\documentclass[a4paper]{article}
\usepackage{amssymb}
\pagestyle{empty}
\hoffset=-40pt
\voffset=-20pt
\textwidth 15.3cm
\textheight 22cm % to fit our printers

\begin{document}

\begin{center}{\Huge On the Ziegler Spectrum of the First Weyl
Algebra}\end{center}
\begin{center}{\Large Mike Prest}\end{center}
\begin{center}{\Large University of Manchester}\end{center}

\bigskip
\Large

Ziegler associated to an arbitrary 
ring a topological space, now called the
Ziegler spectrum, whose closed sets
parametrise complete theories of modules 
over that ring.   The first part
of the talk will be an attempt to give an
overview of what is known about this space 
in general, what is known about
it over particular types of rings and
applications to the theory of modules.   
In the second part of the talk I
will present some recent results on the
Ziegler spectrum of the first Weyl algebra.

The first Weyl algebra $A_1(k)$ over a field $k$, 
which we will take to be
of characteristic $0$, is the ring of
differential operators on the $k$-algebra 
$k[X]$ (the algebra of
polynomials in a single indeterminate $X$ and with
coefficients from $k$).   Explicitly, it is 
the $k$-algebra generated by
elements $x$ and $y$ subject to the relation
$yx-xy = 1$ and a natural action on $k[X]$ 
is defined by having $x$ act as
multiplication by $X$ and $y$ act as
differentiation with respect to $x$.   This 
ring is a simple (i.e. no
non-zero proper two-sided ideals) domain all of
whose simple (=irreducible) modules are 
infinite-dimensional over $k$.

It turns out that the Ziegler 
spectrum of $A_1(k)$ offers some interesting
contrasts to previously investigated
Ziegler spectra.   The starting 
point of our investigations is the
observation by Bavula and Herzog that results of
McConnell and Robson imply that the 
simple (and, more generally, finitely
generated torsion) modules over $A_1(k)$
have rather nice model-theoretic properties.

\end{document}