We will watch a talk given by Michael Ching. We might pause to discuss or discuss afterwards.
Abstract: The calculus of homotopy functors provides a systematic way to approximate a given functor (say from based spaces to spectra) by so-called 'polynomial' functors. Each functor $F$ that preserves weak equivalences has a 'Taylor tower' (analogous to the Taylor series of ordinary calculus) which in turn is built from homogeneous pieces that are classified by certain 'derivatives' for $F$. I will review this material and consider the problem of how the Taylor tower of $F$ can be reconstructed from its derivatives. We will discuss some important examples built from mapping spaces. If time permits, I will use this approach to give a classification of analytic functors from based spaces to spectra and try to describe some connections to the Goodwillie-Weiss manifold calculus.
The video and some notes are available here: https://www.msri.org/workshops/685/schedules/17878