High-Dimensional Dynamical Systems

本项目开发无序物理系统的数学模型。在这个项目中研究的主要应用是湍流流体，玻璃的微观结构，以及在簇聚集和闪电形状中出现的分支结构。正如经常发生的那样，所研究的数学模型也解释了与特定物理系统无关的现象，并且该项目还包括几种广泛使用的数值算法的性能统计方法。本研究的主要目的是发展数学方法，以提高对重要现象的理解，特别是流体中的湍流和玻璃的老化，这是物理学和材料科学中众所周知但知之甚少的系统。该项目还通过培养研究生来促进STEM劳动力的发展。这个项目的主要焦点是发展数学方法的动力系统是强非线性，涉及随机性，并有大量的自由度。分析这些系统的数学工具来自于可积系统、动力学理论、偏微分方程和概率论。考虑了四个项目：(a)不可压缩流体中的随机等距嵌入和湍流；(b)带分支的保形过程；(c)自旋玻璃老化；(d)数值线性代数算法的统计行为。项目(a)和(b)提供了关于随机几何的严格结果。在项目(a)中，湍流启发式用于改进微分几何中等距嵌入问题的已知临界指数。这种结构也揭示了在各向同性均匀湍流中产生的随机流体流动。在项目(b)中，研究了一个新的随机偏微分方程，其解描述共形嵌入树。项目(c)考虑材料科学中玻璃体系统的基本模型。项目(d)的目的是量化计算机科学中几种广泛使用的算法的“困难概率”。这两个项目是统一的研究玻璃的行为。特别是，所研究的算法可以被视为具有指数级多临界点的高维动力系统。它们的性能反映了玻璃态的行为，如老化和亚稳态。相反，这些算法可以作为数值实验室，阐明物理玻璃的老化动力学。

This project develops mathematical models for disordered physical systems. The primary applications studied in this project are turbulent fluids, the microstructure of glasses, and branched structures that arise in the aggregation of clusters and the shape of lightning. As often happens, the mathematical models studied also explain phenomena unrelated to specific physical systems, and this project also includes a statistical approach to the performance of several widely-used numerical algorithms. The primary purpose of this research is to develop mathematical methods to improve understanding of important phenomena, particularly turbulence in fluids and the aging of glasses, which are well-known but poorly-understood systems in physics and materials science. This project also contributes to the development of the STEM workforce through the training of graduate students.The main focus of this project is the development of mathematical methods for dynamical systems that are strongly nonlinear, involve randomness, and have a large number of degrees of freedom. The mathematical tools to analyze these systems are drawn from integrable systems, kinetic theory, partial differential equations and probability theory. Four projects are considered: (a) random isometric embeddings and turbulence in incompressible fluids; (b) conformal processes with branching; (c) aging of spin glasses; and (d) statistical behavior of algorithms in numerical linear algebra. Projects (a) and (b) provide rigorous results on random geometry. In project (a), turbulence heuristics are used to improve the known critical exponents for the isometric embedding problem in differential geometry. This construction also sheds light on random fluid flows arising in isotropic homogeneous turbulence. In project (b), a new stochastic partial differential equation, whose solutions describe conformally embedded trees, is investigated. Project (c) considers a basic model for vitreous systems in materials science. The purpose of project (d) is to quantify the `probability of difficulty' for several widely used algorithms in computer science. Both these projects are unified by the study of glassy behavior. In particular, the algorithms studied may be viewed as high-dimensional dynamical systems with exponentially many critical points. Their performance reflects glassy behavior such as aging and metastability. Conversely, the algorithms serve as numerical laboratories that shed light on aging dynamics in physical glasses.