## Séminaire Janine GUESPIN

Jeudi 10 Novembre 2016 (14h30, Salle G001)

Janine Guespin, Université de Rouen.

Titre : Le complexe, une révolution scientifique qui re-forme la pensée.

Résumé :  Les sciences des systèmes complexes rassemblent des scientifiques, (de toutes les disciplines) qui travaillent sur des systèmes dont la compréhension nécessite la modélisation mathématique ou des simulations informatiques. D'autres chercheurs travaillent sur des systèmes qu'ils disent complexes sans utiliser ces outils (cf par exemple, Edgar Morin ou un certain nombre de ceux qui se réclament de la systémique). Je fais l'hypothèse que ce qui donne sa cohérence à l’ensemble de ces démarches - ce qui en fait une révolution scientifique - c'est une profonde transformation de la forme (méthode) de pensée qui émerge de cette révolution, que j'appelle la pensée du complexe.

## Séminaire Danielle Hilhorst

Jeudi 3 novembre 2016 (14h30, Salle G001)

Danielle Hilhorst, CNRS, Université Paris-Sud.

Titre : Generation of interface for solutions of the mass conserved Allen-Cahn equation.

Résumé :  We consider the mass conserved Allen-Cahn equation, which has been proposed by Rubinstein and Sternberg to model the phase separation in a binary mixture. It does not possess any comparison theorem, which makes its study very difficult. We prove a generation of interface property in the case that the coefficient of the nonlocal reaction term tends to infinity.

This is joint work with Hiroshi Matano, Thanh Nam Nguyen and Hendrik Weber.

Slides (pdf).

Jeudi 17 Mars 2016 (15h00, Salle G001)

Titre : Sur les m-uples Diophantiens et les partitions d'un entier.

Résumé :  The Greek mathematician Diophantus of Alexandria first studied the problem of finding four numbers such that the product of any two of them increased by unity is a perfect square. He found {1/16; 33/16; 17/4; 105/16} , a set of four positive rationals verifying this property. However, the first set of four positive integers with the above property, {1; 3; 8; 120} , was found by Fermat. Euler found the infinite family of such sets, {a; b; a + b + 2r; 4r(r + a)(r +b)g} , where ab + 1 = r^2 . He was also able to add the fifth positive rational ,777480/8288641 , to the Fermat's set. In January 1999, the first example of a set of six positive rationals with the property of Diophantus was found by Gibbs {11/192; 35/192; 155/27; 512/27; 1235/48; 180873/16} . These examples motivate the following definitions:

Definition 1 A set of m positive integers {a1; a2; ... ; am} is called a Diophantine m-tuple if aiaj + 1 is a perfect square for all 1  i \leq  j \leq  m.

It is natural to ask how large these sets, can be?

This question will be discussed and some results linked to integer

partitions will be presented.

## Séminaire Grégory Dumont

Jeudi 17 Mars 2016 (14h00, Salle G001)

Grégory Dumont, LNC, Group for Neural Theory, ENS (Ulm).

Titre : An age structural model for neural networks.

Résumé : Neurons are strongly noisy, and stochastic models are almost always required when a system is driven by random events. In accordance with the origin of variability, the sources of noise are classified as intrinsic or extrinsic, and give rise to distinct mathematical frameworks. While the external variability is generally treated by the use of a Wiener process, the internal variability is mostly expressed via random firing events and a non-homogenous Poisson process. Those distinct stochastic processes are completely determined by their probability density function obtained via partial differential equations such as the Fokker-Plank equation and the von-Foerster-McKendrick system. In the first part, we investigate in what way those partial differential equations are related and how their respective solutions can be mapped one to another via integral transforms. In the second part, we investigate the accommodation of finite size fluctuations into the model when the synaptic coupling is taken into account. Thanks to the tau leaping formula, a trick first popularized by Gillespie, we approximate the number of firing events during a time increment as a Poisson random variable. From there, we are able to derive a corrected field equation that encompasses the presence of fluctuations proportional to the mean number of firing events, and therefore fully retains the randomness character of the spike initiation. Our new description is such that it reduces, in the thermodynamic limit, to the classical deterministic mean field equation known as the refractory density equation or von-Foester-McKendrick system. With such a tool in hand, we are capable of computing several statistical information regarding the network’s firing activity in the asynchronous regime.

## Séminaire François Lemaire

Jeudi 10 Mars 2016 (14h00, Salle G001)

François Lemaire, CRISTAL, Université de Lille.

Titre : Application of differential algebra to the quasi-steady state approximation in Biology and Physics.

Résumé : The quasi-steady state approximation (QSSA) is a technics for approximating the evolution of a dynamical system which involves both slow and fast dynamics. It can be used when the fast dynamics tend to an equilibria which slowly drifts due to action of the slow dynamics.