What is the geometry of our universe? I have some questions about the notion of space in current cosmological theories.  I have been trying to decipher a few papers written on this topic such as this one, but I keep running into some trouble.
Most of the papers I have been skimming talk about anti-de Sitter space (AdS) and about how the best definition of nothing is 

the limit of anti-de Sitter space in which the curvature length goes
  to zero.

This is great and all, but I have read on Wikipedia that AdS implies a negative cosmological constant, which would imply that dark energy should be attractive.  Is my reasoning here incorrect?  If not, could someone explain why cosmologists spend so much time devoted to studying AdS when the universe is clearly expanding?  I have read somewhere about local geometry vs. global geometry, but I wasn't sure if that is what is going on here.  The papers are so complex.
 A: First of all, the AdS space is a hyperboloid-like maximally symmetric space so it doesn't ever "collapse". In general, a negative cosmological constant may speed up the collapse into a Big Crunch but what exactly happens depends on the distribution of matter and/or initial boundary conditions, too.
The Universe around us isn't an AdS space; it has a positive, not negative, cosmological constant. But the AdS space is critically important for theoretical physics, especially because of the AdS/CFT correspondence that gives us another, equivalent way to describe quantum gravitational theories (vacua of string/M-theory in the most general sense) in the AdS space. The results of the AdS/CFT calculations aren't directly applicable to our Universe but certain properties of space and objects in it are the same whether the space is positively or negatively curved so physicists have learned a lot about our Universe from AdS/CFT, too. For example, if the AdS curvature radius is much longer than the black hole radius, the AdS black hole ("small black hole") behaves pretty much identically as it does in the flat space.
One may also give intuitive explanations why it's the AdS space that exhibits holography most directly, via the AdS/CFT correspondence. It's because holography in that space doesn't violate the intuition that "degrees of freedom scale like the volume". If you regulate (cut) the AdS hyperboloid so that the volume of its spatial slices is finite, then this volume is equal to the surface times a constant (curvature radius). So while holography says that the actual number of degrees of freedom scales like the surface area, here it scales like the volume because they're proportional to one another thanks to the negative curvature.
