Let $\mu$ be a probability measure on a metric space $(E,d)$ and $N$ a positive integer.
The {\em quantization error} $e_N$ of $\mu$ is defined as the infimum over all subsets ${\cal{E}}$
of $E$ of cardinality $N$ of the average distance w.~r.~t.~$\mu$ to the closest point in the set
${\cal{E}}$. We study the asymptotics of $e_N$ for large $N$. We concentrate on the
case of a Gaussian measure $\mu$ on a Banach space. The asymptotics of $e_N$ is closely related to
{\em small ball probabilities} which have received considerable interest in the past decade.
The quantization problem is motivated by the problem of encoding a continuous signal
by a specified number of bits with minimal distortion. This is joint work with Steffen Dereich,
Franz Fehringer, Anis Matoussi and Michail Lifschitz.

This talk explores the potential of new geometric multiscale ideas in the area of partial differential equations. We present a recently developed multiscale system - curvelets - based on parabolic scaling, in which basis functions are supported in elongated regions obeying the relation width ~length^2. This system provides optimally sparse representations of the solution operators for a large class of symmetric systems of linear hyperbolic differential equations - such as the wave propagation operator. This has important implications both for analysis, and for numerical applications, where sparsity allows for faster algorithms. In the second part of the talk, we report on preliminary calculations which suggest that it is possible to derive accurate solutions to a wide range of differential equations in O(N log N) where N is the number of voxels; this complexity holds for arbitrary initial conditions. This is joint work with Laurent Demanet (Caltech)