Ballmann, Bridson-haefliger, Eberlein 0000 (copyright Holder)

The study of nonpositively curved spaces goes back to the discovery of hyperbolic space, the work of Hadamard around 1900, and Car-tan's work in the 20's. These spaces play a signiicant role in many areas: Lie group theory, combinatorial and geometric group theory, dynamical systems, harmonic maps and vanishing theorems, geometric topology, Kleinian group theory, and Teichm uller theory. In some of these contexts {for instance in dynamics and in harmonic map theory { nonpositive curvature turns out to be the right condition to make things work smoothly, while in others such as Lie theory, 3-manifold topology, and Teichm uller theory, the basic objects of study happen to be nonpositively curved spaces. With so many closely related interdependent elds, nonpositive curvature has been a very active topic in the last twenty years. To get an idea of the scope of the activity, consider some of the highlights: One point of view which has been quite innuential in recent years is that it is fruitful to work with \synthetic" conditions which are equivalent to nonpositive sectional curvature in the Riemannian case, rather than sectional curvature itself. Though this idea (and the analog for spaces with lower curvature bounds) goes back to A. D. Alexandrov, it was Gromov Gro87] who brought it to the attention of a much wider