# Seminar on functor calculus and chromatic methods

**Venue:** HIM lecture hall, Poppelsdorfer Allee 45, Bonn (unless stated otherwise)

Organizers: Rosona Eldred, Gijs Heuts, Akhil Mathew, Lennart Meier

## Wednesday, November 2

11:00 - 12:00 Gregory Arone: Calculus of functors and homotopy theory, Lecture 1: Goodwillie tower of the identity and the unstable homotopy of spheres

Abstract: The Goodwillie tower of the identity functor interpolates between stable and unstable homotopy groups. When applied to spheres, it has some rather striking properties that go beyond what is predicted by the general theory. This lecture is mostly based on my old work with Mark Mahowald, but I will also discuss how recent advances influence our perspective.

15:00 - 16:00 Sarah Yeakel: Derivatives of the Identity Functor

Abstract: We'll discuss the benefits and pitfalls of defining the derivatives of the identity functor of topological spaces using the indexing category I of finite sets and injective maps.

## Thursday, November 3

11:00 - 12:00 Gregory Arone: Calculus of functors and homotopy theory, Lecture 2: Taylor towers in situations when Tate homology vanishes

Abstract: The derivatives of a functor have a bimodule structure over a certain operad. If the Tate homology of the derivatives vanish, then the functor can be recovered from this bimodule structure. One obvious example of this situation is rational homotopy theory. We can say that rationally, the theory of Taylor towers is equivalent to the theory of bimodules over the Lie operad. Another example is given by functors to L_{n}-local spectra. Here we recover Kuhn’s result on the splitting of Taylor towers of endo-functors of spectra, and extend it to functors from spaces to L_{n}-local spectra. We can also recover the theorem of Behrens and Rezk, which expresses the Bousfield-Kuhn functor of spheres in terms of Andre-Quillen homology. This is joint work with Michael Ching.

16:30 - 17:30 Tomer Schlank: Ultra-Products and Chromatic Homotopy Theory

Abstract: Let C_{p,n} be the E(n)-local category at height n and prime p. These categories are of great interest to the stable homotopy theorist since they serve as a the "bounded pieces" of the chromatic filtration on the category of spectra. It is a well known observation that for a given height n certain "special" phenomena happen only for small enough primes. Further, in some sense, the categories C_{p,n} become more regular and algebraic as p goes to infinity for a fixed n. The goal of this talk is to make this intuition precise.

In a '96 unpublished paper Jens Franke defined for every height n and a prime p the category Fr_{n,p} of Quasi-periodic complexes on the stack MFG_{(p)}^{≤n} as an algebraic analog of C_{p,n}. Now given an infinite sequence of mathematical structures, logicians have a method to construct a limiting one by using "ultra-products". We shall define a notion of "ultra-product of categories". Then for a fixed height n we prove:

This is a joint project with N. Stapleton and T. Barthel.

## Friday, November 4

11:00 - 12:00 Gregory Arone: Calculus of functors and homotopy theory, Lecture 3: Orthogonal calculus and (time permitting) the stable rank filtration

Abstract: We will survey how many results about the Goodwillie tower of the identity functor have analogues for the Taylor tower of the functor V → BU(V). Here the Taylor tower is understood in the sense of Michael Weiss’s orthogonal calculus. If there is time I will say something about the connection with the stable rank filtration and the Whitehead conjecture and its bu-analogue (the last part is joint work with Kathryn Lesh).

## Friday, November 11

14:00 - 15:00 Lukas Brantner: The Lubin-Tate Theory of K(n)-local Lie Algebras

Venue: HIM seminar room (in the basement), Poppelsdorfer Allee 45

Abstract: We compute the operations which adhere to the homotopy groups of K(n)-local Lie algebras over Lubin-Tate space and analyse the partition complex using discrete Morse theory.

## Tuesday, November 15

15:00 - 16:00 Wolfgang Lück: Equivariant Chern characters for proper actions for discrete groups

Abstract: Given an equivariant homology for proper equivariant CW-complexes and discrete groups, we construct an equivariant Chern character which identifies it with the Bredon homology of the equivariant CW-complex with coefficients given by the values of the equivariant homology theory on homogeneous spaces, provided that this coefficients theorem extends to a Mackey functor in a certain sense. In many examples of interest this condition is satisfied and allows to give explicite rational computations of the left hand side of various Isomorphism Conjectures, such as the one of Baum-Connes and Farrell-Jones. An interesting question is what the existence of the equivariant Chern characters means for the equivariant spectra which describe the equivariant homology theories.

## Wednesday, November 16

13:15 - 14:15 Dustin Clausen: Continuous algebraic K-theory

Abstract: Let A be a topological ring. One can consider A as a discrete ring, and take its algebraic K-theory K^{alg}(A). If A lives over the real numbers, one can also consider its topological K-theory K^{top}(A). The algebraic theory K^{alg} gives more refined invariants, but is computationally mysterious; whereas K^{top} only sees homotopy invariants, but is generally calculable. I will describe a theory which lives in between these two, and seems in examples to have the favorable properties of both. It is likely related to Karoubi's "multiplicative K-theory" when A lives over the real numbers, but the definition is completely different, and it seems to give sensible answers in the p-adic case as well.