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.

Many concrete deterministic dynamical systems exhibit apparently random
behaviour. This puzzle is studied my finding a time invariant probability
measure and discussing the statistical phenomenon using this measure. In
this way various systems (e.g. given by PDE's) can be said to be
"completely random" or "completely deterministic".

This leads to the project of classifying invariant probability measures.
The ergodic decomposition theorem shows that the basic building blocks of
these measures are the ergodic measures, which form a dense G_\delta set.
The equivalence relation of isomorphism is given by a Polish group action.
Thus the tools of descriptive set theory directly apply and one can show
that the action is "turbulent" and complete analytic. This precludes any
kind of recognizable classification.

This is joint work with Dan Rudolph and Benjy Weiss.