Elliptic PDE's in probability and geometry:: Symmetry and regularity of solutions

We describe several topics within the theory of linear and nonlinear second order elliptic Partial Differential Equations. Through elementary approaches, we first explain how elliptic and parabolic PDEs are related to central issues in Probability and Geometry. This leads to several concrete equations. We classify them and describe their regularity theories. After this, most of the paper focuses on the ABP technique and its applications to the classical isoperimetric problem for which we present a new original proof, the symmetry result of Gidas-Ni-Nirenberg, and the regularity theory for fully nonlinear elliptic equations.