ENS Lyon

The first reaction-diffusion equation developed and studied is the Fisher-KPP equation.  Introduced in 1937, it accounts for the spatial spreading and growth of a species.  Understanding this population-dynamics model is equivalent to understanding the distribution of the maximum particle in a branching Brownian motion.  Various generalizations of this model have been studied in the eighty years since its introduction, including a model with non-local reaction for the cane toads of Australia introduced by Benichou et. al.  I will begin the talk by giving an extended introduction on the Fisher-KPP equation and the typical behavior of its solutions.  Afterwards, I will describe the model for the cane toads equations and give new results regarding this model.  In particular, I will show how the model may be viewed as a perturbation of a local equation using a new Harnack-type inequality and I will discuss the super-linear in time propagation of the toads.  The talk is based on a joint work with Bouin and Ryzhik.

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Thanks to the p-adic local Langlands correspondence for GL_2(Q_p), one "knows" all admissible unitary topologically irreducible representations of GL_2(Z_p). In this talk I will focus on some elementary properties of their restriction to GL_2(Z_p): for instance, to what extent does the restriction to GL_2(Z_p) allow one to recover the original representation, when is the restriction of finite length, etc.

A decade later, he realized that a family of independent random matrices invariant in law by conjugation by unitary matrices are asymptotically free. This phenomenon is called asymptotic freeness. It had a deep impact in operator algebra and probability and has been generalized in many directions. A simple particular case of Voiculescu's theorem gives an estimate, for N large, of the spectrum of an N by N Hermitian matrix H_N = A_N + 1/\sqrt N X_N, where A_N is a given deterministic Hermitian matrix and X_N has independent gaussian standard sub-diagonal entries.

Nevertheless, it turns out that asymptotic freeness does not hold in certain situations, e.g. when the entries of X_N as above have heavy-tails. To infer the spectrum of a larger class of matrices, we go further into Voiculescu's approach and introduce the distributions of traffics and their free product. This notion of distribution is richer than Voiculescu's notion of distribution of non commutative random variables and it generalizes the notion of law of a random graph. The notion of freeness for traffics is an intriguing mixing between the classical independence and Voiculescu's notion of freeness. We prove an asymptotic freeness theorem in that context for independent random matrices invariant in law by conjugation by permutation matrices.

The purpose of this talk is to give an introductory presentation of these notions.

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This is joint work with Iain S. Duff.

Parallel sparse direct solvers are an interesting alternative to iterative methods for some classes of large sparse systems of linear equations. In the context of a parallel sparse multifrontal solver (MUMPS), we describe a new dynamic scheduling strategy aiming at balancing both the workload and the memory usage. More precisely, this hybrid approach balances the workload under memory constraints. We show that the peak of memory can be significantly reduced, while we have also improved the performance of the solver.

Then, we present preliminary work concerning a parallel out-of-core extension of the solver MUMPS, enabling to solve increasingly large simulation problems.

This is joint work with P.Amestoy, A.Guermouche, S.Pralet and E.Agullo.