MARK KAC SEMINAR

There are several systems of interest in physics and biology which are related to free boundary problems in PDE's. In particular I will refer to Brunet–Derrida models for biological selection mechanisms and to the Fourier law for particles systems in domains changing in time due to the action of reservoirs fixing the current (and not the densities as in the traditional setting). In both cases the hydrodynamic (or continuum) limit is described by parabolic equations with free boundaries. In this talk I will give a short survey on the methods used to study such particle systems and the free boundary problems for the corresponding PDE's.

We consider a system of interacting Wright–Fisher diffusions with seed-bank. Individuals are of two types, live in colonies and are subject to resampling and migration as long as they are active. Each colony has a seed-bank into which individuals can retreat to become dormant, suspending their resampling and migration until they become active again. As geographic space we consider ${\mathsf{\mathbb{Z}}}^{\mathsf{d}}$. Our goal is to identify the change in behaviour induced by the seed-bank. In particular we want to establish the dichotomy between clustering, a mono-type equilibrium, and coexistence, a multi-type equilibrium. To analyze the dichotomy we first show that the seed-bank model has a dual. This dual allows us to establish the dichotomy via a random walk argument. It turns out that the seed-bank affects the dichotomy if the seed-bank is large and the wake up times of individuals have a fat tailed distribution. Joint work with Andreas Greven and Frank den Hollander.

Subgraphs contain crucial information about network structure and function. For inhomogeneous random graphs with infinite-variance power-law degrees, we count the number of times a small connected graph occurs as an induced subgraph (graphlet counting). We introduce an optimization problem to identify the dominant structure of any given subgraph. The unique optimizer describes the degrees of the vertices that together span the most likely subgraph. We find that every subgraph occurs typically between vertices with specific degree ranges. In this way, we can count and characterize all subgraphs.