Wed, 17 Jul 2013

10:15 - 11:15
OCCAM Common Room (RI2.28)

Dispersion of particles dropped on a liquid

Benoit Darrasse
(Ecole Polytechnique)
Abstract

The good use of condiments is one of the secrets of a tasty quiche. If you want to delight your guests, add a pinch of ground pepper or cinnamon to the yellow liquid formed by the mix of the eggs and the crème fraiche. Here, is a surprise : even if the liquid is at rest, the pinch of milled pepper spreads by itself at the surface of the mixture. It expands in a circular way, and within a few seconds, it covers an area equal to several times its initial one. Why does it spread like that ? What factors influence this dispersion ? I will present some experiments and mathematical models of this process.

Thu, 06 Jun 2013

14:00 - 15:00
Gibson 1st Floor SR

Hamiltonian propagation of monokinetic measures with rough momentum profiles (work in collaboration with Peter Markowich and Thierry Paul)

François Golse
(Ecole Polytechnique)
Abstract

Consider in the phase space of classical mechanics a Radon measure that is a probability density carried by the graph of a Lipschitz continuous (or even less regular) vector field. We study the structure of the push-forward of such a measure by a Hamiltonian flow. In particular, we provide an estimate on the number of folds in the support of the transported measure that is the image of the initial graph by the flow. We also study in detail the type of singularities in the projection of the transported measure in configuration space (averaging out the momentum variable). We study the conditions under which this projected measure can have atoms, and give an example in which the projected measure is singular with respect to the Lebesgue measure and diffuse. We discuss applications of our results to the classical limit of the Schrödinger equation. Finally we present various examples and counterexamples showing that our results are sharp.

Thu, 21 Feb 2013
12:00
Gibson 1st Floor SR

1D Burgers Turbulence as a model case for the Kolmogorov Theory

Alexandre Boritchev
(Ecole Polytechnique)
Abstract

The Kolmogorov 1941 theory (K41) is, in a way, the starting point for all

models of turbulence. In particular, K41 and corrections to it provide

estimates of small-scale quantities such as increments and energy spectrum

for a 3D turbulent flow. However, because of the well-known difficulties

involved in studying 3D turbulent flow, there are no rigorous results

confirming or infirming those predictions. Here, we consider a well-known

simplified model for 3D turbulence: Burgulence, or turbulence for the 1D

Burgers equation. In the space-periodic case with a stochastic white in

time and smooth in space forcing term, we give sharp estimates for

small-scale quantities such as increments and energy spectrum.

Thu, 27 Sep 2012

09:30 - 10:30
L1

$\ell$-adic representations of etale fundamental group of curves

Anna Cadoret
(Ecole Polytechnique)
Abstract

I will present an overview of a series of joint works with Akio Tamagawa about l-adic representations of etale fundamental group of curves (to simplify, over finitely generated fields of characteristic 0).
More precisely, when the generic representation is GLP (geometrically Lie perfect) i.e. the Lie algebra of the geometric etale fundamental group is perfect, we show that the associated local $\ell$-adic Galois representations satisfies a strong uniform open image theorem (ouside a `small' exceptional locus). Representations on l-adic cohohomology provide an important example of GLP representations. In that case, one can even provethat the exceptional loci that appear in the statement of our stronguniform open image theorem are independent of $\ell$, which was predicted by motivic conjectures.
Without the GLP assumption, we prove that the  associated local l-adic Galois representations still satisfy remarkable rigidity properties: the codimension of the image at the special fibre in the image at the generic fibre is at most 2 (outside a 'small' exceptional locus) and its Lie algebra is controlled by the first terms of the derived series of the Lie algebra of the image at the generic fibre.
I will state the results precisely, mention a few applications/open questions and draw a general picture of the proof in the GLP case (which,in particular, intertwins via the formalism of Galois categories, arithmetico-geometric properties of curves and $\ell$-adic geometry). If time allows, I will also give a few hints about the $\ell$-independency of the exceptional loci or the non GLP case.

Tue, 08 May 2012

10:15 - 11:15
OCCAM Common Room (RI2.28)

Stability of periodic structures: from composites to crystal lattices

Nicolas Triantafyllidis
(Ecole Polytechnique)
Abstract

Stability plays an important role in engineering, for it limits the load carrying capacity of all kinds of structures. Many failure mechanisms in advanced engineering materials are stability-related, such as localized deformation zones occurring in fiber-reinforced composites and cellular materials, used in aerospace and packaging applications. Moreover, modern biomedical applications, such as vascular stents, orthodontic wire etc., are based on shape memory alloys (SMA’s) that exploit the displacive phase transformations in these solids, which are macroscopic manifestations of lattice-level instabilities.

The presentation starts with the introduction of the concepts of stability and bifurcation for conservative elastic systems with a particular emphasis on solids with periodic microstructures. The concept of Bloch wave analysis is introduced, which allows one to find the lowest load instability mode of an infinite, perfect structure, based solely on unit cell considerations. The relation between instability at the microscopic level and macroscopic properties of the solid is studied for several types of applications involving different scales: composites (fiber-reinforced), cellular solids (hexagonal honeycomb) and finally SMA's, where temperature- or stress-induced instabilities at the atomic level have macroscopic manifestations visible to the naked eye.

Fri, 06 May 2011
14:15
DH 1st floor SR

Stochastic expansions for averaged diffusions and applications to pricing

Prof Emmanuel Gobet
(Ecole Polytechnique)
Abstract

We derive a general methodology to approximate the law of the average of the marginal of diffusion processes. The average is computed w.r.t. a general parameter that is involved in the diffusion dynamics. Our approach is suitable to compute expectations of functions of arithmetic or geometric means. In the context of small SDE coefficients, we establish an expansion, which terms are explicit and easy to compute. We also provide non asymptotic error bounds. Applications to the pricing of basket options, Asian options or commodities options are then presented. This talk is based on a joint work with M. Miri.

Mon, 25 Jan 2010
15:45
Eagle House

Stochastic nonlinear Schrodinger equations and modulation of solitary waves

Anne De Bouard
(Ecole Polytechnique)
Abstract

In this talk, we will focus on the asymptotic behavior in time of the solution of a model equation for Bose-Einstein condensation, in the case where the trapping potential varies randomly in time.

The model is the so called Gross-Pitaevskii equation, with a quadratic potential with white noise fluctuations in time whose amplitude tends to zero.

The initial condition is a standing wave solution of the unperturbed equation We prove that up to times of the order of the inverse squared amplitude the solution decomposes into the sum of a randomly modulatedmodulation parameters.

In addition, we show that the first order of the remainder, as the noise amplitude goes to zero, converges to a Gaussian process, whose expected mode amplitudes concentrate on the third eigenmode generated by the Hermite functions, on a certain time scale, as the frequency of the standing wave of the deterministic equation tends to its minimal value.

Mon, 19 Oct 2009

17:00 - 18:00
Gibson 1st Floor SR

Diffractive behavior of the wave equation in periodic media

Grégoire Allaire
(Ecole Polytechnique)
Abstract

We study the homogenization and singular perturbation of the

wave equation in a periodic media for long times of the order

of the inverse of the period. We consider inital data that are

Bloch wave packets, i.e., that are the product of a fast

oscillating Bloch wave and of a smooth envelope function.

We prove that the solution is approximately equal to two waves

propagating in opposite directions at a high group velocity with

envelope functions which obey a Schr\"{o}dinger type equation.

Our analysis extends the usual WKB approximation by adding a

dispersive, or diffractive, effect due to the non uniformity

of the group velocity which yields the dispersion tensor of

the homogenized Schr\"{o}dinger equation. This is a joint

work with M. Palombaro and J. Rauch.

Fri, 16 May 2008
14:15
Oxford-Man Institute

Some solvable portfolio optimization problems with max-martingales

Nicole El Karoui
(Ecole Polytechnique)
Abstract

Many portfolio optimization problems are directly or indirectly concerned with the current maximum of the underlying. For example, loockback or Russian options, optimization with max-drawdown constraint , or indirectly American Put Options, optimization with floor constraints.

The Azema-Yor martingales or max-martingales, introduced in 1979 to solve the Skohorod embedding problem, appear to be remarkably efficient to provide simple solution to some of these problems, written on semi-martingale with continuous running supremum.

Mon, 16 Feb 2004
15:45
DH 3rd floor SR

Exponents of Growth for SPDEs

Thomas Mountford
(Ecole Polytechnique)
Abstract

We discuss estimating the growth exponents for positive solutions to the

random parabolic Anderson's model with small parameter k. We show that

behaviour for the case where the spatial variable is continuous differs

markedly from that for the discrete case.

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