Collaborative Research: Revealing the Geometry of Spatio-temporal Chaos with Computational Topology: Theory, Numerics and Experiment

我们所经历的天气是由对流驱动的，阳光使地球变暖，从而加热了被外层空间寒冷温度冷却的大气。大多数人对微观行为（例如空气中单个分子的行为）和宏观行为（例如全球平均温度）都不感兴趣。感兴趣的是介观模式，例如导致局部温度变化的天气锋。这种兴趣在介观，而不是微观或宏观特征，大规模的系统发生在各种各样的复杂的大规模的物理现象，如发动机燃烧，在海洋中的生物质动力学，心室纤颤在人类心脏等这些介观模式采取了许多不同的形状和大小，并随时间变化，有时缓慢，有时迅速。这些模式的形式以及它们如何随时间演变通常非常依赖于参数。新技术极大地提高了我们测量和模拟这些物理现象的能力，产生了巨大的数据集，但我们提取和量化这些信息的能力并没有跟上理解，预测和控制这些系统的步伐。我们将探索使用新的数学工具来解决这个问题。Rayleigh-Bénard对流的空间和时间复杂性产生高维时间序列数据。 一个相对较新的代数拓扑工具，称为持久的同源性将被用来提供新的工具，非线性降维。为了确保这些方法的适用性和物理上重要的介观特征的动态保存，他们将被开发与精心控制的高精度对流实验和国家的最先进的，大规模的，高分辨率的数值模拟的Boussinesq方程的进一步发展。这包括协变李雅普诺夫指数的几何分析。在这项工作中开发的新的计算工具应该找到广泛的应用在各种各样的问题，涉及复杂的非平衡系统在自然界（海洋和大气流动，气候和天气预报）和技术（非线性光学系统，燃烧和化学反应）的理解和预测的复杂行为是必要的。

The weather we experience is driven by convection, sunlight warms the earth which heats the atmosphere which is cooled by the cold temperatures of outer space. Most people are not interested in microscopic behavior, for example the behavior of the individual molecules in the air, nor macroscopic behavior, such as worldwide average temperature. What is of interest are mesoscopic patterns, for example weather fronts which result in local changes in temperature. This interest in mesoscopic, as opposed to micro- or macroscopic features, of large scale systems occurs in a wide variety of complex large scale physical phenomena such as combustion in engines, dynamics of biomass in the oceans, ventricle fibrillation in a human heart, etc. These mesoscopic patterns take on many different shapes and sizes and change with time, sometimes slowly and sometimes rapidly. The form of these patterns and how they evolve in time is often very dependent on parameters. New technologies are greatly increasing our abilities to measure and simulate these physical phenomena, resulting in enormous data sets, but our ability to extract and quantify this information in a way that leads to understanding, predictability, and control of these systems is not keeping pace. We will explore the use of new mathematical tools to address this problem.The spatial and temporal complexity of Rayleigh-Bénard convection produces high dimensional time series data. A relatively new algebraic topological tool called Persistent Homology will be used to provide new tools for nonlinear dimension reduction. To ensure the applicability of these methods and that physically important mesoscopic features of the dynamics are preserved they will be developed in conjunction with the further development of carefully controlled high precision convection experiments and state-of-the-art, large scale, high-resolution numerical simulations of the Boussinesq equations. This includes the analysis of the geometry of covariant Lyapunov exponents. The new computational tools developed in this work should find broad application in a wide variety of problems involving complex nonequilibrium systems in nature (oceanic and atmospheric flows, climate and weather forecasting) and in technology (nonlinear optical systems, combustion and chemical reactions) where understanding and prediction of complex behavior is desired.