## 7th session, October 27th & 28th 2014

__Monday 27th__

*EHESS, 105 bd Raspail, 75006 Paris, Room 1*

__This is a CAMS-ReaDi joint session__

15h - 17h : Andrea Bertozzi

*(P*

*rofessor of Mathematics, Betsy Wood Knapp Chair for Innovation and Creativity, Director of Applied Mathematics, University of California Los Angeles)*

*Mathematics of Crime*There is an extensive applied mathematics literature developed for problems in the biological and physical sciences. Our understanding of social science problems from a mathematical standpoint is less developed, but also presents some very interesting problems, especially for young researchers. This lecture uses crime as a case study for using applied mathematical techniques in a social science application and covers a variety of mathematical methods that are applicable to such problems. We will review recent work on agent based models, methods in linear and nonlinear partial differential equations, variational methods for inverse problems and statistical point process models. From an application standpoint we will look at problems in residential burglaries and gang crimes. Examples will consider both “bottom up” and “top down” approaches to understanding the mathematics of crime, and how the two approaches could converge to a unifying.

__Tuesday 28th__

*ReaDi-EHESS, 190-198 av. de France, 75013 Paris, Room 466*

**

9h30 : Coffee

10h - 11h : Juliette Bouhours

*(EHESS, soon Edmonton)*

**A variational approach to reaction diffusion equation with forced speed in dimension 1: application to population dynamics in the presence of climate change**In this presentation, we consider a scalar reaction diffusion equation with a non linear reaction term depending on x-ct. Here c is a prescribed parameter modelling the speed of climate change and we wonder whether the population will survive or not to a shift of its favourable environment du to climate change. This problem has been solved recently when the nonlinearity is of KPP type. We consider in the presentation, general reaction terms, that are only assumed to be negative at infinity.

We will start this presentation by explaining the model. Using a variational approach we construct two thresholds for c determining the existence or non existence of travelling wave solutions. We then study the long time behaviour of the solution of the initial-value problem and prove that it converges either to 0 or to a travelling wave. We also discuss the behaviour of the solution depending on the linear stability of 0. Lastly we illustrate our result and discuss several open questions through numerics.

11h15 - 12h15 : Laurent Dietrich (

*Toulouse*)

*Velocity enhancement of reaction-diffusion fronts by a line of fast diffusion*In 2012, Berestycki, Roquejoffre and Rossi highligthed a mechanism of propagation enhancement in Fisher-KPP equations through a line of fast diffusion. Part of their proof uses explicit computations, specific to the KPP case, which raises the question of the robustness of this phenomenon. In this talk, I will show how to generalise this result to a reaction term with a threshold, by looking at the velocity of the travelling fronts. We will see that this velocity grows like the square root of the diffusivity on the line, with a ratio that can be characterized as the unique admissible velocity for an hypoelliptic limiting system. We will also discuss some extensions about the dynamics of these fronts.