New and old sub-Riemannian challenges bridging analysis and geometry

Simone Verzellesi - PhD in Mathematics, University of Trento

Abstract: 
We propose a systematic exposition of some analytic and geometric problems arising from the study of sub-Riemannian geometry, Carnot-Carathéodory spaces and, more broadly, anisotropic metric and differential structures. We begin with a very quick overview of recent results in the areas of calculus of variations and PDEs. After that, we focus on a major open problem in sub-Riemannian geometry, namely the Bernstein problem in sub-Riemannian Heisenberg groups. We give an exposition of known results and remaining open issues, and we present a solution in the second sub-Riemannian Heisenberg group, highlighting the new tools that we employed and the remaining open questions.