The main problem addressed in the talk is the
characterization of Fantappie transform of positive
measures in the unit ball of C\\^n. The analogous real
results were obtained by G.M.Henkin and A.A. Shananin,
in the line of the classical theorem of Bernstein on the
line. I will propose an approach based on Hilbert spaces
of analytic functions. This will provide, among other things,
a novel proof of Martineau\'s duality theorem.
Based on joint work in progress with John McCarthy.

The initial-boundary value formulation for the Einstein equations has
a number of special features when compared with that for other partial
differential equations. These issues are briefly discussed, and an
approach to prove local in time existence is presented. The main idea
is to rewrite Einstein\'s equations into an equivalent form, called Weyl
system. This work follows the main idea in, though is simpler than,
the work by Friedrich and Nagy, Comm. Math. Phys. 201, 619, (1999).

Given positive integers $n$ and $d$, let $A_2(n,d)$ denote the maximum
size of a binary code of length $n$ and minimum distance $d$. The well
known Gilbert-Varshamov bound asserts that $A_2(n,d) \\geq 2^n/V(n,d-1)$,
where $V(n,d) = \\sum_{i=0}^d {n \\choose i}$ is the volume of a Hamming
sphere of radius $d$. We show that, in fact, there exists a positive
constant $c$ such that
$$
A_2(n,d) \\geq c {2n \\over (n,d-1)}
$$

whenever $d/n \\le 0.499$. The result follows by recasting the Gilbert-
Varshamov bound into a graph-theoretic framework and using the fact that
the corresponding graph is locally sparse. Generalizations and extensions
of this result will be briefly discussed.
*joint work with T. Jiang, Math Department, University of Miami

Many very large graphs that arise in Internet and telecommunications
applications share various properties with random graphs (while some
differences remain). We will discuss some recent developments and mention
a number of problems and results in random graphs and algorithmic design
suggested by the study of these \"massive\"" graphs.

A classical problem in number theory is to count the number of ways a
quadratic form can be represented by another. The generating function for
these numbers turn out to be a modular form (the so-called theta
functions). In this talk, I will discuss an analogous problem involving
cubic forms, and what sort of modular forms it leads to.

The Word Problem is known to be undecidable. Many interesting problems in computational mathematics, however, lead to variants of the Word Problem. This talk will examine the problem of producing an algorithm to decide if a relation in a ring is a consequence of some given relations.

It is reasonable to expect that some of the techniques
used in road traffic theory would apply to modeling of
traffic on the World Wide Web. We review the
derivation and use of entropy maximizing models for the
traffic distribution problem, which calls for the
solution of a matrix balancing problem, and then apply
a similar approach to estimating traffic on the WWW,
which results in a hybrid matrix balancing
model. Recent work has shown that a more general
non-linear interior-point optimization algorithm is
also surprisingly efficient for these very large-scale
problems.