At this colloquium, we are happy to welcome:

Peter Morfe (Max-Planck-Institut Leipzig)

Anomalous Diffusion in a Critical Drift-Diffusion Equation: Convection Enhancement Revisited

I will discuss the motion of a passive tracer in a random incompressible fluid flow, a fundamental but simple model from fluid mechanics.  The talk will begin by recalling two classical examples, which show that incompressible drifts always increase ("enhance") the diffusivity, and can possibly lead to anomalous superdiffusive behavior at large spatiotemporal scales.  I will then proceed to describe recent work on a paradigmatic 2D example, where the superdiffusive scaling can be identified through a scale-by-scale renormalization-style argument built using concepts from homogenization theory.  Joint work with G. Chatzigeorgiou, F. Otto, C. Wagner, and L. Wang.

Feliks Nüske (Max-Planck-Institut Magdeburg)

Modeling Molecular Systems with Koopman Operators and Random Features

Koopman operator theory has emerged as a powerful modelling approach for complex dynamical systems arising in physics, chemistry, materials science, and engineering. The basic idea is to leverage existing simulation data to learn a linear model that allows to predict expectation values of observable functions at future times. Though the algorithm is conceptually quite simple, its underlying mathematical structure (the Koopman operator semigroup) is very rich, and can be used for different purposes including control, coarse graining, or the identification of metastable states in complex molecules and materials.
A critical modeling decision in this context is the choice of a finite-dimensional basis set (called dictionary). Kernel methods, which are well-known in other application areas of machine learning, have recently been shown to provide a powerful model class for Koopman learning, requiring only little prior information. The price to pay is that the dictionary size scales with the data size, leading to large-scale linear algebra problems that can become challenging to solve in practice. In this talk, we demonstrate that stochastic low-rank approximations based on Random Fourier features lead to reduced linear algebra problems that can be solved at much lower cost. We also show that hyper-parameters of the kernel can be tuned efficiently based on physical principles, allowing for an effective identification of metastable states in molecular systems.

Maite Wilke Berenguer (Humboldt-Universität Berlin)

Weak and rare selection

In population genetics, biological phenomena are best encoded in individual-based models. To facilitate the analysis of these models, one then considers large-population size limits, with time rescaled appropriately. I will present a model for (weak and) rare selection which is exemplary for the Wright-Fisher set-up: We begin with a random graph describing the evolution of a population both forward and backwards in time, with selection being driven by a random environment. As is classic, we trace  two processes on this graph: forward in time, we consider a bi-allelic population and trace the frequency of the weakest allele, backwards in time, we start with a sample and trace its genealogy.  The random graph  gives us a sampling duality relation between these two processes. This carries through to their respective  scaling limits: a diffusion with rare selection causing jumps (forward in time) and a branching-coalescing  process where branching is not independent (backward in time) which are moment-duals of each other.
Using the diffusion we give a criterion to characterise the strength of selection vs. the strength of genetic drift.