A central challenge in applied mathematics is to extract predictive structure from data generated by complex dynamical systems. Koopman operator methods provide a principled framework for this task by embedding nonlinear dynamics into a linear operator acting on observables, reducing analysis and forecasting to questions about spectral approximation.

In this talk, I will present recent results on the analysis of data-driven Koopman methods, with an emphasis on when spectral quantities can be reliably approximated from finite data. I will describe a general framework that connects operator-theoretic properties of the Koopman operator with the behaviour of practical algorithms, clarifying phenomena such as spectral pollution and the role of continuous spectra. I will also discuss fundamental limitations: there exist classes of dynamical systems for which finite data cannot recover meaningful spectral information, placing intrinsic constraints on what Koopman-based approaches can achieve. Building on this, I will show how spectral approximation errors translate into quantitative bounds for forecasting, capturing how approximation and statistical errors propagate over time and ultimately limit long-term prediction. These results have implications for applications including fluid dynamics, molecular systems, and geophysical flows. I will conclude by highlighting open problems at the intersection of operator theory, numerical analysis, and scientific machine learning.

Insects use flight as far more than a means of getting from A to B. Flight creates an aeiral theatre for interaction, whether between species or among members of the same species. For example, a male dragonfly must hunt for food, fend off rival males, and pursue evasive females in order to reproduce, tasks that all revolve around chasing fast-moving targets. Despite the remarkable diversity of insect species and their aerial behaviours, common patterns emerge in how they exploit speed and manoeuvrability to achieve these goals. Simple geometric guidance laws can describe these flight trajectories with surprising accuracy, revealing shared strategies that underpin insect aerial combat.