PhD thesis on PDE and metric aspects of nonlocal operators

This PhD project centers on the study of HamiltonJacobiBellman equations, with particular emphasis on the theory of viscosity solutions and innite-horizon control problems. It also investigates several components of weak KAM theory, including its extension to general metric spaces, as well as abstract notions of descent such as the De Giorgi slope, global slope, and various notions of average slope. A central aim of the project is to develop and clarify the connections between these dierent framework