Explore functional inequalities on sub-Riemannian manifolds through the lens of Quasi-Curvature-Dimension (QCD) conditions in this 44-minute lecture by Emanuel Milman. Delve into the challenges of obtaining Poincaré and log-Sobolev inequalities on domains in sub-Riemannian manifolds, and discover why traditional Curvature-Dimension conditions are insufficient for these spaces. Learn about the innovative QCD(Q, K, N) approach, which introduces a slack parameter Q to overcome limitations. Examine the extension of the localization paradigm to general interpolation inequalities and the comparison of QCD densities with their "CD upper envelope." Gain insights into quantitative estimates for Lp-Poincaré and log-Sobolev inequalities on ideal sub-Riemannian manifolds and corank 1 Carnot groups, including the application of the Li-Yau / Zhong-Yang spectral-gap estimate to Heisenberg groups of arbitrary dimension.