Abstract
In the early 80's Li and Yau obtained their fundamental gradient estimates for positive
solutions of the heat equation on complete Riemannian manifolds which Ricci tensor bounded from
below. Such estimates imply a host of global results in Riemannian geometry, such as intrinsic
Harnack inequalities, Gaussian upper bounds, Liouville theorems, etc. In this lecture I will give an
overview of a program which, starting from a new notion of Ricci lower bounds in sub-Riemannian
geometry based on the heat semigroup, proceeds to establishing several results of a global nature
linking curvature lower bounds to various quantitative estimates of solutions of the relevant heat
equation. The approach which will be presented is novel even in the Riemannian setting. The
character of my lecture will be completely self-contained.
