Turbulence is characterized by its nonlinear and multiscale behaviour, self-organization into coherent structures and generic randomness. The number of active spatial and temporal scales involved increases with the Reynolds number, therefore it soon becomes prohibitive for direct numerical simulation. However, observations show that for a given flow realization these scales are not homogeneously distributed, neither in space nor in time, which corresponds to the flow intermittency. To be able to benefit from this property, a suitable representation of the flow should reflect the lacunarity of the fine scale activity, in both space and time.

A prominent tool for multiscale decompositions are wavelets. A wavelet is a well localized oscillating smooth function, i.e. a wave packet, which is dilated and translated. The thus obtained wavelet family allows to decompose a flow field into orthogonal scale-space contributions. The flow intermittency is reflected in the sparsity of the wavelet representation, i.e. only few coefficients, the strongest ones, are necessary to represent the dynamically active part of the flow. We will illustrate this by considering different 2D and 3D turbulent flows, either computed by direct numerical simulation (DNS) or measured by particle image velocimetry (PIV).

To compute the evolution of turbulent flows we have proposed the Coherent Vortex Simulation (CVS), which is based on the wavelet filtered Navier-Stokes equations. At each time step the turbulent fluctuations are split into two parts: the first corresponding to the coherent vortices which are kept, and the second to an incoherent background flow corresponding to turbulent dissipation which is discarded. We will present several simulations of 2D and 3D turbulent flows and show that CVS preserves their nonlinear dynamics.

Related publications can be downloaded from the following web pages:

http://wavelets.ens.fr
http://cmi.univ-mrs.fr/~kschneid

In this lecture, we present a new proof of Perelman's collapsing theorem for 3-manifolds with boundary which is needed for his work on Thurston's Geometrization Conjecture. Among other things, we use an observation of Hamilton-Perelman on incompressible tori boundary for Ricci flow with surgery on thick part of a 3-manifold. Starting from incompressible tori boundary of thin part of 3-manifold, we found that there is an injective F-structure in the sense of Cheeger-Gromov. Consequently, the part of a 3-manifold for Ricci flow with surgery becomes an aspherical graph-manifold, Perelman's collapsing theorem for 3-manifolds follows.

There are several analogues of the theory of one complex variable, when the values of the functions are taken in the division algebra H of quaternions, or in a suitable Clifford algebra. These theories rely on the construction of operators which somehow imitate the Cauchy-Riemann operator; in the quaternionic case one uses the Cauchy-Fueter operator, and in the Clifford case one uses the Dirac operator. The extension to several variables has remained elusive for a long time, but it can in fact be achieved if one considers these systems from the point of view of their algebraic properties. The analysis of such operators from the point of view of the Palamodov-Ehrenpreis Fundamental Principle allows the construction of a non-trivial theory in several variables. This talk will discuss the strength of this approach, as well as some of the questions which remain open, and will be concluded with a new twist on these theories.