I will describe our recent proof of localization at the bottom of the spectrum for Schrodinger operators with Poisson random potentials. Poisson random potentials are the most natural model for describing a material with impurities. This has been a longstanding open problem. I will give a very informal talk on work in progress.
In recent years, I have been thinking about
Mathematical Visualization, and developing a
program that does high quality, customized
visualizations of mathematical objects and
processes. I will demonstrate this program
and discuss some of the interesting and
unexpected ways that certain mathematical
theorems turned out to be just what was
needed to find fast and efficient algorithms
that solve a number of difficult rendering
problems. (Some of these rendering problems
involve seeing 3D objects in stereo, and I
will bring along the red/green glasses needed
for the demonstation.)
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 minimum distance is one of the most important combinatorial
characterizations of a code. The maximum likelihood decoding problem is
one of the most important algorithmic problems of a code. While these
problems are known to be hard for general linear codes, the techniques
used to prove their hardness often rely on the construction of artificial
codes. In general, much less is known about the hardness of the specific
classes of natural linear codes. In this paper, we show that both problems
are NP-hard for algebraic geometry codes. We achieve this by reducing a
well-known NP-complete problem to these problems using a randomized
algorithm. The family of codes in the reductions have positive rates, but
the alphabet sizes are exponential in the block lengths.
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.
