CAREER: Statistical mechanics of superconductors and other macroscopic phenomena

PI将致力于与量子系统的统计力学以及铁磁体和超导体的自旋模型相关的问题。数学物理中的一个基本问题是从微观粒子相互作用的第一原理来解释宏观现象。这个项目将建立在PI及其合作者在玻色-爱因斯坦凝聚和Gross-Pitaevskii方程的统计力学、量子系统的Gibbs测量、铁磁性的自旋模型以及家族谱系关系的概率算法方面的先前结果的基础上。提出的问题包括发展量子多体系统的新的Stein方法；将最近的方法扩展到其他问题，如量子系统的大偏差；研究XY中的亚稳性和相关模型；建立新的超导自旋模型并分析其性质；以及找到与Ginzburg-Landau和Gross-Pitaevskii唯象理论的联系。PI将把她的研究专长整合到概率课程中，并培训学生学习对各种职业有价值的分析和计算技术。PI还将继续她的兴趣，鼓励女性和其他在数学中代表不足的群体。凝聚态物质的数学物理学存在根本性的挑战，凝聚态物质是物质在绝对零度附近的冷却和不寻常的相，在物理和工程中有重要的应用。其一是理解凝聚态物质的独特性质是如何从许多微观粒子的大规模相互作用中显现出来的。例如，超导体允许电流无损耗地自由流动，并可以驱逐磁场；超导磁体被用于粒子加速器和核磁共振机。数学物理学家想要从微观量子多体系统开始解释这些现象(以及相关的方程)，但要做到这一点，需要在数学和物理之间开发出创新的数学工具。这个项目的主要目标是开发这样的工具，制定具有有趣的相变的新模型，并找到与现象学理论的联系。其他目标是为了互惠互利而加强与物理和工程的联系，将研究专业知识整合到概率课程中，并培训学生在各种职业中有价值的分析和计算技术。国际学生联合会还将继续鼓励女性和其他在数学方面代表性不足的群体。

The PI will work on problems related to the statistical mechanics of quantum systems and spin models for ferromagnets and superconductors. A fundamental problem in mathematical physics is to explain macroscopic phenomena from the first principles of interacting microscopic particles. This project will build on previous results of the PI and collaborators on the statistical mechanics of Bose-Einstein condensation and the Gross-Pitaevskii equation, Gibbs measures for quantum systems, spin models of ferromagnetism, and probabilistic algorithms for family pedigree relationships. The proposed problems include developing a new Stein's method for quantum many-body systems; extending recent methods to other questions such as large deviations for quantum systems; studying metastability in XY and related models; formulating new spin models of superconductivity and analyzing their properties; and finding connections to the phenomenological theories of Ginzburg-Landau and Gross-Pitaevskii. The PI will integrate her research expertise into probability courses and train students in analytical and computational techniques valuable for a variety of careers. The PI will also continue her interest in encouraging women and other underrepresented groups in math.There are fundamental challenges in the mathematical physics of condensed matter, the cool and unusual phases of matter near absolute zero that have important applications in physics and engineering. One is to understand how the unique properties of condensed matter emerge from the large-scale interaction of many microscopic particles. Superconductors, for instance, allow current to flow freely with no loss and can expel magnetic fields; superconducting magnets are used in particle accelerators and MRI machines. Mathematical physicists would like to explain these phenomena (and the associated equations) starting from microscopic quantum many-body systems, but to do so, will need innovative mathematical tools developed at the interface between mathematics and physics. The main goals of this project are to develop such tools, to formulate new models with interesting phase transitions, and to find connections to phenomenological theories. Other goals are to strengthen ties with physics and engineering for mutual benefit, integrating research expertise into probability courses, and training students in analytical and computational techniques valuable for a variety of careers. The PI will also continue her interest in encouraging women and other underrepresented groups in math.