CAREER: Statistical mechanics of superconductors and other macroscopic phenomena

PI将研究与量子系统统计力学以及铁磁体和超导体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基本原理来解释宏观现象。该项目将建立在PI和合作者先前在玻色-爱因斯坦凝聚的统计力学和格罗斯-皮塔耶夫斯基方程、量子系统的吉布斯测量、铁磁性的自旋模型和家庭谱系关系的概率算法方面的成果之上。提出的问题包括开发量子多体系统的新斯坦方法；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的亚稳态及相关模型；建立新的超导自旋模型并分析其性质；寻找与金兹堡-朗道和格罗斯-皮塔耶夫斯基现象学理论的联系。PI将把她的研究专长整合到概率课程中，并培养学生对各种职业有价值的分析和计算技术。PI还将继续她对鼓励女性和其他在数学领域未被充分代表的群体的兴趣。凝聚态物质是一种接近绝对零度的冷而不寻常的物质，在物理和工程中有着重要的应用。一个是了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产生的。例如，超导体允许电流自由流动而没有损耗，并且可以驱逐磁场；超导磁体用于粒子加速器和核磁共振成像仪。数学物理学家希望从微观量子多体系统开始解释这些现象（以及相关方程），但要做到这一点，将需要在数学和物理之间的接口上开发出创新的数学工具。该项目的主要目标是开发这样的工具，制定具有有趣相变的新模型，并找到与现象学理论的联系。其他目标是加强与物理和工程的联系以实现互利，将研究专业知识整合到概率课程中，并培养学生在各种职业中有价值的分析和计算技术。PI还将继续她对鼓励女性和其他在数学领域未被充分代表的群体的兴趣。

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.