A Hadamard manifold – or more generally a CAT(0) space – is said to have higher

rank if every geodesic line lies in a flat plane. If a higher rank Hadamard manifold

admits finite volume quotients, then it has to be a symmetric space or split as a direct

product. This is the content of Ballmann’s celebrated Rank Rigidity Theorem, proved

in the 80s. It has been conjectured by Ballmann that his theorem generalizes to the

synthetic setting of CAT(0) spaces. In the talk I will discuss Ballmann’s conjecture

and report on recent progress.