, knowledge in Riemannian geometry and or PDEs is helpful.