Departmental Colloquium
Peter Shalen
UIC
Applying topology to hyperbolic geometry
Abstract: Hyperbolic geometry is the non-Euclidean geometry discovered by
Lobachevsky, Bolyai and Gauss. In hyperbolic space, the area of a
triangle is determined by the sum of its angles, and more generally
the volume of a configuration is determined by its shape. This
phenomenon, which contrasts with the Euclidean situation, persists in
hyperbolic manifolds, which are metric spaces locally isometric to
hyperbolic space, and are natural objects from the point of view of
differential geometry, complex analysis and number theory. The volume
of a hyperbolic manifold is determined by its topological type.
For hyperbolic manifolds of dimension at least 3, even more is true:
it follows from the Mostow rigidity theorem that when n is at least 3,
a compact hyperbolic n-manifold is determined up to isometry by its
topological type. In dimension 3 the situation is better still: over
the last 30 years, through the efforts of Thurston, Perelman and
others, a complete unification between the topology of 3-manifolds and
the geometry of hyperbolic 3-manifolds has been achieved.
In view of this it is not surprising that topological methods are
powerful in the study of hyperbolic 3-manifolds. My talk will be
focused on a couple of recent results of my own in which topological
methods from classical 3-manifold topology are brought to bear on a
geometric question. In the hyperbolic setting these techniques are
seen to be even richer and more powerful than might have been imagined
when the topology of 3-manifolds was a relatively self-contained and
isolated field.
In the course of describing this work I will try to hint at what a
rich field of research 3-dimensional hyperbolic geometry has become,
involving interactions among geometry, topology, algebra, number
theory and analysis.
Friday September 3, 2010 at 3:00 PM in SEO 636