By Sun-Yung Alice Chang
Non-linear elliptic partial differential equations are an incredible instrument within the examine of Riemannian metrics in differential geometry, specifically for difficulties in regards to the conformal switch of metrics in Riemannian geometry. in recent times the function performed through the second one order semi-linear elliptic equations within the research of Gaussian curvature and scalar curvature has been prolonged to a family members of absolutely non-linear elliptic equations linked to different symmetric features of the Ricci tensor. A case of specific curiosity is the second one symmetric functionality of the Ricci tensor in measurement 4 heavily regarding the Pfaffian. In those lectures, ranging from the historical past fabric, the writer reports the matter of prescribing Gaussian curvature on compact surfaces. She then develops the analytic instruments (e.g., better order conformal invariant operators, Sobolev inequalities, blow-up research) which will resolve a completely nonlinear equation in prescribing the Chern-Gauss-Bonnet integrand on compact manifolds of size 4. the fabric is acceptable for graduate scholars and examine mathematicians drawn to geometry, topology, and differential equations. disbursed in the Americas via the yank Mathematical Society.