CAREER: Statistics of Extrema in Complex and Disordered Systems

涉及多个相互作用的主体的自然和社会系统可以用概率模型准确地描述。这是凝聚态物理、数据科学、金融、计算机科学和纯数学中的一大类系统的情况。这种复杂系统的行为在很大程度上取决于可能对其动力学产生重大影响的罕见事件或极端情况。这个项目是对出现在呈现许多不同极端的广泛的复杂系统中的基本模式的严格研究。其目的是发展一般的定量方法，以理解这些系统的极值统计，并与数学和物理中的相关问题建立新的联系。它还包括与学术界和工业界的从业人员合作，对学生进行数据科学、金融和计算机科学中概率应用的高级培训。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。从严格的角度来看，对这类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，自旋玻璃)，它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种情况下开发新的概率方法：统计力学中的现实自旋眼镜和数论中的Riemann Zeta函数。特别是，它建立在描述自旋玻璃无限维模型的最新进展的基础上，提出了一种新的严格的方法来研究现实(有限维)自旋玻璃的行为，由于存在许多极值能量，这些自旋玻璃表现出不寻常的磁性行为。该程序还通过将Riemann Zeta函数作为一个无序系统进行研究，概述了纯数学研究的新方向。其目的是发展统计力学的方法来研究Riemann Zeta函数在临界条的短区间内的极值，并探索其在数论中的含义。

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.