In the 1990s, Goodwillie developed a theory of calculus for homotopical functors. His idea was to approximate a functor by a tower of ‘polynomial functors’, similar to how one approximates a function by its Taylor series. The role of linear polynomials is played by functors that behave like homology theories, in the sense that there is a Mayer-Vietoris sequence computing their homotopy groups. As such, the Goodwillie tower interpolates between stable and unstable homotopy theory. The theory has application to the computation of the homotopy groups of spheres, higher algebra, and algebraic K-theory. In my talk, I will give an introduction to this topic. In particular, I will explain that Goodwillie's calculus reveals a deep connection between the homotopy theory of spaces and Lie algebras and how this is related to a chain rule for the derivatives of functors.
 

I will talk about the problem of recognising when a manifold admits a finite cover that fibres over the circle, with emphasis on the case of hyperbolic manifolds in odd dimensions. I will survey the state-of-art, and discuss the role that group theory plays in the problem. Finally, I will discuss a recent result that sheds light on the analogous group-theoretic problem, that is, virtual algebraic fibring of Poincaré-duality groups. The final theorem is joint with Sam Fisher and Giovanni Italiano.