Partial Differential Equations Under Curvature Bounds

Curvature and PDEs have a long history together. PDEs can be used to give an answer to geometric questions, e.g. Yamabe's problem. Conversely, geometry, specifically curvature, appears when studying qualitative properties of solutions of certain PDEs, such as the symmetry of extremal functions in optimal functional inequalities or the convergence to equilibrium of nonlinear diffusion flows. The course aims at providing a gentle introduction to optimal functional inequalities, Gamma calculus, Otto's calculus, n-conformal geometry and elliptic regularity theory.