Urtzi Buijs (Universitat de Barcelona): Rational homotopy of mapping spaces. An L1-algebra approach.
In this talk we recall the different approaches to the study of the
rational homotopy type of mapping
spaces, from the work of René Thom [6] back in the fifties, to the L1
models developed on [3, 4],
without forgetting Haefliger-Brown-Szczarba model [5, 1].
As an application of the L1-algebra approach, we show how these models
allows us to improve some
known bounds for the rational Whitehead-length of mapping spaces.
References
[1] E. H. Brown and R. H. Szczarba, On the rational homotopy type of
function spaces, Trans.
Amer. Math. Soc. , 349 (1997), 4931--4951.
[2] U. Buijs, Y. Félix and A. Murillo, Lie models for the components
of sections of a nilpotent
fibration, Trans. Amer. Math. Soc. 361(10) (2009), 5601--5614.
[3] U. Buijs, Y. Félix and A. Murillo, L1 models of mapping spaces,
to appear in J. of the Math.
Soc. of Japan.
[4] U. Buijs, An explicit L1 structure for the rational homotopy type
of mapping spces components,
Preprint (2010).
[5] A. Haefliger, Rational Homotopy of the space of sections of a
nilpotent bundle, Trans. Amer.
Math. Soc., 273 (1982), 609--620.
[6] R. Thom, L’homologie des espaces fonctionnels, Colloque de
topologie algébrique, Louvain,
(1957), 29--39.
Antonio Díaz (Universidad de Málaga): Tate's theorem for fusion systems.
It was a question of Atiyah whether the condition that the mod-p
cohomology restriction map between a finite group G and its Sylow
p-subgroup is an isomorphism was sufficient for the existence of a normal
p-complement in G. Tate's theorem gives a positive answer to this
question. In this talk we will explain a fusion system version of this
theorem which proof involves the notions of p-group residuals and transfer
maps in cohomology for fusion systems. This is a joint work with Adam
Glesser, Sejong Park and Radu Stancu.
John Greenlees (University of Sheffield): Rational equivariant cohomology theories, Hasse squares and rigidity of derived categories.
The main result (with Brooke Shipley) is a complete and calculable
algebraic model of rational equivariant cohomology theories for tori.
The formal framework of the result may be of wider interest, since it is
based on patching together rigidity statements for commutative DGAs
with polynomail cohomology (for example this applies to derived
categories of sheaves over an elliptic curve).
Radha Kessar (University of Aberdeen): Finiteness conjectures in modular representation theory and Hochschild cohomology.
Many problems in modular representation theory of finite groups are concerned with
the relationship between p-local invariants of blocks of finite group algebras
and global representation theoretic invariants of the blocks. In this context, I will present joint work with M. Linckelmann on bounding the dimensions of Hochschild cohomology groups of blocks in terms of the underlying defect.
Birgit Richter (Universität Hamburg): Brauer groups for commutative S-algebras.
This is a talk on joint work with Andy Baker and Markus Szymik. We investigate a notion of Azumaya algebras in the context of structured ring spectra and give a definition of Brauer groups. In the talk I will present some of their Galois theoretic properties and discuss examples of Azumaya algebras arising from Galois descent such as a quaterionic extension of real topological K-theory and if time permits give examples that are related to topological Hochschild cohomology of group ring spectra.
Bruno Vallette (Université de Nice and MPIM Bonn): Homotopy Batalin-Vilkovisky algebras.
In this talk, I will survey the recent developments on the
homotopy theory of Batalin-Vilkovisky algebras. For instance, I will
give three different resolutions of the operad encoding BV-algebras:
the Koszul one, the minimal one and a relative one. I will give
applications on double loop spaces, Topological Conformal Field
Theories, vertex algebras, Kontsevich Formality moduli space of curves
and Gromov-Witten invariants.
Fei Xu (Universitat Autònoma de Barcelona): Transporter categories and transfer.
Let G be a finite group and P a finite G-poset. We consider the Grothendieck construction over P, a finite category G*P which will be called a transporter category. There exists a canonical functor from G*P to G, and it allows us to compare the cohomology of these two categories (a group is a category of a single object). As an example, if P=G/H for some subgroup H, the transporter category G*(G/H) is equivalent to H itself (indeed H is the skeleton of G*(G/H)). Suppose k is a field. Using Kan extensions and representations of category algebras, we establish a version of the Becker-Gottlieb transfer: for any kG-modules m and n, there are induced k(G*P)-modules M and N such that we have a restriction and a transfer maps as follows
Ext*_{kG}(m,n) --> Ext*_{k(G*P)}(M,N) --> Ext*_{kG}(m,n),
which compose to the scalar multiplication by the Euler characteristic of P. This way of construction, borrowing ideas from Dwyer-Wilkerson, is entirely analogues to the classical situation for H < G.
