Advances In Homotopy Theory

29 juin 2015, 10:00 → 1 juil. 2015, 13:00 Europe/Paris

Advances in Homotopy Theory - a conference in honour of Hans-Werner Henn on the occasion of his 60th birthday

The conference will be held in Strasbourg (France), from June 29 (10:00) to July 1st 2015 (13:00).

Invited speakers Mark Behrens (Notre Dame) Paul Goerss (Northwestern) John Greenlees (Sheffield) Jean Lannes (Paris 7) Gerd Laures (Bochum) Niko Naumann (Regensburg) Bob Oliver (Paris 13) Charles Rezk (Illinois at Urbana-Champaign) Lionel Schwartz (Paris 13) Vesna Stojanoska (MPIM Bonn) Peter Symonds (Manchester) Christine Vespa (Strasbourg)

Registration

All participants for the conference are required to register. Registration is free and closes 22 June 2015.

Financial support

A limited amount of funding is available for young participants (PhD students and young post-docs). Requests for financial support must be made (by contacting one of the organizers) before 30 April 2015.

Accommodation

Participants should arrange their own accommodation. A list of hotels is available at http://www-irma.u-strasbg.fr/article423.html. Amongst the convenient hotels are:

• Hôtel Esplanade - close to the University
• EtC - in the town centre
• Cap Europe - for those wishing to share an appartment.

Organizing commitee

Christian Ausoni (Paris 13), Gaël Collinet (Strasbourg), Pierre Guillot (Strasbourg), Geoffrey Powell (Angers), Stefan Schwede (Bonn)

Partenaires

lundi 29 juin

10:00 Registration AND Coffee

1 Resolutions and duality in K(2)-local homotopy theory

Some time ago, Goerss, Henn, Mahowald, and Rezk produced a topological resolution of the K(2)-local sphere at the prime 3. Along with the emergence of topological modular forms, this was one of the events that helped organize our thinking about the entire K(2)-local category. The occasion of this conference seems a good time to revisit these constructions and some of the theorems that followed, both new and old.

Orateur: Paul Goerss (Northwestern)

2 Invertible modules over higher real K-theory spectra

For a finite subgroup G of the Morava stabilizer group of height n at a prime p, there is an associated ring spectrum EO = (E_n)^{hG} of homotopy fixed points of Morava E-theory E_n under its G-action. The spectra EO carry some tractable information about the K(n)-local sphere. We study their Picard groups, and show that, when n=p-1, Pic(EO) is always cyclic. This is joint work with Drew Heard and Akhil Mathew.

Orateur: Vesna Stojanoska (MPI Bonn)

13:00 Lunch

3 Characterististic classes in TMF_0(3)

We develop a theory of characteristic classes for the cohomology of topological modular forms with \Gamma_0(3)-level structures. These classes are supposed to play an important role in the determination of the string bordism ring. It turns out that the TMF_0(3)-cohomology of BString is freely generated by Pontryagin classes and one extra class which relates to the theory of cubical structures for line bundles on elliptic curves.

Orateur: Gerd Laures (Bochum)

16:00 Coffee

4 Calculating with power operations for Morava E-theory at height 2

There is a robust theory of power operations for Morava E-theory. At height 2, this theory is sufficiently robust to allow complete calculations. We provide some examples.

Orateur: Charles Rezk (UIUC)

mardi 30 juin

5 Group actions on rings and group cohomology

We consider a finite group G acting on a polynomial ring k[V] and try to understand the multiplicity of a given indecomposable kG-summand in terms of commutative algebra. We then try and do the same for cohomology of a group, considering it as a Mackey functor on the subgroups of G.

Orateur: Peter Symonds (Manchester)

10:00 Coffee

6 Homology of functors from groups to abelian groups

Functor homology (i.e. homological algebra in functor categories) on a suitable category allow us to compute some stable homology of linear groups, orthogonal groups or symplectic groups with twisted coefficients. Little is known concerning stable homology of automorphism groups of free groups with twisted coefficients. We have cancellation results, several computations in small degree by Satoh and explicit classes constructed by Kawazumi. Homology of functors from groups to abelian groups should help us to better understand stable homology of automorphism groups of free groups with twisted coefficients. In this talk, I will explain the previous motivation for the study of homology of functors from groups to abelian groups and I will give several results concerning this functor homology.

Orateur: Christine Vespa (Strasbourg)

7 Automorphisms of p-completed classifying spaces of groups of Lie type

(See PDF.)

Orateur: Bob Oliver (Paris 13)

summary abstract_Oliver.pdf

13:00 Lunch

8 Descent up to nilpotence in equivariant stable homotopy theory

We will report on joint work (partially in progress) with Akhil Mathew and Justin Noel about descent in an equivariant context. This allows to revisit and extend various classical results of Quillen, Hopkins/Kuhn/Ravenel, and Thomason and leads to a number of interesting questions and conjectures.

Orateur: Niko Naumann (Regensburg)

16:00 Coffee

9 On some resolutions in the category HV_{fg}-U

See PDF.

Orateur: Jean Lannes (Paris 7)

summary Lannes_HWH.pdf

19:30 Banquet

mercredi 1 juillet

10 Chromatic Gorenstein descent

It is elementary that ku (with coefficients Z[v]) is Gorenstein of shift -3. It follows that KU (with coefficients Z[v, 1/v]) is Anderson self-dual (with shift one more) and by descent that ko is Gorenstein of shift -5 and KO is Anderson self-dual (of shift one more). There are a number of similar examples arising from topological modular (tmf(2), tmf(3)) and automorphic forms. The talk will describe the general context, how to predict the change in shifts under descent, and some particular cases (Joint work with Vesna Stojanoska).

Orateur: John Greenlees (Sheffield)

10:00 Coffee

11 The ring of cooperations for 2-primary tmf

This talk represents joint work with Kyle Ormsby, Nat Stapleton, and Vesna Stojanoska. I will describe 3 different perspective on tmf_*tmf (localized at 2). (1) the E_2-term of the Adams spectral sequence for tmf_*tmf decomposes as a direct sum of ext groups associated to bo-Brown-Gilter modules, (2) modulo torsion, tmf_*tmf embeds into the ring of 2-variable modular forms, and (3) modulo v_2-torsion, tmf_*tmf embeds into a product of TMF_0(N)_*, for various values of N, and how they are related.

Orateur: Mark Behrens (Notre Dame)

12 Around Takayasu's cofiber sequences

S. Takaysu introduced certain cofiber sequences which generalize the identification of a stunted infinite projective space with a certain Thom space. These sequences occur within the context of the Arone-Goodwillie tower for example. We will describe a new construction of these sequences. The construction uses certain properties of Lannes' T functor which will be discussed. (joint work with Nguyen Dang Ho Hai)

Orateur: Lionel Schwartz (Paris 13)