Random Structures and Integrable Systems: Analysis and Applications

随机结构与可积系统：分析与应用

• 批准号: 1615921
• 负责人: Nicholas Ercolani
• 金额: $ 36.5万
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-06-01 至 2020-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1615921&HistoricalAwards=false
• 关键词: Random; Structures; Integrable; Systems; Analysis

The overarching goal of the projects supported by this award is to develop mathematical tools that can describe the structure and properties of networks and surfaces that model random evolution in random environments. Examples of this are contact processes which model infections that spread by contact with an infected neighbor. Other example are voter models which describe the spread of opinions where an individual's opinions are affected by his or her neighbor's opinions. Other motivations arise from extending methods of statistical mechanics classically done on regular lattices to the setting of random lattices which we may think of as random graphs or networks on a surface. In this work the PI will provide an improved understanding of such models in terms of random metrics. One may imagine these metrics are determined by patterns of disease propagation or social contact. Interestingly the general study of random metrics arose from physical investigations of two-dimensional (2D) quantum gravity.The central problem of 2D quantum gravity, in mathematical terms, is to rigorously construct a measure on the space of metrics on a Riemann surface. This is a long-standing problem of both geometric and physical relevance. But aside from, and perhaps even beyond this, it serves as a rich source of novel problems and ideas that are at the interface between random structures and integrable systems. The focus of this proposal is on emerging crosscurrents of research between probability theory, with an emphasis on random geometry and combinatorics, and integrable systems theory with an emphasis on classical analysis, complex function theory, dynamical systems and conservative partial differential equations.

该奖项支持的项目的总体目标是开发数学工具，这些工具可以描述网络和表面的结构和属性，这些网络和表面可以模拟随机环境中的随机演化。这方面的例子是接触过程，其对通过与受感染的邻居接触而传播的感染进行建模。其他例子是选民模型，它描述了个人的意见受到他或她邻居的意见影响的意见传播。其他的动机来自于扩展统计力学的方法，这些方法经典地在规则格上完成，以设置随机格，我们可以认为随机格是表面上的随机图或网络。在这项工作中，PI将在随机指标方面提供对此类模型的更好理解。人们可以想象这些指标是由疾病传播或社会接触的模式决定的。有趣的是，随机度规的一般研究起源于二维量子引力的物理研究。二维量子引力的中心问题，用数学术语来说，是在黎曼曲面上的度规空间上严格构造一个测度。这是一个长期存在的几何和物理问题。但除此之外，也许甚至更重要的是，它还是随机结构和可积系统之间界面上新问题和想法的丰富来源。这项建议的重点是在概率论，重点是随机几何和组合学，和可积系统理论，重点是经典分析，复杂的功能理论，动力系统和保守的偏微分方程之间的研究新出现的交叉潮流。