Multiplicative Ergodic Theory, Dynamics and Applications

我的研究计划使用遍历理论的思想（具有不变测度的动力系统的长期行为的研究）来验证和通知环境数据测量方案的创建。在环境动力系统中，与系统其余部分缓慢混合的区域（例如海洋环流和大气中的气旋）对系统其余部分有重大影响。由于难以获得细粒度数据，定位这些特征并测量其属性是一项重大挑战。我的工作涉及使用粗粒度数据测量的数学技术以及复杂的后处理来恢复信息，否则必须使用更精细（因此更昂贵）的测量来找到这些信息。这些想法在实践中被证明是非常有效的，尽管存在理论上的差距，因为没有数学证明被测量的东西与物理现实相对应。我的研究是关于缩小理论和实践之间的差距。沿着，我期望能够在数据重建技术方面做进一步的改进。这项研究的数学基础本身就具有相当大的意义。研究中涉及的关键工具之一，李雅普诺夫指数，是混沌理论的主要组成部分。李雅普诺夫指数测量动力系统中的拉伸速率（并且是我们可以期望天气预报准确多久等问题的基础）。几十年来，在快速计算机的广泛应用的推动下，对这一领域进行了大量的研究，一个非常重要的数学问题是这些李雅普诺夫指数稳定的程度。也就是说，如果对一个系统做一个小的改变，那么李雅普诺夫指数也会发生小的变化吗？令人惊讶的是，在某些重要情况下，这个问题的答案是“不”：系统的一个非常小的变化可能导致李雅普诺夫指数的根本变化。如果试图从数据中重建真实世界动力系统中的李雅普诺夫指数，这是有问题的。我的研究的一个主要部分是理解导致李雅普诺夫指数稳定的系统变化的场景和类型。事实上，似乎有一个重要的连接之间的李雅普诺夫指数和上述环境动力系统的缓慢混合区域。

My research program uses ideas from ergodic theory (the study of the long-term behaviour of dynamical systems that have an invariant measure) to validate and inform the creation of measurement schemes for environmental data. In environmental dynamical systems, regions that mix slowly with the rest of the system (e.g. ocean gyres and cyclones in the atmosphere) have a significant impact on the remainder of the system. Locating these features and measuring their properties is a major challenge due to the difficulty of obtaining fine-grained data. My work involves mathematical techniques for using coarse-grained data measurements together with sophisticated post-processing to recover information that would otherwise have to be found using much finer (and therefore much more expensive) measurements. These ideas have proved highly effective in practice, although there is a theoretical gap in that there is no mathematical proof that what is being measured corresponds to physical reality. My research is concerned with reducing the gap between theory and practice. Along the way, I anticipate being able to make further improvements in the data reconstruction techniques. The mathematical underpinnings of this research are of very considerable interest in their own right. One of the key tools involved in the study, Lyapunov exponents, are a major component of chaos theory. Lyapunov exponents measure the rates of stretching in dynamical systems (and underlie such questions as for how long we can expect the weather forecast to be accurate). There has been a large amount of research on this field over decades, spurred by the widespread availability of fast computers, and a question of very considerable mathematical interest is the extent to which these Lyapunov exponents are stable. That is, if a small change is made to a system, is it the case that the Lyapunov exponents also undergo a small change? Surprisingly, the answer to this question in some important cases is "no": a very small change to the system can result in radical changes to the Lyapunov exponents. This is problematic if one is trying to reconstruct the Lyapunov exponents in a real-world dynamical system from data. A major part of my research is understanding scenarios and types of changes to the system that do lead to stability of Lyapunov exponents. Indeed, there seems to be an important connection between Lyapunov exponents and the slowly mixing regions of environmental dynamical systems mentioned above.