From the Jones polynomial to Khovanov homotopy
Since its introduction in the 1980s, the Jones polynomial and its refinements have repeatedly revolutionized knot theory. This talk will start by recalling some basic notions and questions about knots. We will then recall the Jones polynomial and some of its applications to these questions. We will then discuss two refinements of the Jones polynomial: Khovanov homology and the Khovanov spectrum, and some of their recent applications, particularly to questions at the overlap of 3- and 4-dimensional topology. Most of the talk focuses on work of other people, but part is joint with Tyler Lawson and Sucharit Sarkar.