CAREER: Integrating Geometric, Probabilistic, and Topological Methods for Phase Space Transport in Dynamical Systems

本学院早期职业发展（Career）项目的研究目标是为相空间输运问题的理论和计算分析奠定坚实的统一基础。许多物理系统可以被建模为具有一个潜在的动态骨架，它组织所有可能的行为是如何相互关联的。对于已知的具有简单时间依赖性的解析系统，这个骨架由众所周知的数学结构组成，如不变集和渐近接近或偏离不变集的集。但是对于非周期性的、基于数据的、只在短时间内观察到的系统，通常没有这样的结构。提出的研究将找到这些结构的适当类似物，以扩大动力系统结果对现实世界数据的适用性。在流体流动的背景下，我们将开发使用几何、概率和拓扑方法的工具，包括相干集、几乎不变集和符号动力学。该项目的成功完成将为概念化、可视化和提取有关系统可能行为的信息提供一种新的、富有成效的方法，适用于流体力学和其他领域，例如由气象、金融、心理或人口观测定义的数据集中不同性质的行为之间的边界和过渡。教育、外展和传播是项目活动的组成部分，针对四种受众：(1)本科生将参与研究并为(2)高中生开发学习材料，他们将了解混乱运输的作用及其在环境和生物学中的相关性；(3)研究生和教师将通过基于项目的课程探索动力系统工具在新问题领域的应用；全球研究人员将受益于在线传播的数字工具，这些工具可能会揭示动态数据集中以前隐藏的结构。

The research objective of this Faculty Early Career Development (CAREER) project is to develop a firm unified foundation for the theoretical and computational analysis of the phase space transport problem. Many physical systems can be modeled as having an underlying dynamical skeleton which organizes how all the possible behaviors are related. For systems known analytically with simple time dependence, this skeleton consists of well-known mathematical structures such as invariant sets and sets which asymptotically approach or depart from such sets. But for systems of increasing interest which are non-periodic, based on data, and observed over only a short time, there are no such structures in general. The proposed research will find appropriate analogs to these structures to expand the applicability of dynamical systems results to real world data. Working in the context of fluid flows, we will develop tools using geometric, probabilistic, and topological approaches, including coherent sets, almost-invariant sets, and symbolic dynamics.Successful completion of this project will provide a new and fruitful approach for conceptualization, visualization, and extraction of information regarding the possible behaviors of a system, with applicability to fluid mechanics and beyond, such as boundaries and transitions between qualitatively different kinds of behavior in data sets defined by meteorological, financial, psychological, or population observations. Education, outreach and dissemination are integral to the project activities, aimed at four audiences: (1) undergraduates will participate in research and develop learning materials for (2) high school students, who will learn about the role of chaotic transport and its relevance in the environment and biology; (3) graduate students and faculty will explore applications of tools from dynamical systems to new problem domains through a project-based course; and the (4) global community of researchers will benefit from numerical tools disseminated online which may reveal previously hidden structure in dynamic data sets.