AWM Emmy Noether Lecture
Outer space was introduced in the mid-1980s as a tool for studying the group Out(Fn) of outer automorphisms of a finitely-generated free group. The basic philosophy is that one should think of an automorphism of a free group as a topological object, either as a homotopy equivalence of a finite graph or as a diffeomorphism of a suitable three-manifold with free fundamental group. There are compelling analogies between the action of Out(Fn) on Outer space and the action of an arithmetic group on a homogeneous space or the action of the mapping class group of a surface on the associated Teichmuller space. In this talk I will first describe Outer Space and explain how it is used to obtain algebraic information about Out(Fn). I will then indicate how Outer Space is related to other areas, from infinite-dimensional Lie algebras to the mathematics of phylogenetic trees, and how ideas from Outer Space are currently expanding in new directions.