Dev Sinha: Calculus of functors and knot theory

ABSTRACT:

The calculus of functors was first invented to study pseudo-isotopy in differential topology. The formal structure first led to the calculus of homotopy functors, but then was adapted by Weiss to address questions about embeddings and other isotopy functors. In settings where it fully applies, for example to the rational homology of spaces of embeddings of a codimension three manifold, it has been remarkably successful. But one case where it does not fully apply is for classical knot theory.     In this setting, all knowledge points to a conjecture that the Goodwillie-Weiss tower serves as a universal finite-type knot invariant over the integers. I will give the background to make this conjecture meaningful (and significant), and then share recent progress and potential next steps in a longstanding program to address this conjecture.