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Programme scientifique

Scientific program

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.

(joint with Hadi Zare)

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

Phillip
Linke

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.