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High-Dimensional Dynamical Systems

高维动力系统

基本信息

批准号：
1714187
负责人：
Govind Menon
金额：
$ 33.7万
依托单位：
Brown University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2020-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1714187&HistoricalAwards=false
关键词：
Dimensional Dynamical Systems

项目摘要

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.

这个项目为无序的物理系统开发数学模型。该项目研究的主要应用是湍流流体、玻璃的微观结构以及星团聚集时出现的分支结构和闪电的形状。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，该项目还包括对几种广泛使用的数值算法的性能的统计方法。这项研究的主要目的是开发数学方法来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，这些都是物理学和材料科学中众所周知但鲜为人知的系统。该项目还通过培养研究生来促进STEM队伍的发展。该项目的主要重点是为具有强非线性、包含随机性和大量自由度的动力系统开发数学方法。分析这些系统的数学工具来自于可积系统、动力学理论、偏微分方程组和概率论。(A)不可压缩流体中的随机等距嵌入和湍流；(B)具有分支的共形过程；(C)自旋玻璃的老化；以及(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。在方案(A)中，湍流启发式被用来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。在方案(B)中，研究了一类新的随机偏微分方程解描述共形嵌入树。项目(C)考虑了材料科学中玻璃体系统的基本模型。项目(D)的目的是对计算机科学中广泛使用的几种算法的“困难概率”进行量化。这两个项目都通过对玻璃行为的研究而统一起来。特别地，所研究的算法可以被视为具有指数多个临界点的高维动力系统。它们的表现反映了老化和亚稳定等呆板的行为。相反，这些算法充当了数字实验室，揭示了物理眼镜的老化动力学。