This is a mini-course on functor calculus with a particular emphasis on surveying applications of some of the different brands of calculus to problems in homotopy theory and differential/manifold topology. A rough outline of the mini-course is provided below. The precise content will be decided as we go based on participants' interests.
Friday 9 February: Introduction to functor calculus.
I will explain the basic ideas behind functor calculus and give a more detailed construction of Weiss calculus. There should be ample examples, most of which you can do "by hand", including ways to construct topological K-theory and classify certain types of vector bundles by computing derivatives in Weiss calculus.
Friday 16 February: Applications to (stable) homotopy theory.
Potential topics may include Miller's stable splittings of Stiefel manifolds, Mitchell-Richter/Bott filtrations and associated multiplicative properties, the EHP sequence, the J-homomorphism, and the Whitehead conjecture.
Friday 23 February: Applications to chromatic homotopy theory.
Potential topics may include the existence of finite type n spectra and the corresponding v_n-self maps (following Arone, not Mitchell), and computations of v_n-periodic homotopy groups of spheres from their Morava E-cohomology.
Friday 1 March: Applications to differential topology.
Potential topics may include the insensitivity of embedding calculus to exotic smooth structures, the rational homology of certain embedding spaces, the rational homotopy groups of diffeomorphisms of discs, and rational Pontryagin classes.
Notes (last updated 04/03/24)
If you spot any typos, please let me know!