# Chromatic homotopy theory working seminar

**Venue:** MPIM seminar room, Vivatsgasse 7

**Time:** Wednesdays, 3 - 4 pm

**Organizers:** Tobias Barthel and Nathaniel Stapleton

The purpose of the seminar was to discuss directions and open problems in chromatic homotopy theory. The participants were assumed to be familiar with the subject.

## Wednesday, June 3

15:30 - 16:30 Nathaniel Stapleton: Morava E-theory - Height 1 vs Height 2

Abstract: I will discuss some new directions in Morava E-theory while putting an emphasis on the differences between height 1 and height 2. I hope to draw connections to representation theory and arithmetic geometry.

## Wednesday, June 10

15:00 - 16:00 Paul Goerss: Gross-Hopkins duality

Abstract: What is usually called Gross-Hopkins duality is actually an equivalence of two dualities in the K(n)-local category: the Spanier-Whitehead and Brown-Comenetz dualities. I'll explain what this means, when it works and when it does not, and use this to pose some questions for consideration.

## Friday, June 19

14:00-15:00 Doug Ravenel: G-spectra for the working algebraic topologist

Abstract: I will carefully explain the point set definition of G-spectra used to solve the Kervaire invariant problem. This will include a discussion of the Day convolution and, if time permits, the new model category structure that we use.

## Wednesday, June 24

15:00 - 16:00 Gerd Laures: On decompositions of string bordism

Abstract: We will discuss the Thom spectrum MString of the 7-connected cover of BO. Its homotopy groups are the string bordism groups. It is known that they have 2 and 3 torsion but a complete calculation is still missing. We will describe characteristic classes in TMF_{0}(3) and possible decompositions of MString also in the K(n)-local E_{∞}-spectra setting.

## Wednesday, July 1

No seminar this week.

## Wednesday, July 8

15:00 - 16:00 Haynes Miller: Some local computations in homotopy theory

Abstract: I will describe some of the machinery needed to carry out several computations – some old, some new – of the homotopy groups of telescopes of self-maps of finite spectra. What is new is joint work with Michael Andrews.

## Wednesday, July 15

15:00 - 16:00 Mark Behrens: The structure of the Morava E-cohomology of QX

Abstract: I will describe what we know about the structure of the Morava E-cohomology of QX, as an algebra over the Morava E-theory Dyer-Lashof algebra. This is part of a project with Charles Rezk.

This talk takes place in the HIM lecture hall.

## Wednesday, July 22

15:00 - 16:00 Andy Baker: E-infinity ring spectra built from small complexes

Abstract: I will discuss some examples of E-infinity ring spectra built form small cell complexes using the free algebra functor. These also turn out to be E-infinity Thom spectra with orientations for HZ, kO or tmf. I will end with some conjectures about their homotopy types.

This talk takes place in the HIM lecture hall.

## Wednesday, July 29

15:00 - 16:00 John Greenlees: Gorenstein duality and Gross-Hopkins duality

Abstract: The talk will start by recalling the notion of Gorenstein duality, illustrating it with chromatic examples and explain how Gross-Hopkins duality fits into this framework. This is based on joint work with Dwyer and Iyengar from a few years ago: it is an advertisement for the framework and will contain little novelty.

## Wednesday, August 5

15:00 - 16:00 Hans-Werner Henn: K(2)-local resolutions at the prime 2

## Wednesday, August 12

15:00 - 16:00 Agnes Beaudry: Chromatic splitting: the way things work at p=2

Abstract: Recent computations show that the decomposition of L_{1}L_{K(2)}S predicted by the chromatic splitting conjecture does not hold at p=2. I will explain how the situation differs and propose a different decomposition for L_{1}L_{K(2)}S in this case.