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Variational Methods in Mathematical Physics and Analysis

数学物理与分析中的变分方法

基本信息

批准号：
1954995
负责人：
Rupert Frank
金额：
$ 32.23万
依托单位：
California Institute of Technology
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-08-01 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1954995&HistoricalAwards=false
关键词：
Variational Methods Mathematical Physics Analysis

项目摘要

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.

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。正在研究的模型起源于凝聚态物理和量子信息理论，描述了从固体中的准粒子到超导以及熵和纠缠的测量的各种物理概念。除了了解在某些特定系统中起作用的数学和物理原理外，还将开发出超越这些具体问题背景的数学有趣的分析工具，并将用于回答纯数学中的问题。该项目为研究生和本科生提供了研究培训机会。更具体地说，PI将重温极化子问题，将研究超导电性的微观理论和宏观理论之间的联系，并研究交换和非交换环境中的泛函不等式。人们的注意力将集中在调和分析和变分之间的一个新出现的领域。所有这些问题的一个统一特征是，从某种(几乎)最佳性性质，人们想要得出结论，所讨论的对象具有相对简单的结构。实现这一点的数学工具通常与低正则性环境下的半经典分析有关，主要目标是量化基本的非对易，这是量子物理固有的。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。