Analytical Methods in Mathematical Physics

本项目以数学与物理相结合的方式，加强以数学方法对自然界复杂现象的定性与定量理解。它由几个模型组成，例如，由物质的奇异状态、超导性和流体力学的研究驱动。本研究中使用的方法来自数学分析技术，数学分析本身就是微积分的复杂抽象。虽然模型是具体的，但它们显示了某些一般属性，需要开发的方法将与这些具体案例的上下文相关。更具体地说，PI打算通过探索其与谐波分析中的限制问题的联系，通过所谓的新形式的Strichartz不等式来量化费米气体的色散特性。本文旨在从微观的BCS理论中严格推导出宏观的金兹堡-朗道超导理论，并特别强调迈斯纳效应。数学工具包括半经典分析和变分的非交换演算。本文将研究非线性、非局部方程的基态及其稳态和动态作用，特别是分析其全局唯一性和局部唯一性。此外，将尝试找到两个函数不等式的尖锐形式，它们显示保形不变性，并具有物理和几何内容。

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.