CAT(0) spaces originated from Riemannian geometry in the 50s as synthetic generalizations of Hadamard manifolds --
simply connected Riemannian manifolds of non-positive sectional curvature.
Already Riemannian geometry provides very interesting examples, like symmetric spaces of non-compact type. However,
the actual universe of CAT(0) spaces is much larger. While it contains many nice non-smooth examples which
carry additional combinatorial structures, like
triangulations or cubulations, mostly coming from group theory, a general CAT(0) space is simply badly singular.
Given this wild zoo of spaces, one inevitably seeks a form of classification.
A CAT(0) space is said to have higher rank if every geodesic lies in a flat plane.
Otherwise it is said to have rank 1.
Ballmann's Rank Rigidity Conjecture classifies CAT(0) spaces X with a cocompact group action by a group G according to their rank.
If X has rank 1, then it displays hyperbolic dynamics: G has many free subgroups, G-periodic geodesics are dense,
the geodesic flow has a dense orbit...
On the other hand, if X has higher rank, then it carries special geometry:
it must be isometric to a symmetric space or a euclidean building or split as a product.
In the talk, I will discuss the conjecture and report on recent progress.