On Thurston’s vision in geometry, topology, and dynamics

Abstract: since the 1980’s, Bill Thurston has done fundamental work in apparently quite different areas
of mathematics: in particular, on the geometry of 3-manifolds, on automorphisms of surfaces, and on
holomorphic dynamics. In all three areas, he proved deep and fundamental theorems that turn out to be
surprisingly closely connected both in statements and in proofs.

In all three areas, the statements can be expressed that either a topological object has a geometric structure
(the manifold is geometric, the surface automorphism has Pseudo-Anosov structure, a branched cover of
the sphere respects the complex structure), or there is a well defined topological-combinatorial obstruction
consisting of a finite collection of disjoint simple closed curves with specific properties. Moreover, all three
theorems are proved by an iteration process in a finite dimensional Teichmüller space (this is a complex
space that parametrizes Riemann surfaces of finite type).

I will try to relate these different topics and at least explain the statements and their context. I will also try
to outline current work on extending this work from rational to transcendental dynamics (joint with John
Hubbard, Mitsuhiro Shishikura, and Bayani Hazemach).