Variational Methods in Mathematical Physics and Analysis

这个项目的主要重点是对自然界中发生的复杂现象进行数学分析。研究中的模型源于凝聚态物理和量子信息理论，并描述了各种物理概念，从固体中的准粒子到超导性以及熵和纠缠的测量。除了理解在某些特定系统中工作的数学和物理原理外，还将开发出超出这些具体问题背景的数学上有趣的分析工具，并将其应用于回答纯数学中的问题。本项目为研究生和本科生提供科研训练机会。更具体地说，PI将重新审视极化子问题，将研究微观和宏观超导理论之间的联系，并研究交换和非交换条件下的泛函不等式。注意力将集中在谐波分析和变分学之间的一个最近出现的领域。所有这些问题的一个统一特征是，从某种（几乎）最优性属性中，人们想要得出这样的结论：所讨论的对象具有相对简单的结构。实现这一目标的数学工具通常与低正则设置中的半经典分析有关，主要目标是量化潜在的非交换性，这是量子物理固有的。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The main focus of this project is the mathematical analysis of complex phenomena occurring in nature. The models under investigation originate from condensed matter physics and quantum information theory and describe various physical concepts ranging from quasiparticles in solids, to superconductivity and measures of entropy and entanglement. Besides understanding the mathematical and physical principles at work in some specific systems, analytical tools will be developed which are mathematically interesting beyond the context of these concrete problems and which will be applied to answer questions in pure mathematics. The project provides research training opportunities for graduate students and undergraduate students. More specifically, the PI will revisit the polaron problem, will investigate the connection between a microscopic and a macroscopic theory of superconductivity and study functional inequalities in both a commutative and a noncommutative setting. Attention will focus on a recently emerging area at the interface between harmonic analysis and calculus of variations. A unifying feature of all these problems is that from a certain (almost) optimality property one wants to conclude that the objects in question have a relatively simple structure. Mathematical tools to achieve this are often related to semiclassical analysis in a low regularity setting, and a major goal is to quantify the underlying non-commutativity, which is inherent to quantum physics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.