A variational approach to nonlocal interactions: discrete-to-continuum analysis, ground states and geometric evolutions

This thesis deals with variational models for systems governed by nonlocal interactions. In particular, we analyze systems of hard spheres governed by attractive Riesz potentials, surface energies related to fractional perimeters and gradient flows of such energies leading to local and nonlocal geometric evolutions; eventually, we consider similar problems for densities governed by Gagliardo-type seminorms, focussing on fractional heat flows.