MIT Topology Seminar: Knots, Gropes and von Neumann rho-invariants (Tim Cochran)

MIT Topology Seminar

Monday, December 6, 2004
Room 2-131, 4:30pm

Tim Cochran will speak on:

Knots, Gropes and von Neumann ρ-invariants

Abstract: I will give a leisurely talk will be about how some very topological questions can be approached with the help of noncommutative algebra and functional analysis. We address the classical question: When does a knotted circle in the 3-sphere bound an embedded 2-dimensional disk in the 4-ball? We "filter" this question by asking when such a knot is the boundary of a "Grope" of height n. Gropes are 2-complexes that are constructed inductively from surfaces glued together. (In 3-dimensions, this filtration is related to the Kontsevitch integral !). After reviewing a little history of this problem, we will indicate how noncommutative algebra and von Neumann algebras arise naturally through the study of equivariant homology groups and equivariant intersection forms, and how these give new information about this classical problem. The work described is primarily joint with Peter Teichner (also Kent Orr and Taehee Kim).