Áreas de Investigación
I am interested in Calculus of Variations
, Geometric Measure Theory
, with special emphasis on interactions involving Probability
, and on applications to Physics
and Computer Science
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.
Related to the previous point is also the study of asymptotics of a large number of quantum particles, which is relevant to the computations of shapes of large molecules via Density Functional Theory. Here a multimarginal Optimal Transportation problem with an exotic cost appears, and I'm interested in the asymptotics and behavior as the number of marginals grows to infinity.
The appearance of collective behavior allows to make rigorous the link between micro- to macroscopic properties in fluids, solids and gases. Here the goal is to rigorously deduce the macroscopic properties in physically realistic situations, such as for a moving droplet of liquid.