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CAREER: Statistics of Extrema in Complex and Disordered Systems

职业：复杂无序系统中的极值统计

基本信息

批准号：
1653602
负责人：
Louis-Pierre Arguin
金额：
$ 44.6万
依托单位：
CUNY Baruch College
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2023-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1653602&HistoricalAwards=false
关键词：
CAREER Statistics Extrema Complex Disordered

项目摘要

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.

涉及许多相互作用主体的自然系统和社会系统可以用概率模型精确地描述。对于凝聚态物理、数据科学、金融、计算机科学和纯数学中的大量系统来说，情况就是如此。这种复杂系统的行为主要取决于可能对其动力学产生重大影响的罕见事件或极端事件。这个项目是一个严谨的基本模式的研究，出现在广泛的复杂系统表现出许多不同的极端。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实践者合作，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，复杂系统可以看作是具有强相关随机变量的随机过程。从严格的角度来看，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（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玻璃和数论中的黎曼ζ函数。特别是，它建立在描述无限维自旋玻璃模型的最新进展的基础上，提出了一种新的严格方法来研究现实（有限维）自旋玻璃的行为，这些自旋玻璃由于存在许多极端能量而表现出不寻常的磁性行为。该计划还概述了纯数学研究的新方向，通过研究黎曼ζ函数作为一个无序系统。目的是发展统计力学方法来研究临界带短区间内黎曼ζ函数的极值，并探讨其在数论中的意义。