REGULARITY AND SINGULARITY OF SOLUTIONS TO GEOMETRIC VARIATIONAL PROBLEMS

Solutions of geometric variational problems describe equilibrium configurations of physical systems or provide preferred representatives in homology and homotopy classes. Their singularities can be linked to concentration of energy or to topological obstructions, and their study is then of fundamental importance in their applications to Geometry and Physics.This project intends to leverage on a series of novel techniques introduced by the PI in recent years to significantly improve our knowledge of geometric variational problems. The project will address a series of fundamental questions concerning the regularity and the structure of singularities of solutions to geometric variational problems whose answer will enhance our understanding of their behavior. This will be done according to three deeply interrelated lines of research: Regularity of minimizers and critical points in geometric measure theory, Singular structure of PDE constrained measures, Structure of free boundaries and of solutions to spectral optimization problems.