Schedule: 

Abstracts: 

Samuel Punshon-Smith (Tulane University)

Lower Bounds on Lyapunov Exponents for SPDE

I will present recent work with M. Hairer, T. Rosati, and J. Yi on the maximum rate of mixing in randomly stirred fluids. By analyzing the top Lyapunov exponent for the advection-diffusion and linearized Navier-Stokes equations, we prove that the decay rate cannot be infinitely fast. Our main result establishes a quantitative lower bound on this rate that depends on a negative power of the diffusion parameter κ. This finding provides the first rigorous lower bound on the Batchelor scale and partially resolves a conjecture by Doering and Miles. To do this, we introduce the concept of “high-frequency stochastic instability,” a mechanism where random stirring prevents energy from getting trapped in fine-scale patterns where it would dissipate too quickly.

Vincent Goverse (Imperial College London)

Transition to chaos for the random dissipative standard map

We establish the existence of intermittent two-point dynamics and infinite stationary measures for the random dissipative standard map with zero Lyapunov exponent.

Hugo Chu (CMAP, École Polytechnique)

Rigorous enclosure of Lyapunov exponents of stochastic flows

I present a method for obtaining rigorous and accurate upper and lower bounds for Lyapunov exponents of stochastic flows, using computer-assisted tools, the adjoint method, and ergodicity of diffusion processes. The method requires only mild assumptions and applies beyond perturbative regimes, enabling treatment of systems previously inaccessible. We illustrate this by establishing positivity of the top Lyapunov exponent in several examples. Joint work with Maxime Breden, Jeroen Lamb, and Martin Rasmussen.

Stefanie Winkelmann (Zuse Institut Berlin)

Data-driven Galerkin approximation of stochastic clustering dynamics

I present a data-driven framework for analyzing stochastic clustering dynamics using transfer-operator theory. The approach combines Galerkin projection with diffusion-map embeddings to identify low-dimensional collective variables and estimate coarse-grained Markov models capturing slow dynamics.

Rishabh Gvalani (ETH Zürich)

Separation of time scales in interacting diffusions

Anna Shalova (University of Amsterdam)

Nonlinear diffusion limit of non-local interactions on a sphere

We study an aggregation PDE with competing attractive and repulsive forces on a high-dimensional sphere. In particular, we consider the limit of localized repulsion with a constant attraction term. We prove convergence of solutions of such a system to solutions of the aggregation-diffusion equation with porous-medium-type diffusion. The proof combines variational techniques with harmonic analysis on a sphere, including a characterization of the square root of the convolution operator in terms of spherical harmonics. The system naturally extends a toy model of transformers (Geshkovski et al., 2024) and provides insight into mechanisms behind modern language models. Joint work with Mark Peletier.

Michael Scheutzow (TU Berlin)

Attractors for monotone random dynamical systems

We study monotone Markov processes and monotone random dynamical systems (RDS) taking values in a partially ordered Polish space EEE. We provide sufficient conditions for the existence of a random weak set or weak point attractor as well as weak or strong synchronization of the RDS in terms of ergodic properties of the associated monotone Markov process, and provide sufficient conditions for these ergodic properties to hold.

Ale Jan Homburg (University of Amsterdam & Leiden University)

Metric attractors with intermingled basins

I construct novel and elementary examples of dynamics with metric attractors possessing intermingled basins. A key ingredient is the introduction of random walks along orbits of a given dynamical system. Joint work with Abbas Fakhari.

Dennis Chemnitz (ETH Zürich)

Stochastic Flows for SDEs with Strong Shear

I discuss ongoing joint work with Maximilian Engel and Michael Scheutzow showing that sufficiently fast-growing shear terms can cause SDEs to lose global stochastic flows (“shear-induced blow-up”). I also introduce a new condition for SDEs with additive noise guaranteeing that the time-one map satisfies stable/unstable manifold theorems. The condition is easy to verify and applies even to systems exhibiting shear-induced blow-up, generalizing results from Synchronization by Noise (Flandoli–Gess–Scheutzow, 2017).

Tobias Hurth (Swiss Academy of Medical Sciences)

Entropy for Conditioned Random Dynamical Systems

We study topological and metric entropy for linear expansions on the circle with additive noise, conditioned to avoid a certain segment. We highlight an ergodic invariant measure for conditioned random dynamical systems (Castro, Chemnitz, Chu, Engel, Lamb & Rasmussen, 2022). Based on joint work with Maximilian Engel.

Holger Spellmann (Ulm University)

Preserving Detailed Balance for Multiscale Chemical Reaction Networks

We investigate how time-scaling structures influence the preservation of detailed balance under limiting procedures in chemical reaction networks with general kinetics.

Enrique Carro Garrido (University of Amsterdam)

On the role of heterogeneities in reaction-diffusion equations

Introducing spatial heterogeneity into reaction–diffusion equations raises intriguing mathematical questions. In the Nagumo equation, such heterogeneities:

1. alter travelling wave speeds, making them spatially dependent;

2. can cause localized pinning of waves;

3. break translational symmetry, eliminating the usual zero eigenvalue of the linearised operator.

A numerical simulation of the Barkley model illustrates how a localized heterogeneity can trigger spiral wave breakup through pinning. These findings relate to spiral wave dynamics in electrocardiograms and cardiac arrhythmias.

Giuseppe Tenaglia (Imperial College London)

Density of random horseshoes for non-uniformly expanding random dynamical systems with bounded diffusive noise

We consider random dynamical systems with bounded diffusive noise, for which all Lyapunov exponents are positive. For such a class, we prove density of random horseshoes and existence of random Young towers with annealed exponential tail.

Iacopo P. Longo (Imperial College London)

Nonautonomous Differential Equations in the Presence of Bounded Noise

Nonautonomous systems with bounded noise are common in applied contexts, especially when parameters vary in time and may induce tipping. Their dynamics can be represented via deterministic set-valued dynamical systems, but analysis is difficult because they act on the space of compact sets. We introduce a nonautonomous generalisation of the boundary map, enabling the study of nonautonomous invariant sets by tracking boundaries and producing a deterministic single-valued dynamical system in higher dimension.

Guillermo Olicón-Méndez (Universität Potsdam)

Tails of stationary densities in random maps: an early-warning signal for bifurcations

We consider a large class of i.i.d. one-dimensional random difference equations, where the noisy perturbations are sampled from an absolutely continuous distribution supported on [-1,1]. When the map is smooth, and each partial derivative is positive, the right boundary of the support of any stationary distribution, $x_+$, is an equilibrium of the extremal map $h_+(x):=h(x,1)$. Although it has been shown that any stationary density is flat at $x_+$, we present asymptotic rescalings which describe more precisely its shape around $x_+$. In particular, we show that the rescaling expression depends on the Lyapunov exponent of $x_+$ (in the case it is hyperbolic), or on its first nonvanishing term in its Taylor expansion (when it is nonhyperbolic). For the hyperbolic setting, we introduce a numerical scheme which allows estimating the Lyapunov exponent of $x_+$ from a time series $(x_n)_{n=1}^N$, and thus predicting how close the system is to a bifurcation. This is joint work with Martin Rasmussen, Jeroen S.W. Lamb, and Wei Hao Tei (Imperial College London).

Getting around: 

From the main train station to the university: You can take the M85 bus (in the direction of Lichterfelde Süd) to Kurfurstenstr., then transfer to the U3 (in the direction of Krumme Lanke) to Dahlem-Dorf. From there you can walk to Arnimallee 6.
Alternatively, you can take the S7 (in the direction of Potsdam Hbf) to Berlin Zoologischer Garten, then transfer to the U9 (in the direction of Rathaus Steglitz) to Spichernstr., then transfer to the U3 (in the direction of Krumme Lanke) to Dahlem-Dorf. 

From the Südkreuz train station to the university: Take the S41 (from gate 11, upstairs) to Heidelbergerplatz, then the U3 (in the direction of Krumme Lanke) to Dahlem Dorf. 

From the airport to the university: Take the S9 (in the direction of Berlin-Spandau) to Warschauer Strasse, then transfer to the U3 (in the direction of Krumme Lanke) to Dahlem Dorf. 

From the hotel to the university: Take the X83 (in the direction Königin-Luise Straße/Clayallee or U Dahlem Dorf) to Arnimallee.

Tickets can be bought at automated systems in most stations, or on the "Deutsche Bahn" app, which we also recommend for planning your trips around the city!