Analytical Methods in Mathematical Physics

这个关于数学和物理的交界点的项目将通过数学方法加强对自然界中发生的复杂现象的定性和定量理解。它由几个模型组成，这些模型的动机是研究物质的奇异状态、超导和流体力学。本研究使用的方法来自于数学分析中的技术，而数学分析本身就是微积分的复杂抽象。虽然这些模型是特定的，但它们显示了某些一般性质，并且需要开发的方法将超出这些具体情况的范围。更具体地说，PI旨在通过探索它们与调和分析中的限制问题的联系，通过新形式的所谓Strichartz不等式来量化费米气体的色散性质。从微观的BCS理论出发，特别强调Meisner效应，严格推导出宏观的Ginzburg-Landau超导理论。数学工具包括半经典分析和非对易变分。研究了非线性、非局部方程的基态及其定态和动力学作用，特别是分析了它们的全局唯一性和局部唯一性，并试图找到两个具有共形不变性且具有物理和几何内容的泛函不等式的尖锐形式。

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.