Let G be a transient graph, and flip a fair coin at each vertex.
This gives a distribution P. Now start a random walk from a vertex v, and
retoss the coin at each visited vertex, this time with probability 0.75
for heads and probability 0.25 for tails. The eventual configuration of
the coins gives a distribution Q. Are P and Q absolutely continuous w.r.t.
each other? are they singular? (i.e. can you tell whether a random walker
had tampered with the coins or not?) In the talk I'll answer to this
question for various graphs and various types of random walk. Based on
joint work with Y. Peres.

A novel approach to exploiting multipath for communications and radar is time-reversal focusing. In the basic time-reversal process, a signal from a beacon location is recorded at one or more receiving antennas. The received signal is then time-reversed, and retransmitted from the antennas used to receive the initial beacon signal. A portion of the time-reversed signal will retrace the initial pathincluding multipath reflectionsand focus at the location of the original beacon transmitter. In a rich multipath environment, the size of the focused spot can be of order one-half wavelength at an arbitrary distance from the antenna or array. This is referred to as super-resolution since the size of the focused spot would normally be limited by the numerical aperture resulting from the size of the array and the distance to the focal point. With suitable modifications to the time-reversed signals, it is possible to create a situation where the multiple paths interfere destructively at the beacon location, resulting in a null rather than a focused spot. These techniques can also be used to improve radar performance in clutter by focusing energy on the target.
In this presentation, super-resolution focusing and nulling experiments are described based on multipath-enhanced time-reversal techniques. Using these techniques, two independent 2.45 GHz signals focused at locations separated by 5 cm at a distance of 6.7 m are successfully demodulated. Experiments are also described showing how the time-reversal technique can be used to improve the radar detection of targets in clutter.

Motivated by the construction made by Goppa on curves, we present some error-correcting codes on algebraic surfaces. A surface whose Neron-Severi group has rank 1 has a "nice" intersection property that allows us the construction of a good code. We will verify this on specific examples. Surfaces with many points and rank 1 are not easy to find. We were able, though, to find also surfaces with low rank and many points, and these gave us good codes too. Finally, we present a decoding algorithm for these codes. It is based on the realization of the code as a LDPC code, and it is inspired on the Luby-Mitzenmacher algorithm.