For generalized Reed-Solomon codes, it has been proved
that the problem of determining if a
received word is a deep hole is co-NP-complete.
The reduction relies on the fact that
the evaluation set of the code can be exponential
in the length of the code --
a property that practical codes do not usually possess.
In this talk, we first present a much simpler proof of
the same result. We then consider the problem for standard
Reed-Solomon codes, i.e. the evaluation set consists of
all the nonzero elements in the field.
We reduce the problem of identifying deep holes to
deciding whether an absolutely irreducible
hypersurface over a finite field
contains a rational point whose coordinates
are pairwise distinct and nonzero.
By applying Cafure-Matera estimation of rational points
on algebraic varieties, we prove that
the received vector $(f(\alpha))_{\alpha \in \F_p}$
for the Reed-Solomon $[p-1,k]_p$, $k < p^{1/4 - \epsilon}$,
cannot be a deep hole, whenever $f(x)$ is a polynomial
of degree $k+d$ for $1\leq d < p^{3/13 -\epsilon}$.
This is a joint work with Elizabeth Murray.

Tate's work on Rigid Analytic Spaces can be used to obtain the
$p$-adic uniformization of a curve. In this talk, I will describe a
criterion determining which hyperelliptic curves admit this type of
uniformization. Then, we will discuss Mumford curves, which are the
uniformizing spaces, and explain how to approximate the $p$-adic
uniformization of a given totally split hyperelliptic curve.

Let $K$ be a number field and $E/K$ an elliptic curve without
complex multiplication. A well-known theorem of Serre asserts that the
Galois group of $K(E[\ell])/K$ is as all of ${\rm GL}_2(\Z/\ell)$ for any
sufficiently large prime $\ell$. If we replace $E/K$ by a polarized abelian
variety $A/K$ with trivial endomorphism ring, then Serre later showed
that the Galois group of $K(A[\ell])/K$ is also as large as possible, for
all sufficiently large $\ell$, provided $\dim(A)$ is 2,6 or odd. We will
show how to prove a similar result for `most' $A$ and without any
restriction on $\dim(A)$.

Magnetic Resonance Imaging (MRI) uses the resonance of the nucleus of
chemical species to generate images of their spatial distribution. In
medical MRI, a simple model considers tissue as made up of only water
and fat. Most of the clinically relevant information is in the water
signal and the fat signal is considered clutter to be suppressed. The
separation of water and fat based on the difference of their resonance
frequencies using multiple images provides a robust method for fat
suppression in areas where the magnetic field is inhomogeneous. In this
talk, we will show how to propagate the uncertainty due to imperfections
of the magnetic field into our estimate of water and fat. The
optimization of data acquisition based on the Cramer-Rao bound (CRB) for
this nonlinear problem leads to new optimal solutions which do not arise
when the magnetic field is assumed to be homogeneous. A reconstruction
based on maximum likelihood estimation allows us to achieve the CRB for
realistic noise levels which is verified by Monte Carlo simulations,
experimentally and clinically. Our acquisition and reconstruction is
part of a method for chemical species separation currently used by
General Electric MRI scanners.