Geometric Analysis on Riemannian manifolds
Geometric Analysis lies at the intersection of differential geometry, analysis, partial differential equations and mathematical physics, and is having a profound impact on all of these fields, leading to the resolution of many classical conjectures such as Poincaré conjecture via geometric flows, and Willmore and Lawson conjectures in the differential geometry of surfaces.
Ricci flow deforms Riemannian metrics on manifolds via their Ricci tensor, according to an equation that presents many similarities with the heat equation. Other geometric flows, such as, for instance, the mean curvature flow of hypersurfaces, have similar regularization properties. The aim, for many geometric flows, is to produce canonical geometries starting by general initial data on these geometries. Depending on the initial datum, the solutions of these geometric flows may develop singularities, where at a certain time the solution stop to be smooth. In many cases the analysis of the singularities that may arise along the flow is modeled by self-similar solutions. This fact leads to investigate classification and rigidity results for Ricci solitons and for self-shrinkers and translators for the mean curvature flow. The study of these structures brings to consider Riemannian manifolds endowed with a measure that is absolutely continuous with respect to the Riemannian one. A fundamental role on the study is played by the connection between topology, geometry of manifolds with density and analytical properties of the diffusion operator naturally associated to these spaces.