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.

Consider a compact manifold $M$ (e.g. a torus) equipped with
a smooth measure $\mu$ (e.g. Lebesgue measure in the case
of torus) as a probability space $(M,\mathcal M,\mu)$. Consider
an ergodic map $T:M \to M$ along with a smooth function
$p:M \to (0,1)$. Define a random walk along orbits of $T$ as follows:
a point $x$ jumps to $T x$ with probability $p(x)$ and
to $T^{-1} x$ with probability $1-p(x)$.
Is there a limiting distribution of such a random walk for a generic
initial point? Is it absolutely continuous with respect to $\mu$?
We shall present an answer for several essentially different
maps $T$.

We describe a stochastic model for complex networks possessing three
qualitative features: power-law degree distributions, local clustering, and
slowly-growing diameter.
The model is mathematically natural, permits a wide variety
of explicit calculations, has the desired three qualitative features,
and fits the complete range of degree scaling exponents and clustering parameters.
Write-ups exist as a

short version
and as a
long version

We consider the rate of spread of a body of passive tracers moving under the influence of a random evolving vector field.
The vector field is of a type used as a model for ocean currents and was introduced by Kolmogorov. The rate of growth of the diameter of the body is of interest for practical reasons (such as in problems of pollution control) and we specify its rate of growth.

This talk will discuss random walks on percolation clusters.

The first case is supercritical ($p>p_c$) bond percolation in
$Z^d$. Here one can obtain Aronsen type bounds on the transition
probabilities, using analytic methods based on ideas of Nash.

For the critical case ($p=p_c$) one needs to study the incipient
infinite cluster (IIC). The easiest situation is the IIC on trees -
where the methods described above lead to an alternative approach to
results of Kesten (1986). (This case is joint work with T. Kumagai).