Analytical Methods in Mathematical Physics

这个关于数学和物理界面的项目将通过数学方法提高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，例如，通过研究物质的奇异状态，超导性和流体力学。本研究中使用的方法来自数学分析中的技术，数学分析本身是微积分的复杂抽象。虽然这些模型是特定的，但它们显示出某些一般性质，需要开发的方法将超越这些具体案例的背景。更具体地说，PI打算通过探索它们与谐波分析中的限制问题的联系，通过所谓的Hohartz不等式的新形式来量化费米气体的色散性质。它的目的是严格地从微观BCS理论推导出宏观的金斯堡-朗道超导理论，特别强调迈斯纳效应。数学工具包括半经典分析和非交换变分法。本文将研究非线性、非局部方程的基态及其定态和动力学性质，特别是分析它们的全局和局部唯一性，并试图找到两个具有共形不变性的、具有物理和几何内容的泛函不等式的精确形式。

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.