I just heard a report on an NPR station -- it was probably BBC news -- in which freelance mathematician Jeff Weeks (who wrote the book "The Shape of Space") was being interviewed after he had announced that the spatial universe is finite. There was not enough time allotted for him to give even a brief layperson's sketch of his reasoning.
But it seemed to have something to do with the fluctuations in the background radiation from the Big Bang having bounded wavelengths.
Jeff Weeks has been talking about this "shape of space" stuff for quite a while. The original idea was to use the data from the WMAP -- a very high-resolution picture of the cosmic background radiation -- to search for "circles in the sky", a potentially visible trace of the self-intersections of the sphere bounding the visible universe. This sphere is centered at the earth, and its diameter is the distance light has travelled since "decoupling", the point in the cooling of the early universe when light started travelling at the speed of, er, light. If the universe is multiply connected and small enough that the sphere self-intersects, we should be able to notice that we're seeing the circle of self-intersection looking in two different directions. When the data from the WMAP came in, the initial analysis didn't find any circles :-(. But it did notice something very odd, roughly that the autocorrelation function fell off for large angles (over around 60 degrees). Funster Thomas Colthurst probably remembers where to find the initial papers on this; I'm sure they're on the physics arXiv. The dipole can't be measured directly due to interference, but the quadropole is much smaller than it ought to be. Today's story, which made the cover of Nature, is that Weeks and his colleagues have observed that the observed harmonics of the Poincare Dodecahedron are a really good match for this initial data. Shh, don't tell anyone: I put a copy of the letter to Nature at http://people.brandeis.edu/~kleber/nature.pdf But the circles-in-the-sky people have already rebutted this claim, saying they're really sure the circles that the dodecahedron would leave are simply not there. Boy, I'd be delighted if the universe were a Poicare Dodecahedron! (Though I'd been hoping for a nonorientable space: if Weeks is right, then you could go in a straight line for 74 billion light-years and get back to where you started, just rotated by 2pi/5. I'd been hoping you'd get back to where you started mirror imaged.) --Michael Kleber kleber@brandeis.edu