Measure contraction properties for sub-Riemannian structures beyond step 2

In this talk, I will introduce Carnot groups and their quotients as metric measure spaces. These are examples of Carnot–Carathéodory spaces, or sub-Riemannian manifolds, and include the Heisenberg group, the Engel group, and the Martinet flat structure, to name but a few. Once we have a good grasp of these geometric structures, I will outlinesome open problems in the field before shifting focus to the study of curvature and the so-called metric contraction properties. These analytic inequalities aim to define, in a synthetic way, a lower bound on Ricci curvature. The new results I will present show how these properties can be preserved by taking quotients, and how this affects their validity or failure. This is joint work with Luca Rizzi from SISSA.