This series of lectures consists of 9 hours. In the first part, we will survey some recent developments in the study of qualitative properties including symmetry, asymptotic symmetry, and monotonicity of solutions for nonlinear fractional elliptic and parabolic equations. Other typical fractional parabolic operators will be introduced, such as the dual fractional heat operator with Marchaud time derivative and the master operator. The extent of their non-locality will be illustrated by simple examples with pictures. We will also mention some of our recent results on interior regularity estimates for nonnegative solutions to fractional Laplace equations and fully fractional parabolic equations.

In the second part, we will review and discuss some related possible projects.