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### Université de Nantes, May 22-24, 2013 — Conference in Algebra and Topology Programme scientifique Scientific program

The program is chosen by the Scientific Committee

Christian Ausoni
Topological Hochschild Homology of K-theory

Abstract:
I will present a computation of the integral topological Hochschild homology of complex K-theory, inspired by computations for number rings by Hesselholt, Lindenstrauss and Madsen.

Andrew Baker
Commutative S-algebras and topological Andre-Quillen theory

Abstract:
I will discuss properties of topological Andre-Quillen homology and show how to do some calculations with Thom spectra and other examples. These involve old friends such as Dyer-Lashof operations.

Georg Biedermann
Moduli spaces for realizing unstable coalgebras

Abstract:
Singular homology with coefficients in a prime field can be viewed as a functor from topological spaces to unstable coalgebras. Given an unstable coalgebra, one can ask if there exists a topological space whose homology is isomorphic to it, and, if yes, how many different realizations exist. We define moduli spaces of realizations as an inverse limit of moduli spaces of intermediate steps following the approach by Blanc, Dwyer and Goerss for $\Pi$-algebras. The obstructions lie in Andre-Quillen cohomology groups of unstable coalgebras which are part of the $E_2$-term of the unstable Adams spectral sequence.
(joint with G. Raptis and M. Stelzer)

Natàlia Castellana
Normalizable classifying spaces

Abstract:
The notion of a normalizable space (at a prime p) was introduced by Benson, Greenlees, and Shamir. A space X is normalizable if there is a p-compact group Y and a map into BY so that the homotopy ﬁbre is a mod p finite complex. We say that X is complex normalizable if Y is a unitary group. Classical examples are given by classifying spaces of finite groups and compact Lie groups. But classifying spaces of Kac-Moody groups are not complex normalizable.
I will report on projects with L. Morales and J. Cantarero where we study this property for classifying spaces of p-local finite and p-local compact groups. These algebraic structures introduced by Broto, Levi and Oliver model the mod p homotopy theory of classifying spaces of finite and compact Lie groups. As a corollary, classifying spaces of finite loop spaces are normalizable at every prime p.

Takuji Kashiwabara
A cellular construction of the Brown-Peterson spectrum

Abstract:
The Brown-Peterson spectrum BP is known to have torsion-free homotopy groups and homology groups, both concentrated in even degrees, just like the complex cobordism spectrum MU.
As a matter of fact BP was originally constructed as a spectrum that realizes the quotient of the Steenrod algebra by the ideal generated by the bockstein by a series of fibration building upon Eilenberg-MacLane spectrum HZ/p. Later, Quillen showed that it can be split off the complex cobordism spectrum. Then Priddy gave a cellular construction of BP by killing successively all homotopy groups appearing in odd dimensions, which can be considered as a sort of dual construction to the original one, albeit less explicit.
On the other hand, using the Steinberg idempotent, Mitchell and Priddy showed that the spectrum HZ/p can be "filtered'' by a sequence of spectra D(n)'s' realizing the length filtration of the Steenrod algebra, and that M(n)=D(n)/D(n-1) can be split off the classifying space of the elementary abelian group B(Z/p x ... x Z/p), as well as that of the orthogonal group BO(n). They also constructed a "complex counterpart'' of M(n), which they call BP(n), that splits off the classifying space of the n-dimensional torus, as well as that of the unitary group BU(n).
Now, as to MU, aside from the classical cellular structure, there is another cellular structure coming from the spectra MTU(n)'s introduced by Galatius-Madsen-Tillmann-Weiss, the Thom spectra of the virtual bundle orthogonal to the universal bundle over BU(n).
In this talk, we show that we can obtain a cellular structure of BP using the splitting of MTU(n)'s, and a filtration that is "complex counterpart" of D(n), thus realizing the length filtration of reduced powers in terms of Adem-Cartan-Serre basis of the Steenrod algebra on the cohomology of BP.

Nick Kuhn
The Krull and Nilpotent filtrations of the category of unstable modules

Abstract:
In a 1988 paper and in his 1994 book, Lionel Schwarz introduced the Nilpotent and Krull filtrations of U, the category of unstable modules over the Steenrod algebra. Both of these filtrations have many lovely properties and characterizations, allowing one to "slice and dice" unstable modules and algebras in frutiful ways. Their interactions with J. Lannes' T functor has had diverse topological application: to group cohomology, topological realization questions, and the classification of H-spaces.

Jean Lannes
Hecke operators for even unimodular lattices

Computational approach to the Artinian conjecture

Abstract:
What is generic representation theory? When looking at the category $\mathcal{F}_q=\mathrm{Func}(\mathrm{mod}\mathbb{F}_q,\mathrm{Mod}\mathbb{F}_q)$ we obtain that a functor $F\in\mathcal{F}_q$ generically gives rise to representations of $\mathrm{GL}(V)$ for all $V\in \mathrm{mod}\mathbb{F}_q$. By the Yoneda-lemma we know how certain projectives in $\mathcal{F}_q$ look like. For each $V\in \mathrm{mod}\mathbb{F}_q$, $\mathrm{Hom}(V,-)$ is projective. Such a projective is called a standard projective. It turns out that these standard projective even generate the whole category.
In the 1980s Lionel Schwartz conjectured that all the standard projectives would be noetherian. If true this would imply that every finitely generated functor in $\mathcal{F}_q$ admits a projective resolution by finitely generated projectives. There are partial results that back up this conjecture but no solution so far.
In the talk we will not reach quite as far. The aim is to give an idea why the category $\mathcal{F}_q$ is at least coherent. That means that every finitely presented functor admits a resolution by finitely generated projectives. To get to this goal we will use certain combinatorial properties of the dimension function $\phi(F,n)=\mathrm{dim}_{\mathbb{F}_q}F(\mathbb{F}_q^n)$ for a functor $F\in \mathcal{F}_q$.

Nguyen Dang Ho Hai
Division of the Dickson algebra by the Steinberg unstable module

Abstract:
We compute the division of the Dickson algebra by the Steinberg module in the category of unstable modules over the mod-2 Steenrod algebra. We also describe how to derive from this computation some information about the homotopy type of the Spanier-Whitehead dual of a Thom spectrum.

Jérôme Scherer
Realization of conjugation spaces

Abstract:
This is joint work with Wolfgang Pitsch. I will introduce the beautiful subject of conjugation spaces and conjugation manifolds, as defined by Hausmann, Holm, and Puppe. Roughly speaking they are even dimensional spaces (or manifolds) equipped with an involution such that their mod two cohomology is isomorphic to that of the fixed points after dividing degrees by two. I would like to present an application to equivariant Chern classes for Real bundles in the sense of Atiyah. I will also try to give some answers to the problem of realizing a given space (or manifold) as fixed points of a conjugation space.

Stephen Theriault
The nilpotence class of certain finite loop spaces

Abstract:
A method is presented for determining the nilpotence class of certain finite loop spaces, which is applicable in either a p-local or p-complete setting. The loop spaces G in question are such that there is a space A, a map from A to G which induces the inclusion of the generators in mod-p homology, and G is a retract of the loop suspension of A. This condition is satisfied p-locally by a simple compact Lie group provided the prime is big enough, and by most p-compact groups. The method is an effective tool for calculating the nilpotence class when applied to p-local exceptional Lie groups or to sporadic p-compact groups when the group is homotopy equivalent to a product of spheres and sphere bundles over spheres.

Antoine Touzé
Rational, generic and functor cohomology

Abstract:
We will present some recent progress about understanding the difference between strict polynomial functors (which are related to reductive algebraic groups) and polynomial functors (which are related to finite groups of Lie type), and we will explain an application to the construction of some universal classes for algebraic groups.

Victor Turchin
H-principle in the calculus of embeddings

Abstract:
It is well known that the h-principle fails for spaces of maps avoiding multisingularities that depend on more than one point. However I believe that if one applies carefully the h-principle taking into account configurations with at most k points in the source and target manifolds, then the obtained space is going to be exactly the k-th Goodwillie-Weiss Taylor approximation (to the space of maps $M \to N$ without given multisingularities).  I will show how this construction works for spaces of embeddings. The solution to this geometric-homotopy problem has more of an algebraic flavor --  the approximations are described as spaces of maps between truncated right modules over the framed Fulton-MacPherson operad.

Sarah Whitehouse
Derived A-infinity algebras from the point of view of operads

Abstract:
Derived A-infinity algebras are a recent generalisation of A-infinity algebras, due to Sagave. They provide a framework for a minimal model theorem for differential graded algebras over a general commutative ground ring. Joint work with Livernet and Roitzheim gives an operadic interpretation of these structures.