Institut de Mathématiques de Toulouse

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Rencontre LPT-IMT

par Delphine Dallariva - publié le

Programme de la rencontre LPT-IMT
du 10 juin 2016 dans le bâtiment 3R1 au 3ème étage.

9h45-10h00 : ACCUEIL CAFE

10h00-11h00 : MANON COSTA

"Modélisation probabiliste de la co-évolution dans les communautés proies-prédateurs"

Résumé : Dans cet exposé, je présenterai différents modèles permettant d’étudier la co-évolution des phénotypes de proies et prédateurs. Dans un premier modèle, je m’intéresserai à des hypothèses liées à la théorie des dynamiques adaptatives : grande population, mutations rares et petite amplitude. Je présenterai des exemples de phénomènes coévolutifs pouvant être observés sous ces hypothèses. Ensuite, je présenterai un modèle permettant de décrire la co-évolution de communautés asymétriques dans lesquelles les populations de prédateurs évoluent sur un échelle de temps plus rapide que les proies (exemple : les communautés arbres-insectes).

11h00-11h15 : PAUSE CAFE

11h15-12h15 : SYLVAIN PROHLAC

"Finite-time fluctuations for TASEP on the relaxation scale"

The totally asymmetric simple exclusion process (TASEP) is an integrable Markov process describing N particles hopping forward on a one-dimensional lattice of L sites. The periodic model evolving during a time t has been studied recently (PRL 116 090601) on the relaxation time scale when L, N, t go to infinity with finite density rho=N/L, and finite rescaled time tau=t/L^3/2 characteristic of KPZ universality. Exact expressions have been obtained for the average density profile, for the stationary two-point function, and for the probability density of current fluctuations for simple initial conditions. At small tau, the distribution of current fluctuations converges to Tracy-Widom distributions. At large tau, one recovers the stationary large deviation function of the current. The finite tau formulas have a nice interpretation as a functional integral on a field conjugate to the current.

12h15-13H45 : REPAS

13h45-14h45 : STEFAN LE COZ

"On a singularly perturbed Gross-Pitaevskii equation."

We consider the 1D Gross-Pitaevskii equation perturbed by a Dirac potential.
Using a fine analysis of the properties of the linear propagator, we study the well-posedness of the Cauchy Problem in the energy space of functions with modulus 1 at infinity. Then we study existence and stability of standing waves with a combination of variational and perturbation arguments.

14h45-15h45 : CLEMENT SIRE

"Interacting random walkers and fish schools"

I’ll present a model of interacting (with each other and with the wall of the enclosing "tank") random walkers which faithfully reproduces the collective motion of fish in a tank (experiments in G. Theraulaz group at CRCA Toulouse). In addition, in free space (no tank) and for a large number of fish, this model reproduces several collective phases observed in nature : swarming, schooling, milling…
An effective model for one single fish swimming in the mean effective field of other fish will be also presented, a well posed (but certainly non trivial) problem mathematically which could motivate mathematicians !

15h45-16h00 : PAUSE CAFE

16h00-17h00 : REDA CHHAIBI

"The connection between last passage percolation/directed polymers, random matrices and some representation theory"

"In this informal talk, I will present the very close models of last passage percolation and directed polymers, as well as conjectures on the subject. Then I will explain how the relevant quantities in these models are related to random matrices on the one hand and stochastic processes built out of representation theory on the other hand"