Mathematical Analysis of Topological Singularities in some physical problems – MAToS

The central theme of this project lies in the area of nonlinear analysis (nonlinear partial differential equations and calculus of variations). Our main focus will be on the structure and dynamics of topological singularities arising in some variational physical models driven by the Landau-Lifshitz equation (in micromagnetics) and the Gross-Pitaevskii equation (in superconductivity, Bose-Einstein condensation, nonlinear optics). These include vortex singularities, traveling waves and domain walls in magnetic thin films.

The mathematical challenges concern the description of microscopic structures and of phenomena that occur at very different spatial or temporal scales. They often require recently developed mathematical tools and the introduction of new mathematical techniques (such as Gamma-convergence and refined elliptic estimates). We intend to make significant progress in challenging mathematical problems that will give more insight into the corresponding physical phenomena. Moreover, the mathematical methods that we want to develop have their own interest by their richness, the variety of tools and the links they weave between different domains: analysis, partial differential equations and geometry.