Multiplicative Ergodic Theory, Dynamics and Applications

我的研究计划使用遍历理论(研究具有不变度量的动力系统的长期行为)中的想法来验证环境数据测量方案的创建并为其提供信息。在环境动力系统中，与系统其余部分混合较慢的区域(例如大气层中的海洋漩涡和气旋)对系统的其余部分有重大影响。由于获取细粒度数据的困难，定位这些特征并测量它们的属性是一项重大挑战。我的工作涉及使用粗粒度数据测量和复杂的后处理来恢复信息的数学技术，否则必须使用更精细的(因此也更昂贵的)测量来恢复信息。这些想法在实践中被证明是非常有效的，尽管理论上存在一个空白，因为没有数学证据证明所测量的东西符合物理现实。我的研究关注的是缩小理论与实践之间的差距。在此过程中，我预计能够在数据重建技术方面做出进一步的改进。 这项研究的数学基础本身是非常有意义的。李雅普诺夫指数是这项研究的关键工具之一，它是混沌理论的主要组成部分。李亚普诺夫指数衡量的是动力系统中的伸展速度(这也是我们可以预期天气预报准确多久等问题的基础)。在快速计算机的广泛使用的推动下，几十年来在这个领域已经有了大量的研究，而一个非常令人感兴趣的数学问题是这些李亚普诺夫指数的稳定程度。也就是说，如果对一个系统做了一个小的改变，那么李雅普诺夫指数是否也经历了一个小的改变？令人惊讶的是，在一些重要的情况下，这个问题的答案是否定的：系统的一个很小的变化就可能导致李亚普诺夫指数的根本变化。如果有人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了解导致Lyapunov指数稳定的系统变化的场景和类型。的确，李亚普诺夫指数与上述环境动力系统的缓慢混合区之间似乎存在着重要的联系。

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.