CAREER: Statistical mechanics of superconductors and other macroscopic phenomena

PI将研究与量子系统的统计力学和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相互作用的微观粒子的第一原理解释宏观现象。该项目将建立在PI和合作者先前在玻色-爱因斯坦凝聚和Gross-Pitaevskii方程的统计力学，量子系统的吉布斯测度，铁磁性的自旋模型和家族谱系关系的概率算法上的结果之上。提出的问题包括发展一个新的斯坦的量子多体系统的方法;最近的方法扩展到其他问题，如量子系统的大偏差;研究XY和相关模型的亚稳性;制定新的超导自旋模型和分析其属性;并找到连接到金斯堡-朗道和格罗斯-Pitaevskii的唯象理论。PI将把她的研究专长融入概率课程，并培养学生对各种职业有价值的分析和计算技术。PI也将继续她的兴趣，鼓励妇女和其他代表性不足的群体在数学。有基本的挑战，在数学物理学的凝聚态，冷却和不寻常的阶段的物质接近绝对零度，有重要的应用在物理学和工程。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尺度相互作用中产生的。例如，超导体允许电流自由流动而没有损失，并且可以排出磁场;超导磁体用于粒子加速器和MRI机器。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（以及相关的方程），但要做到这一点，需要在数学和物理之间的界面上开发创新的数学工具。这个项目的主要目标是开发这样的工具，制定有趣的相变的新模型，并找到连接到唯象理论。其他目标是加强与物理学和工程学的互利联系，将研究专业知识融入概率课程，并培训学生对各种职业有价值的分析和计算技术。PI还将继续鼓励妇女和其他数学代表性不足的群体。

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.