Abstracts

Denham: A construction originating in work of Davis and Januszkiewicz gives an interesting "combinatorial" family of spaces which are parameterized by a finite simplicial complex and a pair of spaces. With suitable choices, one obtains a range of familiar and less familiar spaces: classifying spaces for right-angled Artin groups, Coxeter groups, and Bestvina-Brady groups; the moment-angle complexes of Buchstaber and Panov; the complements of certain real and complex subspace arrangements. I will try to motivate the construction and describe some interesting aspects of its attendant algebraic topology.

Clementino: In [Van Kampen theorems for categories of covering morphisms in lextensive categories, Top. Appl (1997)] R. Brown and G. Janelidze showed how the classical Van Kampen Theorem can be formulated and generalized using Grothendieck Descent Theory. In this talk I will present the General Van Kampen Theorem of Brown-Janelidze and outline the techniques and results of Topological Descent Theory that can be used to obtain Van Kampen-type theorems.

Karpova: The definition of a well-behaved smash product on the category of symmetric spectra over a model monoidal category C gave birth to algebra of spectra, which offers a generalization of homological algebra, since it makes possible clear categorical definitions of ring, module and algebra spectra. The homotopy theory in this setting is encoded in Quillen model structures. I will present the result of B. Shipley, stating that the category of HZ-algebra spectra is Quillen equivalent to the category of differential graded Z-algebras. The aim will be to explain this statement and to give the essential arguments used in the proof.

Egger: Let k≥2 be an integer. We show that the space of ordered cacti with k lobes is homotopy-equivalent to the k-th ordered configuration space in the complex plane up to translation. We then give an explicit homotopy-equivalence between these two spaces.

Trova: Dendroidal sets have been recently introduced by I. Moerdijk and I. Weiss and should extend the theory of simplicial sets to the context of multicategories. What still lacks is a dendroidal analogue of the geometric realization. I will describe two possible approaches to this problem. In particular I will show how a dendroidal set can be considered as a simplicial set endowed with partial operations. Finally topics and ideas for future research will be outlined and discussed.

Scherer (02.03): L'une des motivations originales de Quillen d'introduire les catégories de modèle était de pouvoir traiter avec les mêmes méthodes l'homotopie classique et l'algèbre homologique. En particulier une résolution injective d'un module peut être vue comme un remplacement fibrant de ce module vu comme un complexe de chaine concentré en degré zéro. Mais comment construire des résolutions injectives (ou projectives) de complexes de chaînes non-bornés? J'expliquerai dans cet exposé les solutions proposées par Spaltenstein, Böckstedt et Neeman, Hovey et d'autres, puis le point de vue que Chacholski, Pitsch et moi-même adoptons, celui des approximations de modèle.

Scherer (09.03): Notre but est de comprendre si l'on peut relativiser les méthodes de constructions de résolutions, injectives ou projectives, de complexes de chaînes non bornés. Nous aimerions donc remplacer la classe des modules injectifs par une nouvelle classe qui jouera le rôle des injectifs et nous permettra de construire des résolutions relatives. Je donnerai quelques exemples pour motiver cela, expliquerai quand notre approche fonctionne, et terminerai par un exemple d'anneau et de classe de modules pour lesquels notre point de vue est trop naïf.

Hausmann: Nous rappellerons la formule classique de Künneth à coefficients dans un corps, avec ses hypothèses, et discuterons de ses généralisations possibles en cohomologie équivariante.

Bujard: D'après un résultat de Dieudonné et Lubin, le groupe Aut_k(F) des automorphismes d'une loi de groupe formel F de hauteur finie n définie sur un corps séparablement clos k de caractéristique positive p est isomorphe au groupe O* des unités de l'ordre maximal de l'algèbre à division centrale D = D(Q_p, 1/n) d'invariant 1/n et de dimension n^2 sur le corps Q_p des nombres p-adiques. On s'intéresse à la classification à conjugaison près des sous-groupes finis de O*=Aut_k(F). Pour ce faire, on commence par établir une classification des classes d'isomorphisme de ces groupes dans D*, avant de l'affiner aux classes de conjugaison, d'abord dans D*, puis dans O*. Cette classification est maintenant achevée et a pour particularité de contredire et invalider les travaux de T. Hewett datant de la fin des années 90 sur la structure des normalisateurs des sous-groupes finis des algèbres à division sur les corps locaux. Les erreurs ont en effet été localisées et on en donnera un contre exemple explicite. Si le temps le permet, on présentera comment cette classification peut être généralisée à la catégorie des lois de groupe formel de hauteur finie définies sur un corps variable parfait de caractéristique positive.

Lukács: The Pontryagin dual of an abelian topological group A is the group of continuous homomorphisms from A into R/Z, equipped with the compact-open topology. By the famous Pontryagin-van Kampen duality, for every locally compact abelian group A, the evaluation map from A into its double-dual is a topological isomorphism. Comfort and Ross introduced in their seminal paper ["Topologies induced by groups of characters"] a duality theory for precompact abelian groups (= dense subgroups of compact abelian groups). Although it is not a secret that the Comfort-Ross duality is based on the Pontryagin-van Kampen one, no satisfactory categorical framework is available in the literature to explain this relationship. (The papers [Barr and Kleisli, "On Mackey topologies in topological abelian groups"] and [Barr, "*-autonomous categories"] witness the difficulty in addressing this relationship.) In this talk, a duality theory of the category of locally precompact abelian groups (= dense subgroups of locally compact abelian groups) is proposed as a unification of the Pontryagin-van Kampen and the Comfort-Ross duality. It turns out that the meaningfulness of the new duality is witnessed by a number of dual properties, that is, pairs (P, Q) of properties such that an object satisfies P if and only if its dual possesses Q.

Shipley: I will discuss various algebraic models, including recent joint work with John Greenlees for G a torus.

Bücher-Karlsson: A classical inequality of Milnor says that the Euler number of flat oriented vector bundles over surfaces (different from the sphere) are at most one half the Euler characteristic of the surfaces. I will discuss generalizations of this seminal inequality and applications to affine manifolds, in particular to a conjecture of Chern saying that an affine manifold has vanishing Euler characteristic. This is joint work with Tsachik Gelander.

Johnson: The calculus of homotopy functors was introduced and developed by T. Goodwillie in a series of three papers published between 1990 and 2003. In these papers he showed that a homotopy functor of topological spaces can be approximated by a sequence of functors and natural transformations, called a Taylor tower, that has properties formally resembling those of a Taylor series for a real-valued function. Since then, the calculus of functors has become an important tool in homotopy theory and algebraic topology. The beginnings of an analogous theory for functors of abelian categories can be found in the work of A. Dold and D. Puppe, and S. Eilenberg and S. Mac Lane. The goals of these lectures are to describe a theory of Taylor towers that extends the work of Dold-Puppe and Eilenberg-Mac Lane, and to show how this theory is related to Goodwillie's work.

Lecture I: The first part of this lecture will be devoted to describing the general properties of Taylor towers. We will then show how to construct Taylor towers for functors of abelian categories using Eilenberg and Mac Lane's notion of cross effects and basic facts about cotriples.

Lecture II: We will establish some key properties (especially conditions for convergence) of the Taylor towers constructed in Lecture 1. The simplicial prolongations of functors of abelian categories are a particularly nice collection of functors with convergent Taylor towers. These functors also play a fundamental role in Dold and Puppe's theory of stable derived functors. We will review this theory and its connections with Taylor towers.

Lecture III: We will present two means of constructing Taylor towers for functors of based spaces. The first of these is Goodwillie's method and the second is an adaptation of the cotriple approach (developed in the first lecture) due to A. Mauer-Oats. We will analyze the differences between these towers and establish conditions under which they agree.

Lecture IV: This lecture will continue the comparison of Goodwillie's towers with towers arising from cotriples. One aspect of this comparison involves the domain categories of the functors. Mauer-Oats' adaptation requires that functors be evaluated on based spaces whereas Goodwillie's techniques do not require that the spaces be based. More generally, N. Kuhn has pointed out that Goodwillie's constructions carry over nicely to topological and simplicial model categories. We will discuss recent work with R. McCarthy and K. Bauer to generalize the cotriple approach to simplicial model categories whose objects are not based.

Müller:

Jeanneret: We determine the homology and cohomology algebras of a regular quotient ring spectrum F of an even E_\infty-ring spectrum R. We show that the homology is a Clifford algebra with respect to a certain bilinear form naturally associated to F. We construct a free and transitive action of the group of bilinear forms Bil(I/I^2[1]) on the set of R-products on F, where F_* is isomorphic to R_*/I. We show that this action induces a free and transitive action of the group of quadratic forms Quad (I/I^2[1]) on the set of equivalence classes of R-products on F. We discuss the examples of the Morava K-theories K(n) and the 2-periodic Morava K-theories K_n.

Dernière mise à jour: 01.06.10