Kathryn Hess: A calculus for knot theory

Schedule

14.00-15.00 Pre-colloquium by Kaj Börjesson (room FB55 at Albanova)
15:15-16:15 Colloquium lecture by Kathryn Hess (lecture hall FD5 at Albanova, not the usual one)
16:15-17:00 SMC social get together with refreshments

Abstract

A mathematical knot consists of an embedding of the circle in 3-space. Knot theory - the mathematical study of knots - has been an active, important field of research for well over 100 years. When studying knots, instead of considering one knot at a time, it can be very fruitful to analyze all at once the entire space of knots, i.e., the set of all knots, together with a notion of when two knots are "close" to each other.

The space of knots is an example of what is known as a space of embeddings. I will start by reviewing the classical theory of spaces of embeddings, including famous results of Hirsch, Smale, and Gromov, and explain the analogy between these results and the theory of linear functions. I will then describe how Goodwillie and Weiss extended this analogy, establishing a "calculus" for spaces of embeddings. In particular their theory enables us to construct "polynomial approximations" to spaces of embeddings, together with "converging Taylor series" under nice conditions. To conclude I will sketch briefly a couple of recent applications of the Goodwillie-Weiss calculus, in work of Arone and Turchin and of Dwyer and myself.