In 2002 and 2003, Grigori Perelman posted two papers to the arXiv which claimed to provide a proof for William Thurston's geometrization conjecture, using Richard Hamilton's theory of Ricci flow.

[7][8] In 2005, Max-K. von Renesse and Karl-Theodor Sturm showed that the a lower bound of the Ricci curvature on a Riemannian manifold could be characterized by optimal transportation, in particular by the convexity of a certain "entropy" functional along geodesics of the associated Wasserstein metric space.

[9] In 2009, Lott and Cédric Villani capitalized upon this equivalence to define a notion of "lower bound for Ricci curvature" for a general class of metric spaces equipped with Borel measures.

[10][11] The papers of Lott-Villani and Sturm have initiated a very large amount of research in the mathematical literature, much of which is centered around extending classical work on Riemannian geometry to the setting of metric measure spaces.

[12][13][14] An essentially analogous program for sectional curvature bounds (from either below or above) was initiated in the 1990s by an article of Yuri Burago, Mikhail Gromov, and Grigori Perelman, following foundations laid in the 1950s by Aleksandr Aleksandrov.