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，以模拟随机环境中随机进化的模型。示例是接触过程，这些接触过程模拟了通过与受感染的邻居接触传播的感染。其他例子是选民模型，描述了个人意见受到邻居意见影响的观点的传播。其他动机是由在常规晶格上传统的统计力学方法扩展到随机晶格的设置，我们可以将其视为表面上的随机图或网络。在这项工作中，PI将根据随机指标提供对此类模型的改进理解。可能会想象这些指标是由疾病传播或社会接触模式决定的。有趣的是，对随机指标的一般研究源于对二维（2D）量子重力的物理研究。用数学术语，2D量子重力的核心问题是严格构建Riemann表面指标空间的度量。这是几何和身体相关性的长期问题。但是，除了甚至之外，它还可以充当随机结构和可集成系统之间界面的新颖问题和想法的丰富来源。该提议的重点是概率理论之间的新兴杂交，重点是随机几何和组合学，以及可集成的系统理论，重点是经典分析，复杂功能理论，动力学系统和保守的部分差分方程。