The Curie-Weiss model is an exchangeable probability measure $\mu$ on $\{0,1\}^n.$
It has two parameters -- the external magnetic field $h$ and the interaction $J$.
A natural problem is to determine when this measure extends to an exchangeable measure
on $\{0,1\}^{\infty}$. We will discuss two approaches to the following result:
$\mu$ can be (infinitely) extended if and only if $J\geq 0$. One of these
approaches relies on the classical Hausdorff moment problem. When $Jn$ can $\mu$ be extended to an exchangeable measure on $\{0,1\}^l$. Our approach
to this question involves an apparently new type of moment problem, which we will
solve. We then take $J=-c/l$, and determine the values of $c$ for which $l$-extendibility
is possible for all large $l$. This is joint work with Jeff Steif and Balint Toth.

The number of solutions to a set of polynomial equations defined over a
finite field of $q=p^a$ elements is encoded by its zeta function, which is a
rational function in one variable. A question of fundamental interest is
how to compute this function efficiently. We describe a method to solve
this problem (due to Wan) using a $p$-adic trace formula of Dwork. We
examine how well this method works in practice on some explicit examples
coming from a certain class of varieties known as $\Delta$-regular
hypersurfaces. Our experience suggests several potential avenues of further
research.

The pressure term has always created difficulties in treating the
Navier-Stokes equations of incompressible flow, reflected in the
lack of a useful evolution equation or boundary conditions to
determine it. In joint work with Bob Pego and Jie Liu, we
show that in bounded domains with no-slip boundary conditions,
the Navier-Stokes pressure can be determined in a such way that
it is strictly dominated by viscosity. As a consequence, in a
general domain with no-slip boundary conditions, we can treat the
Navier-Stokes equations as a perturbed vector diffusion equation
instead of as a perturbed Stokes system. We illustrate the
advantages of this view by providing simple proofs of (i) the
stability of a difference scheme that is implicit only in
viscosity and explicit in both pressure and convection terms,
requiring no solutions of stationary Stokes systems or inf-sup
conditions, and (ii) existence and uniqueness of strong solutions
based on the difference scheme.

A preprint is available at http://arxiv.org/abs/math.AP/0502549