Topology Colloquium

2012 June 26, 16:30, room B3.03

Étienne Ghys (Ens Lyon)

Hardy Lecture: The history of the uniformization theorem

Abstract: The uniformization theorem is a fundamental tool in the geometry of Riemann surfaces. It is easy to state: "simply connected Riemann surfaces are isomorphic to the complex plane, the unit disc or the sphere". Today, most geometers have forgotten that such a simple statement has not always been "obvious". The history of this theorem is long, complicated, and fascinating. It was an opportunity to create many concepts that seem natural to us... today. For algebraic curves, it was formulated by Klein and Poincaré around 1881. I would like to give an outline of this history, mainly focusing on Poincaré's approach which lead to an "almost convincing proof" in August 1881.

2010 June 4, 16:00, room B3.02

Karen Vogtmann (Cornell)

Outer Spaces

Abstract: An "outer space" for a group G is a contractible space with a proper action of the group Out(G) of outer automorphisms of G. Classical examples include homogeneous spaces and Teichmuller spaces. For a free group F of finite rank, an outer space was introduced in the mid-1980s. The basic idea is to think of an automorphism of a free group topologically, as a homotopy equivalence of a finite graph. In this talk, I will describe this outer space and indicate how it is used to obtain algebraic information about Out(F). I will then show how theses ideas have recently led to the construction of outer spaces for other types of groups.

2008 November 13, 14:00, room B3.02

Ulrike Tillmann (Oxford)

The topology of manifolds embedded in Euclidean space

Abstract: Some 25 years ago, Mumford conjectured that the rational cohomology of Riemann's moduli space is a polynomial ring in degrees roughly half the genus of the underlying surface. This was proved by Madsen and Weiss. The key of a simplified proof of this theorem - which also gives a nice statement for dimensions other than 2 - which studies the cobordism category of (d-1) and d-dimensional manifolds embedded in infinite dimensional Euclidean space. The talk will mainly be based on a joint paper with Galatius, Madsen and Weiss.