Analysis of Ginzburg-Landau models

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The goal of the proposed research is the rigorous analysis of nonlinear partial differential equations (PDE) and variational problems arising from models of the physical world. While the Ginzburg-Landau (GL) model was initially introduced to study phase transitions in superconductors, over time it has become ubiquitous in the mathematical description of multi-phase systems in physics and materials science and has lead to a wealth of deep mathematical results and new methods. In a Ginzburg-Landau model the state of a physical system is described as a critical point of a given energy functional, and solves a system of nonlinear elliptic PDE. Heuristic analysis suggests that in certain parameter limits, the solutions may develop geometrical singularities (such as vortices, monopoles, or domain walls,) which give the most salient features of the system and characterize its fundamental interactions and dynamics.