http://www.ams.org/new-in-math/mathnews/universe.html

What is the shape of the universe? Is it finite? These are two of the most
important questions in cosmology today. Mathematics is providing striking
new insights into deducing the shape of the universe from observational
data. These insights, which will be put to the test in the coming decade as
scientists receive data from a new space probe, have the potential to
transform our view of cosmology.

Gazing out into the night sky, one easily gets the impression that the
universe continues forever in all directions. However; this impression is
akin to thinking that the Earth is flat and continues forever in all
directions because that is how it appears when one scans the horizon. Fairly
sensitive measurements are needed to detect the curvature of the Earth. The
same is true for detecting the shape of the universe.

One possible shape the universe might have is analogous to the surface of a
doughnut. Mathematicians call this shape a torus, and it is a fundamental
object of study in the areas of geometry and topology. The torus model has a
weird property: For every object we observe in the universe, we would see
not one but multiple images of the object. These images correspond to light
emitted by the object at different points in time and at different angles.
In fact, mixed in with all the observational data cosmologists collect would
be multiple images of our very own galaxy. Taken together, these images
would provide information about some fundamental properties of the shape of
the universe.

The problem is that we cannot recognize these images of our own galaxy; we
do not even know what it looks like from the "outside". However, the Cosmic
Microwave Background radiation---the after-glow of the Big Bang that
permeates the universe---may provide some clues. For this approach to work,
more detailed data of the CMB is needed. By 2002, NASA's Microwave
Anisotropy Probe will have furnished more accurate data with much better
resolution than is available today.

These ideas are explored in the article "Measuring the Shape of the
Universe" by Neil J. Cornish and Jeffrey R. Weeks, which appeared in the
December 1998 issue of theNotices of the AMS.