Walter Neumann (Columbia University)
The Volume of Three-manifolds
Abstract: The lectures will trace the history of ideas of shape and size of space, following two threads. The dream that we might soon discover the global shape and size of our universe, as Eratostheness measured the spherical shape and size of the earth's surface over 2000 years ago, recently seemed not impossible, although this dream has suffered a setback with the inflation theory of the early history of the universe. At issue is the large scale curvature of the universe, which, if non-zero, would suggest a precise measure of volume in terms of a natural scale of length, together with theories of the origin of the universe that imply this volume should be small. Euclid's attempt to formalize the concept of volume eventually led to Hilbert's 3rd problem, which asked if a regular tetrahedron can be cut into finitely many polyhedral pieces that can be re-assembled to a cube. Although the problem was solved by Max Dehn the year Hilbert posed it, problems arising from it remain an active area of research, with ramifications in fields ranging from geometry to number theory. We shall start with a survey of this history and follow it to recent connections with other invariants of three-manifolds, including the Jones polynomial invariant of knots. At the same time we will discus what is known about the volume of three-manifolds, particularly the smallest ones. The Coble Lectures will be held at 4 p.m. April 15, 16 and 17. A reception will follow the April 16 lecture in General Lounge, Illini Union. |
