Turbulence is a state of flows which is characterized by a combination of chaotic and random behaviours affecting a very large range of scales. It is governed by Navier-Stokes equations and corresponds to their solutions in the limit where the fluid viscosity becomes negligible, the nonlinearity dominant and the turbulent dissipation constant. In this regime one observes that fluctuations tend to self-organize into coherent structures which seem to have their own dynamics.

A prominent tool for multiscale decomposition are wavelets. A wavelet is a well localized oscillating smooth function, e.g. a wave packet, which is translated and dilated. The wavelet transform decomposes a flow field into scale-space contributions from which it can be reconstructed.

We will show how the wavelet transform can decompose turbulent flows into coherent and incoherent contributions presenting different statistical and dynamical properties. We will then propose a new way to analyze and predict the evolution of turbulent flows.

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The presentation will use different results obtained in collaboration with:

Kai Schneider (Universite de Provence, Marseille, France),
Naoya Okamoto, Katsunori Yoshimatsu and Yukio Kaneda (Nagoya University, Japan)

Related publications can be downloaded from the web page
http://wavelets.ens.fr

The Schwarzchild, respectively the Kerr space-times are solutions for the vacuum Einstein equation which model a spherically symmetric, respectively a rotating black hole. In this talk I will discuss the decay properties of solutions to the linear wave equation on
such backgrounds.

Suppose g is a square integrable function on the real line. The principal shift invariant space, , generated by g is the closure of the span of the system
B ={g(.-k): k an integer}. These spaces are most important in many areas of Analysis. This is particulrly true in the theory of Wavelets. We begin by describing a very simple method for obtaining the basic properties of and the systems B.
The systems obtained by applying, in addition to the integral translations, also the integral modulations (these are the multiplication of a function by exp(-2pinx)) are known as the Gabor systems. By using the Zak transform we show how the same methods can be used to study the basic properties of the Gabor systems and their span.
We will define the Zak transform and explain all this
in a very simple way that will be easily understood by all who know only a "smidgeon" of mathematics. A bit more challenging will be the explanation how all this can be extended to general locally compact abelian groups and their duals.
This is joint work with E. Hernandez, H. Sikic and E. N.
Wilson.