CAREER: Statistics of Extrema in Complex and Disordered Systems

自然和社会系统涉及许多相互作用的代理人可以准确地描述概率模型。凝聚态物理、数据科学、金融、计算机科学和纯数学中的一大类系统就是这种情况。这种复杂系统的行为主要取决于可能对其动态产生重大影响的罕见事件或极值。这个项目是一个严格的研究基本模式，出现在广泛的复杂系统表现出许多不同的极值。其目的是发展一般的定量方法来理解这些系统的极值统计，并与数学和物理学中的相关问题建立新的联系。它还涉及与学术界和工业界的从业者合作，对学生进行概率在数据科学，金融和计算机科学中的应用的高级培训。更准确地说，复杂系统可以被视为具有强相关随机变量的随机过程。从严格的观点来看，对这些过程的极值的统计仍然知之甚少。重要的例子包括统计力学中的无序系统（例如，自旋眼镜），这与计算机和数据科学中的优化问题密切相关。该项目的重点是在两个实例中开发新的概率方法：统计力学中的现实自旋玻璃和数论中的黎曼zeta函数。特别是，它建立在自旋玻璃的无限维模型的描述的最新进展，提出了一个新的严格的方法来调查现实的（有限维）自旋玻璃，表现出不寻常的磁性行为，由于存在许多极端的能量的行为。该计划还通过研究作为无序系统的黎曼zeta函数，概述了纯数学研究的新方向。目标是发展统计力学的方法来研究黎曼zeta函数在临界带的短间隔内的极值，并探索其在数论中的意义。

Natural and social systems involving many interacting agents can accurately be described by probabilistic models. This is the case for a large class of systems in condensed matter physics, data science, finance, computer science, and pure mathematics. The behavior of such complex systems depends crucially on rare events, or extrema, that may have a large impact on their dynamics. This project is a rigorous study of the fundamental patterns that arise in a wide range of complex systems exhibiting many different extrema. The aim is to develop general quantitative methods to understand the statistics of extrema of these systems and to draw new connections with related problems in mathematics and physics. It also involves the advanced training of students in the applications of probability in data science, finance, and computer science in partnership with practitioners in academia and in industry. More precisely, complex systems can be viewed as stochastic processes with strongly correlated random variables. The statistics of extrema of such processes are still poorly understood from a rigorous point of view. Important examples include disordered systems in statistical mechanics (e.g., spin glasses), which are closely related to optimization problems in computer and data sciences. The focus of the project is on developing new probability methods in two instances: realistic spin glasses in statistical mechanics and the Riemann zeta function in number theory. In particular, it builds on recent progress in the description of infinite-dimensional models of spin glasses to propose a new rigorous method to investigate the behavior of realistic (finite-dimensional) spin glasses, which exhibit unusual magnetic behavior due to the presence of many extremal energies. The program also outlines new directions of research in pure mathematics by studying the Riemann zeta function as a disordered system. The objectives are to develop methods of statistical mechanics to study the extreme values of the Riemann zeta function in a short interval of the critical strip, and to explore the implications in number theory.