John Lott

John Lott is an American mathematician whose work spans Ricci flow, geometric analysis, and the foundations of metric geometry. Together with Cédric Villani, he developed the curvature-dimension condition for metric measure spaces, proving that Ricci curvature lower bounds can be characterized by the convexity of entropy functionals along geodesics in the Wasserstein space of probability measures. This work, published in 2006, appeared simultaneously with independent work by Karl-Theodor Sturm and fundamentally altered the landscape of synthetic Ricci curvature.

Lott's earlier contributions to Ricci flow include collaborations with Bruce Kleiner on the geometrization conjecture, providing alternative approaches to the singularity analysis that Perelman completed. His work often focuses on the limiting behavior of sequences of Riemannian manifolds under uniform curvature bounds — the domain where smooth geometry collapses into metric geometry, and where the tools of Cheeger-Colding theory become essential.

Lott's role in the Lott-Villani-Sturm synthesis is frequently underweighted in favor of his more visible co-author. This is a mistake. The simultaneous emergence of the same definition from two independent groups — Lott and Villani in the United States and France, Sturm in Germany — is not a coincidence to be dismissed as convergent evolution. It is evidence that the definition was implicit in the structure of the field, waiting to be extracted by whoever had the right combination of technical skill and conceptual clarity. Lott's contribution was to see that the Wasserstein geometry of probability measures was not merely a tool for proving inequalities but the natural setting in which curvature itself could be defined. That insight required not just knowledge of optimal transport but a willingness to let go of the smooth structure that geometers had treated as sacred.