**ABSTRACTS**

**Ariane Trescases**

**Cross-diffusion and competitive interaction in Population dynamics**In Population dynamics, reaction-cross diffusion systems model the evolution of the populations of competing species with a repulsive effect between individuals. For these strongly coupled, often nonlinear systems, a question as basic as the existence of solutions appears to be extremely complex.

We introduce an approach based on the most recent extensions of duality lemmas and on entropy methods. We prove the existence of weak solutions in a general setting of reaction-cross diffusion systems, as well as some qualitative properties of the solutions.

This is a joint work with L. Desvillettes, Th. Lepoutre and A. Moussa.

**Benjamin Aymard**

**Emergence of vascularisation networks**Networks are patterns created dynamically in order to optimize the distribution or the exchange of quantities in an environment.

They appear naturally in nature, for instance in Human Biology (capillaries, nerves, neurons), in Vegetal Biology (tree branches, leaf skeleton, roots), in Entomology (ant trails, insect nests) or in Geology (erosion patterns, rivers, salt ponds).

Despite the fact that these structures are sometimes separated by many physical scales, and rely on very different physics, these structures are very similar and probably share mathematical properties.

The mechanisms underlying the formation of such networks are not entirely known, and we think that self organization could play a major role.

In this talk, we are interested in the formation of the capillary networks.

Capillaries are thin vessels connecting arteries (which bring oxygenated blood into the tissues) to veins (which take back "consumed" blood to the heart). They appear as a response to a need of oxygen within the tissue, process called hypoxia.

Our aim is to understand the mechanisms underlying the formation of such networks.

In order to do that, we define heuristic rules, as simple as possible, and simulate numerically the resulting model.

Finally, we confront these results to actual biological results.

After introducing the biological background, we will present in a comprehensive way the model we have designed.

This is a hybrid continuum/agent-based model, taking into account four actors, namely the blood flow, the oxygen flow, the capillary network and the tissue.

Afterwards we present the numerical method we have designed to approximate the solutions of this system and finally show numerical simulations based on this model

**Claude Bardos**

**About Boltzmann-Maxwell relation and Multiscale Analysis**This is a report on a paper in preparation with F. Golse, R. Sentis and T. Nguyen. In the description of a plasma involving ions and electrons the following formula:

is used to describe the density of electrons. Therefore the present talk is an attempt to justify this relation. It involves an analysis of the different scalings which govern the evolution of the couple system, the proofs of some ” if theorems ” , some rigorous results on the reduced system and some considerations on Arnold type stability of the full system.

**Norbert J Mauser (WPI Univ. Wien)**

**Quantum kinetic equations and the semiclassical equations of solid state physics**We present some aspects of quantum kinetic theory, one of the many fields where P. Degond has significantly contributed. Phase-space formulations of quantum mechanics, that are a reformulation of the density matrix formalism, are widely used e.g. in quantum optics. The "Wigner transform", an extension of the Fourier transform, is the appropriate mathematical tool that has a tremendous impact in homogenization of PDEs, where the "Wigner measures" (a name coined by P.-L. Lions and T. Paul) go beyond the "H-measures" (of L. Tartar). A special variant for the homogenization of periodic problems (in the sense of the pioneering work of A. Bensoussan, J.-L. Lions and G. Papanicolaou) , are the "Wigner series" (a name coined by the late F. Poupaud and the speaker) where the „kinetic variable“ lives in the torus (Brioullin zone). By an appropriate combined „homogenization+classical limit“ one obtains the "semiclassical equations" of solid state physics as an important kinetic equation.

We give a classical pedagocial blackboard lecture explaining this fascinating field from scratch to anyone who has a solid knowledge of PDEs and functional analysis.

**Amic Frouvelle**

**Alignment processes on the circle****Giacomo Dimarco**

*Are tumor cell lineages solely shaped by mechanical forces?*In this talk, we investigate cells proliferation dynamics using an individual agent based model (IBM). In particular, the model is designed to examine the organization of the cell population, the impact of the orientation of the division plan and the cell lineages formation during growth. Our IBM model is based on the hypothesis that cells are growing incompressible objects that divide once a threshold size is reached and that newly born cell progressively form cluster. We perform comparisons of the mathematical model and of the corresponding numerical simulations with the experimental evidences by using several statistical indicators. The results obtained through this comparison suggest that the emergence of particular families structures as well as the influence of the position of the cell on the division process in a proliferating cell population can be explained by simple mechanical interactions.

**Bani Anvari**

*Innovative design of urban transport environments/spaces for mixed-mode use***Rene Carmona**

*Probabilistic mean-field games*We review the paradigm of Mean Field Games introduced by Lasry and Lions, and independently by Caines Huang and Malhame under the name of Nash Certainty Equivalent. We introduce two variants of a probabilistic approach based on the solution of Backward Stochastic Differential Equations. For each of them, we give (without complete proofs) sufficient conditions for the existence of solutions, and we provide examples of applications. These applications are chosen from problems already discussed in the existing literature, but for which no mathematical solution was given. Among other examples, we shall discuss macro-economic growth models (Aiyagari & Krusell-Smith), flocking from generalized Cucker-Smale type models, and exit from a room in the presence of congestion.

**José Antonio Carrillo de la Plata**

*Swarming models with attractive-repulsive effects: Sharp Sensitivity Regions & Phase Transition*I will review results about flocking and milling solutions of attractive-repulsive models for collective behavior. Special attention will be paid to long time stability of solutions for discrete models, rigorous derivation of kinetic models and phase transitions driven by noise.

**Li Wang**

*The uniform convergence of generalized polynomial chaos (gPC)-based numerical methods for transport equation with random input*We consider gPC-based numerical methods for the linear transport equation with random scattering and initial data. In the diffusion regime, these methods suffer from slow convergence due to the small scales. We show that, as long as the numerical scheme is asymptotic preserving for every choice of the random variable, the numerical scheme enjoys a uniform convergence independent of the small parameters. For the stochastic collocation method, the proof relies on the regularity of the solution, whereas for the stochastic Galerkin method, we also need to take care of the position of the scattering term. This is a joint work with Shi Jin and Qin Li.

**Sara Merino Aceituno**

*A new flocking model through body attitude coordination*We present a new model for multi-agent dynamics where each agent is described by its position and body attitude: agents travel at a constant speed in a given direction and their body can rotate around it adopting different configurations. Agents try to coordinate their body attitudes with the ones of their neighbours. This model is inspired by the Vicsek model.

The goal of this talk will be to present this new flocking model, its relevance and the derivation of the macroscopic equations from the particle dynamics.

In collaboration with Pierre Degond (Imperial College London) and Amic Frouvelle (Université Paris Dauphine).

**Pierre Degond**

*Mathematics of the Mob*(Inaugural lecture)Abstract and details of the inaugural lecture here.