Abstracts

Hess: (Joint work with J. Rognes) Let t be a twisting cochain from a connected, coaugmented chain coalgebra C to an augmented chain algebra A over an arbitrary PID R. I'll explain the construction of a twisted extension of chain complexes H(t) of which both the Hochschild complex of an associative algebra and the coHochschild complex of a coassociative coalgebra are special cases. We call H(t) the Hochschild complex of t.

When A is a chain Hopf algebra, I'll give conditions under which H(t) admits an rth-power map extending the usual rth-power map on A and lifting the identity on C. In particular, both the Hochschild complex of any cocommutative Hopf algebra and the coHochschild complex of the normalized chain complex of a double suspension admit power maps. Moreover, if K is a double
suspension, then the power map on the coHochschild complex of the normalized chain complex of K is a model for the topological power map on the free loops on K, illustrating the topological relevance of our algebraic construction.

The first and second lectures will be devoted to recalling the necessary algebraic background material, as well as the history of the rth-power maps in algebra and topology, then to explaining the construction and algebraic properties of the Hochschild construction H(t). In the third lecture I will explain the link with topology.

 

Naïto: Je vais commencer par définir les coalgèbres d'Alexander-Whitney et expliquer en quoi ces coalgèbres sont intéressantes pour un topologue. Ensuite, je donne une description opéradique des coalgèbres d'Alexander-Whitney. Et finalement, je parlerai d'une notion de modèle minimal pour ces coalgèbres.

 

Harper: I will explain Mandell's results concerning the E-infinity algebra structure of the singular cochains on a topological space.

 

Vallette: La notion d'algèbre de Batalin-Vilkovisky joue un role important en physique mathématique (algebres vertex), en topologie des cordes (opérations en homologie) et en topologie algébrique (espaces de lacets itérés, conjecture de Deligne cyclique). Dans chacun des cas, une notion relâchée à homotopie près s'avèrerait très utile. Grâce à une théorie de dualité de Koszul généralisée, je donnerai la définition des algèbres de Batalin-Vilkovisky à homotopie près et démontrerai un théorème de Poincaré-Birkhoff-Witt pour l'opérade associée. J'étudierai ensuite le théorie de déformation ainsi que la théorie homotopique de telles algèbres. Ceci permettra de démontrer une version forte de la conjecture de Lian-Zuckerman sur les algebres vertex. J'expliciterai aussi le lien avec les structures de champs conformes topologiques.

 

Dwyer: Lecture I will be an introduction to homotopy theories and model categories, as well as to homotopy limits and colimits. During Lecture II, I will talk about how to get spaces from categories and about localization. Finally, the subjects of Lecture III will be cohomology of function spaces and maps between classifying spaces.

 

Hoyois: Je vais d'abord parler du caractère de Chern classique, puis je présenterai un cadre général permettant la définition d'un caractère de Chern. J'expliciterai ensuite cette construction dans le contexte de la géométrie algébrique dérivée.

 

Dotto: En un premier temps on va expliquer les bases de théorie de l'indice, et en particulier le théorème du point fixe d'Atiyah-Singer. Cette théorie étudie les représentations d'un groupe de Lie G associées à une certaine classe d'operateurs au dérivées partielles définis entre des espaces de sections de G-fibrés vectoriels. On va ensuite appliquer cette théorie à l'opérateur de Dirac d'une variété de spin. Ceci nous donnera une formule pour le genre de Witten trunqué, que nous utiliserons pour définir le genre de Witten. Nous allons ensuite énoncer une conjecture reliant le genre de Witten d'une
variété de corde M et l'indice d'un opérateur de Dirac sur l'espace des lacets lisses LM. Pour conclure on va considerer un cas équivariant du genre de Witten trunqué, et on verra dans quels cas il est possible de l'utiliser pour définir un genre de Witten équivariant.

 

Dror-Farjoun : (Joint work with Dwyer and Prezma.) The localization and cellularization of a principal fibration G--->X-->B is still a principal fibration under mild conditions, although the nature of the new group that arises is still hard to unlock. One way to approach this problem is by means of "normal maps" of loop spaces, which is a homotopy version of a normal subgroup. We show how to build a Segal-like machine that characterizes normal maps and that enables us to show they are preserved under product-preserving homotopy functors.

 

Skoda : Nonabelian cocycles are related to weak functors between higher categories; thus their combinatorics is difficult when the degree is large, say if already larger than 2. However, in most applications the higher categories themselves can be taken to be strict, which is a great simplification, while the weak functors are still an issue.

The category of strict omega-categories has a Quillen model category structure which has been recently studied. Higher nonabelian cocycles of a space X with coefficients in a presheaf of omega-categories A by our definition form an omega-category which is enriched hom from a cofibrant replacement of some presheaf of omega categories related to X (or a (hyper)cover of X; e.g. fundamental, path and Cech groupoids are examples) to A. Sometimes, one needs to use additional colimit procedures (e.g. for refinement of covers). The cofibrant replacement involves combinatorics similar to Street orientals; however the latter suffice only if the coefficient presheaf A is a presheaf of omega-groupoids. This is a work in progress with U. Schreiber, D. Stevenson and H. Sati.

 

Seal : After reviewing the theory of monads, we will investigate how certain monads on SET can be modified to become monads on categories of Kleisli monoids (these include for example the category ORD of ordered sets, or TOP of topological spaces). The new monads turn out to have the same Eilenberg-Moore category as the original ones, but allow for a more transparent description of the corresponding Eilenberg-Moore algebras. This simplified presentation will lead us to the identification of injective objects in the base category of Kleisli monoids.

 

Shamir : Local cohomology has proved to be a useful tool in commutative algebra. Local cohomology has also made surprising appearances in algebraic topology, as in the works of Benson and Greenlees and others. For example, Benson and Greenlees' results provide a sort of duality statement for the cohomology of certain classifying spaces. I will "define" local cohomology in simple terms (in fact I will use Dwyer and Greenlees' equivalent definition) and explain some of its uses in algebraic topology. I will also offer a way to generalize some of these results into a non-commutative setting.

van der Zypen : Priestley duality is a "dictionary" between distributive lattices and compact, totally order-disconnected topological spaces (Priestley spaces). We give an introduction as well as examples for lattice theoretical problems that are more treatable when "translated" into the language of Priestley spaces. Finally, we briefly discuss some open problems.

 

Ching : I'll explain joint work with Greg Arone that decomposes the Goodwillie tower of a functor from spaces to spaces. We construct an approximation to the tower built from a bimodule structure on the derivatives of the functor, and show that the fibre of the map from the real tower to the approximation can be described in terms of Tate spectra. In particular, in cases where the Tate spectra vanish, such as rationally, we obtain models for the Goodwillie tower explicitly in terms of this bimodule. I'll also mention our plans to apply a result of Nick Kuhn on vanishing Tate spectra to this setting.

 

Bonanomi : Les extensions de Hopf-Galois d'anneaux généralisent les extensions de Galois de corps. Sous certaines conditions, on peut introduire une théorie d'extensions de Hopf-Galois homotopiques dans une catégorie munie d'une structure monoïdale et d'une structure modèle. Je vais en donner un exemple dans la catégorie des algèbres différentielles commutatives graduées rationnelles. L'étude de cette catégorie est essentiel à la compréhension de la théorie de l'homotopie rationnelle.

 

Genton : In order to study the classifying space of a group G, a filtration by principal G-bundles of the universal G-bundle is constructed using the spaces of homomorphisms from the descending central series into G and a simplicial structure similar to the bar construction of classifying spaces. The second stage of this filtration is built upon the commuting elements Z^n -> G. An isomorphism allows us to compute its rational cohomology using the maximal torus of Lie group G. SU(2) is given as an example.

 

Lack : There is a well-known Quillen model structure on the category Cat of categories and functors for which the weak equivalences are the equivalences of categories. This restricts to a model structure on the full subcategory Gpd of Cat consisting of the groupoids, and this provides a model for (not necessarily connected) homotopy 1-types. I will describe how this "n=1" case (1-categories, 1-groupoids, 1-types) can be extended to the cases of n=2 and n=3. In each of these cases there is a model structure on the category of all n-dimensional categories. Once again, there is a restricted model structure on the full subcategory consisting of those n-dimensional categories for which all arrows at all dimensions are invertible; and this provides a model for n-types. In the case n=3, the n-dimensional categories considered are the Gray-categories of the title. I will build up from n=1 to n=2 and n=3 in an inductive sort of way, although I do not know how to continue the induction to deal with higher n.

 

Lesh : I will discuss connections between the calculus of functors and the Whitehead Conjecture, both for the classical theorem of Kuhn and Priddy for symmetric powers of spheres and for the analogous conjecture in topological K-theory. It turns out that key constructions in Kuhn and Priddy's proof have bu-analogues, and there is a surprising connection to the stable rank filtration of algebraic K-theory.

 

Dernière mise à jour: 13.08.09