CAREER: Computational Dynamics and Topology

该项目将动力系统理论和计算拓扑学相结合，构建并利用严格的数值技术来研究模型系统中的动力学。主要的重点是构建、扩展和优化严格的数值技术，并调查可以有效地使用计算拓扑学来测量动力系统的其他设置。在可能的情况下，通常通过引入误差界和使用拓扑工具来严格验证计算结果。建议的项目包括扩展现有的计算机辅助研究和离散时间动力系统的证明技术，以及辫子定常解的强迫定理。虽然已经取得了一些初步进展，但在仍达不到期望的严格结果的领域又提出了两套项目。其中包括测量特定家族中表现出混沌行为的系统的参数集，以及使用计算同源性研究空间和时空模式的形成和演化。在这些领域有许多开放的研究项目(包括短期和长期的)，包括研究一些非常有趣的基本建模问题，涉及模型中适当的空间和其他尺度。这笔资金将用于资助教师-研究员博士后职位、本科生研究项目以及支持教学和研究小组活动。动力系统模型正在全社会得到应用。一些例子包括用于飓风预测的天气模型和用于研究环境对人口数量和持久性的影响的人口模型。目前，许多研究人员使用高性能的计算机模拟和统计技术来研究这样的动力系统。在光谱的另一端，数学家已经能够在更抽象的数学模型中破译高度复杂的动力学。本提案中描述的工作旨在成为这两种方法之间的桥梁。更具体地说，这位研究人员和她的合作者专注于计算技术的开发，这些技术使用复杂的数学工具并产生数学上严格的结果。这些数学工具来自代数拓扑学、分析、数值分析和动力系统理论等领域，可以用来解释所研究系统中一些感兴趣的现象。以前在种群生态学和热对流模型中研究复杂动力学的进展推动了这些继续研究的进行。除了继续与现有合作者合作外，研究人员还将指导一名博士后教师兼研究员，并为这些领域的本科生研究项目提供建议。

This project combines dynamical systems theory and computational topology in constructing and utilizing rigorous numerical techniques for the study of dynamics in model systems. The primary focus is on constructing, extending and optimizing rigorous numerical techniques and investigating additional settings where computational topology may be used effectively to measure dynamical systems. When possible, the computational results are verified rigorously, typically through the incorporation of error bounds and the use of topological tools. The proposed projects include extending existing techniques for computer-assisted studies and proofs for discrete time dynamical systems and forcing theorems for braided stationary solutions. Two more sets of projects are proposed in areas where the desired rigorous results are still out of reach, although some initial progress has been made. These include measuring parameter sets for systems in a specified family which exhibit chaotic behavior, and the study of spatial and spatio-temporal pattern formation and evolution using computational homology. There are many open research projects in these areas (both shorter and longer term) including the study of some very interesting and fundamental modeling questions concerning appropriate spatial and other scales in models. This grant will fund a teacher-researcher postdoctoral position, undergraduate research projects, and support teaching and research group activities.Dynamical systems models are being used throughout society. Some examples include weather models used for hurricane prediction and population models used to study environmental effects on population size and persistence. Currently, many researchers study dynamical systems like these using high powered computer simulations and statistical techniques. On the other end of the spectrum, mathematicians have been able to decipher highly complicated dynamics in more abstract mathematical models. The work described in this proposal aims to serve as a bridge between these two approaches. More specifically, the investigator and her collaborators focus on the development of computational techniques that use sophisticated mathematical tools and yield mathematically rigorous results. The mathematical tools come from the fields of algebraic topology, analysis, numerical analysis, and dynamical systems theory and may be used to decipher some of the phenomena of interest in the studied systems. Prior progress in studying complicated dynamics in models from population ecology and heat convection motivates these continued studies. In addition to continuing work with existing collaborators, the investigator will mentor a postdoctoral teacher-researcher and advise undergraduate research projects in these areas.