Geometric flows such as Ricci flow and mean curvature flow are nonlinear parabolic PDEs that tend to develop singularities in finite time.
Differential Geometry & Geometric Analysis Seminar
The Differential Geometry and Geometry Analysis seminar sees talks most often about interactions between elliptic PDE's and differential geometry. Common topics include (but are not limited to) conformal geometry, minimal surfaces and other variational problems, K\"ahler geometry, CR geometry, elliptic problems from general relativity, nonlinear and/or nonlocal elliptic or parabolic PDE's, and geometric functional inequalities.
University of Oregon
Tian's properness conjectures, the strong Moser-Trudinger inequality, and infinite-dimensional Finsler geometry
In the 90's, Tian introduced a notion of properness in the space of Kahler metrics in terms of Aubin's nonlinear Dirichlet energy and Mabuchi's K-energy and put forward several conjectures on the relation between properness and existence of Kähler-Einstein metrics.
University of Maryland
University of Wisconsin