This lecture by Mattia Magnabosco from Oxford explores the rectifiability properties of CD(K,N) spaces with unique tangents. Dive into the Lott-Sturm-Villani curvature-dimension condition CD(K,N), which provides a synthetic framework for understanding when metric measure spaces have Ricci curvature bounded from below by K and dimension bounded from above by N. Learn about the stability properties of CD(K,N) spaces with respect to measured Gromov-Hausdorff convergence, while examining the still-developing understanding of their geometric and analytic structure. Discover new research results that prove rectifiability for CD(K,N) spaces having a unique metric tangent space almost everywhere, based on joint work with Andrea Mondino and Tommaso Rossi.