Nonlinear elliptic PDEs arise in many physical and geometric contexts, for example in models of soap films, crystal surfaces, and cloud motion. I will discuss some of the questions mathematicians aim to answer about such PDEs, using the minimal surface equation and Monge-Ampere equation as guiding examples.

The notion of viscosity solution was introduced in 1980s by Evans and Crandall/Lions, which is one of the most important developments in the  theory of elliptic equations.  It provides a rigorous mathematical framework to describe the correct ``physical" solution of  first or second order PDEs when classical solutions might not exist. Important examples include first order Hamilton-Jacobi equations or second order degenerate elliptic equations (e.g mean curvature type equations) arising from control theory or front propagation problems in real applications.   In this talk, I will go over basic definitions, some important techniques, fundamental results and interesting examples.