|Location:||MSRI: Simons Auditorium|
I will survey some mathematics around the application of Goodwillie-Weiss Embedding Calculus to knot theory and some then share some related problems. These problems (and related comments and questions) include:
- Are the cochains of the n-disks operad formal, say as diagrams of E_infinity algebras? (This would imply a spectral sequence collapse result, which would in turn imply major results in knot theory and its interface with mathematical physics and Lie theory. What machinery is there to establish formality of cochains integrally? Could intersection
theoretic models serve over the integers some of the roles that differential forms play over the reals?)
- Can one explicitly and geometrically define invariants of homotopy classes of maps between say finite complexes which are associated to a bar construction on their cochains over the Koszul dual of the E_infty operad (that is, the derivatives of the identity functor)? (Rationally, such explicit and geometric "homotopy periods" defined using the bar
construction over the Lie cooperad are a pretty story, useful for applications to geometric questions in which homotopy arises.)
- What are some other consequences of formality of the disks operad, including a strong version in dimension two? (This is less my direct interest and more something to bring to the attention of people doing things with these operads, including in calculus and factorization homology. Some like Kontsevich have gotten quite a lot of mileage from this formality over the reals.)
To lead up to these questions, I'll start with simple models for the Goodwillie-Weiss tower, inspired by Gauss's definition of linking number.