The Friedmann--Lemaitre--Robertson--Walker family of spacetimes are the standard homogenous isotropic cosmological models in general relativity.  Each member of this family describes a torus, evolving from a big bang singularity and expanding indefinitely to the future, with expansion rate encoded by a suitable scale factor.  I will discuss a mixing effect which occurs for the Vlasov equation on these spacetimes when the expansion rate is suitably slow.

 This is joint work with Renato Velozo Ruiz (Imperial College London).

Speaker Lucas Theis will talk about: 'What makes an image realistic ?'

The last decade has seen tremendous progress in our ability to generate realistic-looking data, be it images, text, audio, or video. 
In this presentation, we will look at the closely related problem of quantifying realism, that is, designing functions that can reliably tell realistic data from unrealistic data. This problem turns out to be significantly harder to solve and remains poorly understood, despite its prevalence in machine learning and recent breakthroughs in generative AI. Drawing on insights from algorithmic information theory, we discuss why this problem is challenging, why a good generative model alone is insufficient to solve it, and what a good solution would look like. In particular, we introduce the notion of a universal critic, which unlike adversarial critics does not require adversarial training. While universal critics are not immediately practical, they can serve both as a North Star for guiding practical implementations and as a tool for analyzing existing attempts to capture realism.

This talk discusses various applications of the relative entropy method in the context of fluid mechanics, focusing on weak-strong uniqueness results and asymptotic limits. Particular attention is given to Euler-type equations involving nonlocal interactions. Furthermore, I will present recent results regarding a novel approach to pressureless Euler equations.

Another application of the relative entropy method to be discussed is the unconditional stability of certain radially symmetric steady states for compressible viscous fluids in domains with inflow/outflow boundary conditions. Specifically, we demonstrate that any solution to the associated evolutionary problem, not necessarily radially symmetric, converges to a unique radially symmetric steady state.