Stable homotopy theory: classical calculations and modern structures

Semaine spéciale/Summer school

Strasbourg (France), May 7-11, 2007

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Program

This "summer school type" activity on stable homotopy theory is part of the Graduiertenkolleg "Homotopy and Cohomology" at the universities of Bonn, Bochum and Düsseldorf and the Ecole Doctorale  of the University of Strasbourg.

The program will start on Monday morning at 9:00 am and end on Friday at 3:20pm. Here is the schedule for the week and here you find information  how to get there.

There will be three series of 5 lectures each by

Here is a tentative list of topics for each of the series:

Calculations in stable homotopy theory, from quantitative towards qualitative phenomena   (Here are preliminary notes for these lectures)
- The Classical Adams Spectral Sequence:  Review and definitions, Adams periodicity. 
- Applying the geometry of formal groups: The Adams-Novikov Spectral Sequence.  Hopf algebroids; the E_2 term. Chromatic spectral sequence; change of rings. Image of J. Higher order periodic phenomena.
- Monochromatic Calculations: what formal groups of single height can say: Chromatic convergence and fracture squares. Group cohomology and collapsing results. Some calculations. Topological decompositions.

Structured spectra (The following preprints are particularly relevant for this series: [1] [2])
-  Motivation: reminder of spectra as representing objects and as stable spaces. Things one might want to do with structured spectra  (Geometric topology, algebraic K-theory, brave new algebra, derived algebraic geometry).
- EKMM spectra, symmetric spectra, orthogonal spectra. Diagram spectra. Examples.
- Applications. Possibilities include outline of the use of THH and TC to calculate algebraic K-theory. Completion theorems.  Morita theorems and classification results. Brave new commutative algebra, Gorenstein ring spectra and duality theorems.

Model categories and rigidity in stable homotopy theory (The following papers/preprints are particularly relevant for this series: [3] [4] [5])
- Basics on model categories: axioms, homotopy category, spaces and chain complexes as examples
- More on model categories:  Quillen functors/equivalences, simplicial sets, simplicial model categories, more examples. 
- Spectra as a model category: sequential (aka "Bousfield-Friedlander-") spectra with the stable model structure. Universal property: spectra are "free stable model category on one generator".   
- Rigidity theorem: the stable homotopy category has a  unique model up to Quillen equivalence

In addition to these lecture series there will be a series of talks given by students on topics related to these lectures series.  (Here is more information on these talks.)

Prerequisites

Besides an introductory course into algebraic topology (covering singular homology, cohomology and elementary homotopy theory like fibrations, cofibrations, CW-complexes and the Theorems of Whitehead and Hurewicz) some acquaintance with the following topics is recommended: the homotopy category of spectra, Bousfield localization, the Adams spectral sequence for a generalized cohomology theory, the Steenrod algebra and the classical Adams spectral sequence, complex oriented cohomology theories and formal group laws. (The Lazard ring and Quillen's theorem).

Special Talk

There will also be a talk on Wednesday, May 9, at 2:45 pm by Klaus Volkert (Universität Köln) on "Poincare on his way to his conjecture", which aims at a wider audience and which touches on the historical origin of our subject.

Registration Form and Accomodation

We will be happy to help you finding accomodation if you register before April 16.

Travel Information

Organizers

Hans-Werner Henn (Strasbourg), Stefan Schwede (Bonn)