Phase separation and minimal surfaces, from local to nonlocal

I would like to discuss some questions related to some semilinear equations driven by a nonlocal elliptic operator (for example, the Allen-Cahn equation, in which the classical Laplace operator is replaced by a fractional Laplacian). In particular, I would like to study the qualitative properties of the solutions, such as symmetry, density estimates of the level set, asymptotic behaviors, etc. The limit interfaces of these problems are related to both the local and the nonlocal perimeter functionals on this topic, I would like to discuss some recent rigidity and regularity results and to present some open problems.