L3

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.

Core-annular flows are often proposed to reduce frictional losses in industrial pipeline transport processes. Traditionally, a low-viscosity lubricating film is placed around a more viscous core to reduce the drag on the core. However, maintaining stable pipelining, where the core and the lubricant remain separated has proved challenging.
In this talk we present an alternative approach using three-layer, horizontal core-annular pipe flow, in which two fluids are separated by a deformable elastic solid. In the experiments, an elastic solid created by an in-situ chemical reaction maintains the separation of the core and annular fluids. Corrugations of the elastic interface are observed and stable pipelining, where the elastic shell created separating the two fluids remains intact, is successfully demonstrated even when the core fluid is buoyant. We also develop a theoretical model combining lubrication theory for the fluids with standard shell theory for the elastic solid, to predict the buckling states resulting from radial compression of the shell.
The self-sculpting of the shell by buckling cannot by itself generate hydrodynamic lift owing to symmetry in the direction of flow. Instead, we demonstrate that hydrodynamic lift can be achieved by other elastohydrodynamic effects, when that symmetry becomes broken during the bending of the shell.

The talk will be about a natural percolation model built from the so-called Brownian loop soup. We will give sense to studying its phase transition in dimension d = 2 + epsilon, with epsilon varying in [0,1], and discuss how to perform a rigorous „epsilon-expansion“ in this context. Our methods give access to a whole family of universality classes, and elucidate the behaviour of critical exponents etc. near the (lower-)critical dimension, which for this model is d=2. 

Based on joint work with Wen Zhang.

In the first part of the talk, I will present an overview of recent advances in the description of diffusion-reaction processes and their first-passage statistics, with the special emphasis on the role of the boundary local time and related spectral tools. The second part of the talk will illustrate the use of these tools for the analysis of boundary-catalytic branching processes. These processes describe a broad class of natural phenomena where the population of diffusing particles grows due to their spontaneous binary branching (e.g., division, fission, or splitting) on a catalytic boundary located in a complex environment. We investigate the possibility of the geometric control of the population growth by compensating the proliferation of particles due to catalytic branching events by their absorptions in the bulk or on boundary absorbing regions. We identify an appropriate Steklov spectral problem to obtain the phase diagram of this out-of-equilibrium stochastic process. The principal eigenvalue determines the critical line that separates an exponential growth of the population from its extinction. In other words, we establish a powerful tool for calculating the optimal absorption rate that equilibrates the opposite effects of branching and absorption events and thus results in steady-state behavior of this diffusion-reaction system. Moreover, we show the existence of a critical catalytic rate above which no compensation is possible, so that the population cannot be controlled and keeps growing exponentially. The proposed framework opens promising perspectives for better understanding, modeling, and control of various boundary-catalytic branching processes, with applications in physics, chemistry, and life sciences.

I will introduce the Polchinski dynamics, a general framework to study asymptotic properties of statistical mechanics and field theory models inspired by renormalisation group ideas. The Polchinski dynamics has appeared recently under different names, such as stochastic localisation, and in very different contexts (Markov chain mixing, optimal transport, functional inequalities...) Here I will motivate its construction from a physics point of view and mention a few applications. In particular, I will explain how the Polchinski dynamics can be used to generalise Bakry and Emery’s Γ2 calculus to obtain functional inequalities (e.g. Poincaré, log-Sobolev) in physics models which are typically high-dimensional and non-convex.