The recipient of the de la Vallée Poussin Chair 2023 is Professor Kathryn Hess Bellwald, professor at the Ecole Polytechnique Fédérale de Lausanne.
The general title of the lectures is: "Homotopical perspectives on algebra and geometry".
Photo credit: Eddy Mottaz, Le Temps
Programme :
Wednesday November 29, at 4.30 pm (Auditoire de la Vallée Poussin ) :
Inaugural lecture: Hochschild homology: from classical to modern
Abstract: Hochschild homology of associative rings generalizes the notion of Kähler 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.
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. 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 “shadow”, in the sense of Ponto.
(Based on collaborations with Paul-Eugène Parent, Jonathan Scott, Brooke Shipley, and Nima Rasekh)
Thursday November 30, at 4.30 pm (Auditoire de la Vallée Poussin)
2nd lecture: 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. For example, if a Hopf algebra H 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. 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)
Friday December 1st, at 10.45 am (auditoire Sud 09)
3rd lecture: Configuration spaces for product manifolds
Abstract: For any topological space X, the configuration space Confn(X) of n points in X is the subspace of the iterated product Xn consisting of n-tuples of distinct points in X. Configuration spaces play an important role in many areas of mathematics, in particular low-dimensional topology and homotopy theory. For example, the fundamental group of Confn(R2) is the pure n-stranded braid group, while the orbit space of the natural action of the symmetric group, Confn(R2)/Sn, has fundamental group isomorphic to the entire n-stranded braid group. Moreover, Confn(R2)/Sn is homeomorphic to the space of complex monic polynomials of degree n with exactly n roots.
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. 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)
Friday December 1st at 1 pm (Auditoires A.03, Faculty of Sciences)
(Lecture in french) Exposé grand public: Aventures mathématiques en neurosciences
Abstract: Le cerveau de chacun d'entre nous est composé de centaines de milliards de neurones (ou cellules nerveuses) reliés par des centaines de billions de synapses, qui transmettent des signaux électriques d'un neurone à l'autre. En réaction à un stimulus, des ondes d'activité électrique traversent le réseau de neurones, traitant le signal entrant. Les outils fournis par le domaine des mathématiques appelé topologie algébrique nous permettent de détecter et de décrire la riche structure cachée dans cette tapisserie dynamique.
Au cours de cet exposé, je vous guiderai dans une visite mathématique mystérieuse de ce que la topologie révèle sur la façon dont le cerveau traite l’information.
(Collaboration avec le Blue Brain Project)