In 2005, Ching showed that the derivatives of the Goodwillie tower of the identity functor of spaces form an operad. This implied that the derivatives of all homotopy functors have extra algebraic structure, and was used most notably by Arone and Ching to show that the Goodwillie derivatives satisfy a (higher order) chain rule.

In 2016 Johnson, Osborne, Riehl and Tebbe and I showed that abelian functor calculus (a version of functor calculus established by Johnson and McCarthy) is an example of a cartesian differential category - the type of category which axiomatizes classical calculus. A higher order chain rule followed.

In this talk, I will explain that the derivatives of certain abelian functor calculus towers also form an operad, and explain to what extent this is a result of the fact that the category of abelian categories is a differential category. This is closely related to the work of Cockett and Seely which established a Faa di Bruno formula for differential categories.

In joint work with Markus Spitzweck, we show that over a base field of characteristic $p$, the conjectured form of the mod $p$ dual motivic Steenrod algebra is a retract of the actual answer. I will sketch the proof and possible applications. I will also explain how this problem is closely related to the Hopkins–Morel–Hoyois isomorphism, a statement about the algebraic cobordism spectrum $MGL$.

For a diagram of a 2-stranded tangle in the 3-ball we define a twisted complex of compact Lagrangians in the triangulated envelope of the Fukaya category of the smooth locus of the pillowcase. We show that this twisted complex is a functorial invariant of the isotopy class of the tangle, and that it provides a factorization of Bar-Natan's functor from the tangle cobordism category to chain complexes. In particular, the hom set of our invariant with a particular non-compact Lagrangian associated to the trivial tangle is naturally isomorphic to the reduced Khovanov chain complex of the closure of the tangle. Our construction comes from the geometry of traceless SU(2) character varieties associated to resolutions of the tangle diagram, and was inspired by Kronheimer and Mrowka's singular instanton link homology.

A cobordism between classical knots is a surface, properly embedded in $S^3 \times I $, which cobounds the knots. A cobordism between virtual knots $ K: S^1 \hookrightarrow \Sigma_g \times I $ and $ K': S^1 \hookrightarrow \Sigma_{g'} \times I $ is a pair $ ( S, M ) $, for $M$ a compact oriented $3$-manifold with $ \partial M = \Sigma_{g} \sqcup \Sigma_{g'} $, $ S $ an oriented surface properly embedded in $ M \times I $ with $ \partial S = K \sqcup K' $. If the genus of $S$ is zero then $(S,M)$ is a concordance. We may therefore ask new questions about the complexity of the $3$-manifolds appearing in cobordisms between $K$ and $K'$.

We outline one such question regarding the $3$-manifolds appearing in concordances between virtual knots and the unknot. Roughly, given a virtual knot $K \hookrightarrow \Sigma_g $ and concordance $(S,M)$ from $K$ to the unknot, place a Morse function on $M$; the level sets are surfaces $ \Sigma_l $. If there exists a level set $ \Sigma_l $ such that $ l > g $ then the concordance is ascent. Does there exist a slice virtual knot such that every concordance between it and the unknot is ascent?

As an example, we apply this construction to the functor $F(M) = \text{Emb}(M,W)$ of smooth embeddings in a given manifold $W$ ; in that case our cosimplicial model in degree $p$ is then just the configuration space of all the $p$-simplices of $M$ in $W$ product with a power of a Stiefel manifold. When $\dim(W ) > \dim(M ) + 2$, a theorem of Goodwillie-Klein implies that our explicit cosimplicial space is a model of $\text{Emb}(M,W)$. This generalizes Sinha’s cosimplicial model for the space of long knots which was for the special case when $M$ is the real line. (This is joint work with Pedro Boavida de Brito, Pascal Lambrechts, and Daniel Pryor)