Point-like (or more complicated) topological singularities arising in Nonlinear Sobolev Spaces and Gauge Theory can be interpreted as particles, vortices, charges or defects. 

I study the asymptotic behavior of minimizing configurations as the number of vortices increases. The sharp asymptotics appearing in these studies are relevant in Approximation theory, in Statistical Physics and in Random Matrix theory. 

At the moment I am especially interested in uniformization/crystallization phenomena, where for large numbers of points one can prove/quantify that the configurations show an emergent collective behavior, and come close to forming lattice-like structures. 
New notions of curvature seem to arise in the study of these asymptotics.