I will discuss modular forms on two algebraic groups of type
D4. These two groups are naturally associated to the two octonion
algebras over the rationals. After introducing the basic properties of
modular forms on these two groups, I will discuss a theta-correspondence
between them. This can be thought of as an octonionic generalization of
the Jacquet-Langlands correspondence.

We construct several new statistical zero-knowledge proofs
with$ _efficient_provers_,$ i.e. ones where the prover strategy
runs in probabilistic polynomial time given an NP witness for
the input string.

Our first proof systems are for approximate versions of the
Shortest Vector Problem (SVP) and Closest Vector Problem (CVP),
where the witness is simply a short vector in the lattice or a
lattice vector close to the target, respectively. Our proof
systems are in fact proofs of knowledge, and as a result,
we immediately obtain efficient lattice-based identification
schemes which can be implemented with arbitrary families of
lattices in which the approximate SVP or CVP are hard.

We then turn to the general question of whether all
problems in SZK intersection NP admit statistical zero-knowledge
proofs with efficient provers. Towards this end, we give
a statistical zero-knowledge proof system with an efficient prover
for a natural restriction of Statistical Difference, a complete
problem for SZK. We also suggest a plausible approach to resolving
the general question in the positive.

Joint work with Salil Vadhan (Harvard University).
Talk based on a paper presented at CRYPTO $2003.$

We obtain a tight bound of $O(L^2 log r)$ for the mixing time of the exclusion process in $Z^d/LZ^d$ with $r <= L^d/2$ particles.

The classical conjectures of Gross and Brumer-Stark seem to describe two
completely unrelated properties of special values of Galois equivariant
global L-functions. In this talk, we will develop a general Equivariant
Main Conjecture in Iwasawa Theory which captures the Brumer-Stark and
Gross phenomena simultaneously and works equally well in characteristics $0$ and p. The characteristic p side of the theory draws its main ideas from
Deligne\'s construction of $1-motives$ associated to smooth, projective
curves over finite fields. The characteristic $0$ side of the theory is
based on our new construction of number field analogues of the l-adic
realizations (i.e. l-adic etale cohomology groups) of Deligne\'s $1-motives$
and is deeply rooted in earlier work of Tate and Ritter - Weiss on the
theory of multiplicative Galois module structure. Time permitting, we will
also provide evidence in support of this new equivariant Iwasawa theoretic
statement and discuss its links to l-adic refinements of integral
Rubin - Stark - type conjectures on special values of global L-functions.

Many approaches exist to compute the (approximate) inverse of an operator K to recover an approximation to f from a dataset that represents a noise-corrupted version of Kf. Several approaches have been proposed that are adapted to the special case where f has a sparse wavelet expansion, a case that applies to many types of images or other types of signals; an example of the operator K in this context is, e.g., blurring, the convolution with a known function.
The talk will present an iterative approach to solve this problem, which can be used with respect to arbitrary orthonormal bases. The algorithm is similar to the Landweber algorithm, except that the prior information incorporated into the variational functional uses a weighted $l^p-norm$ of the wavelet coefficients instead of the $l2-norm$, standard for Landweber methods. This iterative approach converges in norm and is stable; some applications will be shown.

This is joint work with Michel Defrise (Vrije Universiteir Brussel) and Christine De Mol (Universite Libre de Bruxelles)