Multiplicative Ergodic Theory, Dynamics and Applications

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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.