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

• Paul Goerss (Northwestern)

Room: Grand Amphi de Maths 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.

Invertible modules over higher real K-theory spectra

• Vesna Stojanoska (MPI Bonn)

Room: Grand Amphi de Maths 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.

Characterististic classes in TMF_0(3)

• Gerd Laures (Bochum)

Room: Grand Amphi de Maths 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.

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

• Charles Rezk (UIUC)

Room: Grand Amphi de Maths 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.

Group actions on rings and group cohomology

• Peter Symonds (Manchester)

Room: Grand Amphi de Maths 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.

Homology of functors from groups to abelian groups

• Christine Vespa (Strasbourg)

Room: Grand Amphi de Maths 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.

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

• Bob Oliver (Paris 13)

Room: Grand Amphi de Maths (See PDF.)

Descent up to nilpotence in equivariant stable homotopy theory

• Niko Naumann (Regensburg)

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.

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

• Jean Lannes (Paris 7)

Room: Grand Amphi de Maths See PDF.

Chromatic Gorenstein descent

• John Greenlees (Sheffield)

Room: Grand Amphi de Maths 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).

The ring of cooperations for 2-primary tmf

• Mark Behrens (Notre Dame)

Room: Grand Amphi de Maths 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.

Around Takayasu's cofiber sequences

• Lionel Schwartz (Paris 13)

Room: Grand Amphi de Maths 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)