Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - Université de Rouen Normandie

The emergence of mean-curvature flow of an interface between different phases or populations is a phenomenon of long standing interest in statistical physics.  
In this talk, we review recent progress with respect to a class of reaction-diffusion stochastic particle systems on an $n$-dimensional lattice.
In such a process, particles can move across sites as well as be created/annihilated according to diffusion and reaction rates.  
These rates will be chosen so that there are two preferred particle mass density levels $a_1$, $a_2$ in `balance'.

Building on the presentation given at the colloquium on September 11, we go into some of the details of the homogenizations needed to obtain the sharp interface limit and associated fluctuations.

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Peer-to-peer (P2P) insurance promises lower costs and better alignment of interests by having small groups of policyholders directly share risk. Yet full pooling can destroy prevention incentives under moral hazard, while self-insurance sacrifices diversification. In this talk, we develop a unified mean–variance framework that endogenizes both the pooling matrix and the effort levels in one joint optimization.

Zajíček's theorem characterizes the sets of non-differentiability of convex functions.
Its statement is sharp, elegant and tremendously powerful.
We provide the proof in a simple case and give some applications.

Based on the mechanical viewpoint that living tissues present a fluid-like behaviour, Partial Differential Equations models inspired by fluid dynamics are nowadays well established as one of the main mathematical tools for the macroscopic description of tissue growth. Depending on the type of tissue, these models link the pressure to the velocity field using either Brinkman’s law (visco-elastic models) or Darcy’s law (porous-medium equations (PME)).