Analytical Methods in Mathematical Physics

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This project on the interface of mathematics and physics will enhance the qualitative and quantitative understanding of complex phenomena occurring in nature by mathematical methods. It consists of several models which are motivated, for instance, by the study of exotic states of matter, superconductivity, and fluid mechanics. The methods used in this study come from techniques in mathematical analysis, which itself is a sophisticated abstraction of calculus. While the models are specific, they show certain general properties and the methods that need to be developed will be relevant beyond the context of these concrete cases.More specifically, the PI intends to quantify dispersive properties of a Fermi gas by novel forms of so-called Strichartz inequalities, via exploring their connection with the restriction problem in harmonic analysis. It is intended to rigorously derive the macroscopic Ginzburg-Landau theory of superconductivity from the microscopic BCS theory with particular emphasis on the Meisner effect. Mathematical tools include semi-classical analysis and a non-commutative calculus of variations. Ground states of non-linear, non-local equations and their stationary and dynamical role will be studied, in particular, their global and local uniqueness properties will be analyzed.In addition, an attempt will be made to find the sharp form of two functional inequalities which display conformal invariance and have both physical and geometric content.