Why are anti-de Sitter spaces so interesting when we believe the universe is expansionary? Perhaps this is a naive question, but in my recent (admittedly limited) readings about AdS spaces, I keep wondering why they seem to be such a hotbed for theoretical research (AdS/CFT correspondence, etc.).  To my understanding, an AdS space has constant negative curvature in a vacuum, which should yield an attractive universe, not one with accelerating expansion.  An AdS space can be thought of as having a negative cosmological constant, while a universe with accelerating expansion would imply that such a constant be positive.  Since we observe that our universe's expansion is accelerating, it seems that if anything, we should be seeking to model it as a de Sitter space.
Am I mistaken?  What aspects of our universe do AdS spaces attempt to model?
 A: An AdS universe can explain the cosmological observations. Our universe can be interpreted as an effective de Sitter brane in an Anti-de Sitter space. Therefore, you have to distinguish between a 5-dimensional cosmological constant from the bulk and the 4-dimensional constant from the brane (which is responsible for an accelerated expansion). Unfortunately, many physicists like to ignore this distinction.
Further reading
Standard Cosmology on the Anti-de Sitter boundary
Class. Quantum Grav., 2021
https://doi.org/10.1088/1361-6382/ac27ee
https://arxiv.org/pdf/2010.03391.pdf
A: The reason why the AdS/CFT correspondence is interesting is not that AdS space is supposed to describe our universe, which, as you have correctly pointed out, would lead to conflicts with experiments. In the context of the correspondence, a four-dimensional (conformal) field theory is mapped to a string theory living in an $AdS_5\times S^5$ space, although there exist generalizations in which the AdS part is of higher or lower dimension than five. 
This duality in principle allows one to carry calculations from one side to the other, making it possible to choose the framework in which the solution to the problem at hand can be found conveniently. 
One key observation in this context is that the duality can map a strongly coupled theory to a weakly coupled one, circumventing the failure of perturbation series. This is especially interesting with respect to QCD, where a a conventional perturbative low energy description is not possible. Even though an exact holographic dual of QCD is yet to be found, there are theories (for example the Sakai-Sugimoto model) that capture important features of QCD surprisingly well. 
One may now ask what is so special about AdS space that allows for such a duality? One way to approach this is to point out the rich symmetry content of this kind of spacetime. The isometry group of Anti-de Sitter space is given by $SO(4,2)$, which is precisely the conformal group in four dimensions.
Regarding de Sitter space: the nature of this spacetime makes it difficult to formulate a correspondence analogous to its positively curved counterpart. See this article for more information.
