# The Rouse Ball Lecture for 2008

## Professor John Morgan

### Columbia University, USA

## Geometry and Topology of 3-dimensional spaces

Tuesday 20th May at 12 Noon.

Room 3, Mill Lane lecture rooms.

All interested are invited to attend.

When he introduced what is now known as Riemannian geometry, Riemann vastly
generalized what had come before. He also explicitly separated geometry from
the topology of the underlying space; geometry became an extra structure
given to a topological space. Around the turn of the 20th century Poincare
put topology on an independent footing as a subdiscipline of mathematics.
He also formulated a question which he considered as central. That question
was to characterize the simplest 3-dimensional space, the 3-sphere.
Poincare's conjecture was generalized to include a classification of all
3-dimensional spaces, technically, compact 3-manifolds, and even higher
dimensional spaces (but that is another story). In the 1980s Thurston
conjectured that 3-dimensional spaces could be classified, and Poincare's
original conjecture could be resolved, by uniting homogeneous Riemannian
geometry and topology in dimension 3, undoing, in a sense for 3-dimensional
spaces, Riemann's separation of topology and geometry. Around the same time,
Richard Hamilton proposed a method of attacking Thurston's conjecture. His
idea was to use a version of the heat equation for Riemannian metrics to
evolve any starting Riemannian metric on the space under consideration to a
nice Riemannian metric. Recently, Perelman has given a complete proof of
Thurston's conjecture along the general lines envisioned by Hamilton.

The talk will introduce ways of thinking about the topology 3-dimensional
spaces and the homogeneous geometries that come into play. The talk will
describe the version of the heat-type equation, called the Ricci flow
equation, for Riemannian metrics. It will then discuss the analytic and
geometric approaches and ideas and some of the difficulties that one must
overcome in order to arrive at a positive resolution of all these
conjectures by these methods.