Sub-Riemannian spaces are geometrical structures that model constrained systems and constitute a vast generalization of Riemannian geometry. They arise in control theory, harmonic and complex analysis, subelliptic PDEs, geometric measure theory, calculus of variations, optimal transport, and potential analysis. In the last 10 years, a surge of interest in the study of geometric and functional inequalities on sub-Riemannian spaces revealed unexpected behaviours and intriguing phenomena that failed to fit into the classical schemes inspired by Riemannian geometry. In this project, I aim to develop a framework of geometric and functional interpolation inequalities adapted to sub-Riemannian manifolds, and to use this theory to tackle old and new problems concerning the geometric analysis of these structures. The project focuses on the following interconnected topics:
(i) the development of a unifying theory of curvature bounds including sub-Riemannian structures,
(ii) the study of measure contraction properties of Carnot groups,
(iii) applications to isoperimetric-type problems,
(iv) applications to the regularity of the sub-Riemannian heat kernel at the cut locus.
The project adopts a unique approach combining methods from geometric control theory, optimal transport and comparison geometry that I developed in recent years, and which already allowed me and my collaborators to obtain important results in the field. The project aims to achieve an ambitious unification program, solve long-standing problems, and explore new research directions in sub-Riemannian geometry, with an impact in several neighboring areas, including geometric analysis on non-smooth spaces, analysis of hypoelliptic operators, geometric measure theory, spectral geometry. My long-term purpose is to build a leading research group in sub-Riemannian geometry, to significantly advance our understanding of geometry under non-holonomic constraints.
