Hydrogels are soft, highly absorbent porous materials which are commonly used in pharmaceutical applications such as in soft contact lenses, drug delivery and wound healing. They are commonly modelled as hyperelastic materials with an additional chemical force driving the influx of water into the gel. In this talk, I will show how taking the “osmosis-dominated limit” (i.e. the regime where chemical forces dominate over elastic, which is the relevant limit for most commonly used hydrogels) can simplify the PDEs governing hydrogel dynamics. In the linear case, I will show the swelling problem can be entirely decoupled from the solid mechanics problem. In the nonlinear case, I will show the coupling is sufficiently weak as to enable a simplified solution procedure by finite element methods.

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.