Abstract: The dynamics of physical, chemical, and biological systems in a broad range of applications has the property of exhibiting coherent behavior embedded within chaotic dynamics. A classical example is the Earth’s climate system, which exhibits coherent oscillations such as the El Nino Southern Oscillation or the Madden-Julian Oscillation despite an extremely large number of active degrees of freedom and chaotic behavior on short (“weather”) timescales. Identifying these patterns from observational data or model output is useful from a UQ standpoint, for instance for assessing limits of predictability, or for providing predictor variables capturing the conditional statistics of quantities of interest. In this talk, we describe how operator-theoretic techniques from ergodic theory, combined with methods from data science, can identify observables of complex systems with two main features: slow correlation decay and cyclicity. These observables are approximate eigenfunctions of Koopman evolution operators, estimated from high-dimensional time series data using kernel methods. We discuss mathematical and computational aspects of these approaches, and illustrate them with applications to idealized systems and real-world examples from climate science.