Colloquium: Tuesday, December 29, 2pm. Speaker: Ayelet Lindenstrauss (Indiana) . Title: Taylor Series in Homotopy Theory.
I will discuss Goodwillie’s calculus of functors on topological spaces. To mimic
the set-up in real analysis, topological spaces are considered small if their nontrivial homotopy
groups start only in higher dimensions. They can be considered close only in relation to a map
between them, but a map allows us to construct the difference between two spaces, and two
spaces are close if the difference between them is small. Spaces can be summed (in different
ways) by taking twisted products of them. It is straightforward to construct the analogs of
constant, linear, and higher degree homogenous functors, and they can be assembled into
\polynomials” and \in nite sums”. There are notions of differentiability and higher derivatives,
of Taylor towers, and of analytic functions.
What might look like a game of analogies is an extremely useful tool because when one looks
at functors that map topological spaces not into the category of topological spaces, but into the
category of spectra (the stabilized version of the category of spaces, which will be explained),
many of them are, in fact, analytic, so they can be constructed from the homogenous functors
of different degrees. And we can use appropriate analogs of calculus theorems to understand
them better. I will conclude with some recent work of Randy McCarthy and myself, applying
Goodwillie’s calculus to algebraic K-theory calculations.
