Series: Mathematics Colloquium

Date: Thursday, December 6, 2001

Time: 4:30 - 5:30 PM

Place: 102 McAllister Building

Host: Dmitri Burago

Refreshments: 4:00 - 4:30 PM, in 212 McAllister

Speaker: Alexander Nabutovsky (University of Toronto)

Title: Geometric Variational Problems and Non-Computable Functions

Abstract: 

We will explain an approach to geometric variational problems based on
non-computability. It can be used to prove the existence of
non-trivial ``thick'' knots of codimension one in high dimensions, and
of contractible closed geodesics on Riemannian manifolds with
complicated fundamental groups.  But the most interesting applications
of this method are quite general results establishing that the Morse
landscapes of many interesting functionals on the space of Riemannian
metrics on a closed manifold M are extremely rugged and have large
scale fractal features, for any M of dimension greater than four.  As
a corollary these functionals tend to have many local minima.  Besides
computability theory one needs quite deep results from topology of
manifolds and, surprisingly, modern number theory in order to prove
these results.  We are going to present an introduction into this
circle of ideas not assuming a previous knowledge of either logic or
Riemannian geometry.