Fully Nonlinear PDEs in Real and Complex Geometry and Optics

This volume contains the notes from the CIME school “Fully Nonlinear PDEs in Real and Complex Geometry and Optics” held at Cetraro (Cosenza, Italy) during the week of July 9–13, 2012. The school consisted of four courses: Extremal problems for quasiconformal mappings in space by Luca Capogna, Fully nonlinear equations in geometry by Pengfei Guan, Monge-Ampère type equations and geometric optics by Cristian E. Gutierrez, and "On the Levi Monge–Ampère equation" by Annamaria Montanari. The purpose of the school was to present current areas of research, arising both in the theoretical and in the applied settings, that involve fully nonlinear partial differential equations. The solution to these problems requires the development of ad hoc techniques arising in the relevant geometrical framework, and in case of the applications, determined by the physical laws governing the studied phenomena. Precisely, the equations presented in the school come from conformal mapping theory, differential geometry, optics, and geometric theory of several complex variables.