小瓢虫传媒

听

听

Cette ann茅e la听Chaire de la Vall茅e Poussin est听attribu茅e 脿 la professeure Kathryn Hess Bellwald, professeure 脿 l鈥櫭塩ole Polytechnique F茅d茅rale de Lausanne.

Le titre g茅n茅ral des expos茅s est le suivant : 芦 Homotopical perspectives on algebra and geometry听禄 .听

听

听

听

听

Cr茅dit photo : Eddy Mottaz, Le Temps

Programme :

Mercredi 29 novembre 脿 16h30 (Auditoire de la Vall茅e Poussin) :
Le莽on inaugurale : Hochschild homology: from classical to modern

Abstract: Hochschild homology of associative rings generalizes the notion of K盲hler differentials from the commutative to the merely associative setting. Hochschild homology, which turns out to be an important tool for non-commutative geometry, satisfies interesting properties such as Morita invariance and provides a useful approximation to algebraic K-theory via the Dennis trace.听

In this lecture, after recalling classical Hochschild homology and motivating its importance, I will describe a multitude of useful generalizations and extensions of this remarkable construction. 听In particular we will see the dual notion of coHochschild homology and a common generalization of Hochschild homology and its dual, explore connections to free loop spaces, and consider analogous constructions in a variety of enriched categories, leading in the end to an understanding of Hochschild homology as a 鈥渟hadow鈥�, in the sense of Ponto.
(Based on collaborations with Paul-Eug猫ne Parent, Jonathan Scott, Brooke Shipley, and Nima Rasekh)
听

Jeudi 30 novembre 脿 16h30 (Auditoire de la Vall茅e Poussin听)
2e le莽on: Homotopical perspectives on Morita and Hopf-Galois theory

Abstract: The study of equivalences between categories of comodules over coalgebras over a field was initiated by Takeuchi and is commonly referred to as Morita-Takeuchi theory. The more general question of when categories of comodules over corings are strictly equivalent categories has also been studied, for corings over k-algebras, for k a commutative ring.

The theory of Hopf-Galois extensions of associative rings, introduced by Chase and Sweedler and by Kreimer and Takeuchi, generalizes Galois theory of fields, replacing the action of a group by the coaction of a Hopf algebra.听For example, if a Hopf algebra H听is the coordinate ring of a quantum group, then an H-Hopf-Galois extension can be viewed as a noncommutative torsor with the quantum group as its structure group. It can moreover be fruitful to study Hopf algebras via their associated Hopf-Galois extensions, just as algebras are studied via their associated modules.

In this talk, I will first address a homotopical version of Morita-Takeuchi theory, which arose out of attempts to understand the relations among homotopic Hopf-Galois extensions, Grothendieck descent, and Koszul duality, each of which can be expressed as asserting a derived equivalence between particular corings. 听I will then describe the theory of homotopic Hopf-Galois extensions, in terms of the homotopical Morita theory of comodules, providing in particular a descent-type characterization of homotopic Hopf-Galois extensions of finite-type differential graded algebras over a field.
(Joint work with A. Berglund)听

听

Vendredi 1er d茅cembre 脿 10h45 (auditoire Sud 09)
3e le莽on : Configuration spaces for product manifolds

Abstract:听For any topological space X, the configuration space Conf_n(X)听of n听points in X听is the subspace of the iterated product Xⁿ听consisting of n-tuples of distinct points in X. 听Configuration spaces play an important role in many areas of mathematics, in particular low-dimensional topology and homotopy theory. 听For example, the fundamental group of Conf_n(R²)听is the pure n-stranded braid group, while the orbit space of the natural action of the symmetric group, Conf_n(R²)/S_n, has fundamental group isomorphic to the entire n-stranded braid group. 听Moreover, 听Conf_n(R²)/S_n听is homeomorphic to the space of complex monic polynomials of degree n听with exactly n听roots. 听

In this talk I will provide a brief overview of the theory of configuration spaces, then describe the connection between configuration spaces and little disks operads, which encode operations and relations among operations in iterated loop spaces. 听To conclude I will explain a method for computing homotopy invariants of the configuration space of a product of two closed manifolds in terms of the configuration spaces of each factor separately that exploits this relationship with operads.
(Joint work with W. Dwyer and B. Knudsen)听

听

Vendredi 1er d茅cembre 脿 13h听 (Auditoires de Sciences A.03)
Expos茅 grand public :

Abstract: Le cerveau de chacun d'entre nous est compos茅 de centaines de milliards de neurones (ou cellules nerveuses) reli茅s par des centaines de billions de synapses, qui transmettent des signaux 茅lectriques d'un neurone 脿 l'autre. En r茅action 脿 un stimulus, des ondes d'activit茅 茅lectrique traversent le r茅seau de neurones, traitant le signal entrant. Les outils fournis par le domaine des math茅matiques appel茅 topologie alg茅brique nous permettent de d茅tecter et de d茅crire la riche structure cach茅e dans cette tapisserie dynamique.听

Au cours de cet expos茅, je vous guiderai dans une visite math茅matique myst茅rieuse de ce que la topologie r茅v猫le sur la fa莽on dont le cerveau traite l鈥檌nformation.听
(Collaboration avec le Blue Brain Project)

听