Random Structures and Integrable Systems: Analysis and Applications

该奖项支持的项目的总体目标是开发能够描述随机环境中模拟随机进化的网络和表面的结构和特性的数学工具。这方面的例子是接触过程，它模拟了通过与受感染的邻居接触而传播的感染。另一个例子是选民模型，它描述了一个人的观点受到他或她邻居的观点的影响。其他动机来自于将统计力学的经典方法在规则格上进行的扩展到随机格的设置，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将根据随机度量提供对这些模型的更好理解。人们可能会认为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，随机度量的一般研究起源于二维量子引力的物理研究。用数学术语来说，二维量子引力的中心问题是在黎曼曲面上的度量空间上严格地构造一个测度。这是一个长期存在的几何和物理相关问题。但除此之外，甚至可能除此之外，它还可以作为随机结构和可积系统之间的界面上的新问题和想法的丰富来源。本提案的重点是在概率论（强调随机几何和组合学）和可积系统理论（强调经典分析）、复函数理论、动力系统和保守偏微分方程之间新兴的交叉研究潮流。

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.