Institut de recherche mathématique avancée

Résumé : Travail avec V. Dotsenko. On obtient une base de Grobner et une description combinatoire complète d'un type d'algebre introduit récemment par A. Das. On décrira en détail la combinatoire, basée sur les partitions non-croisées sans singletons, et quelques propriétés.

Résumé : Dans différentes questions géométriques ou analytiques les objects compacts se comportent mieux. Je commencerai par donner des exemples de ce phénomène, pour ensuite expliquer comment en arithmétique on essaie de l'imiter.

La majorité de l'exposé sera niveau "mémoire de Licence" (autour des formes quadratiques). Vers la fin j'essaierai de dire un mot sur des aspects plus modernes (théorie de l'intersection...), en m'efforçant de rester compréhensible.

Résumé : A commutative ring A is called a binomial ring if it is torsion free over Z and is binomially closed: namely binomial coefficients of all elements of A (viewed as elements of A tensor Q) still lie in A. I will talk about the joint work https://arxiv.org/abs/2308.01110 with Georgii Shuklin and Sasha Zakharov where we studied a derived version of this notion. In the derived context (derived) binomial ring structure is really an extra structure and not a property, and using it can really make a difference. Non-trivial examples of derived binomial rings are given by Z-valued singular cohomology of topological spaces. It turns out that the natural functor X --> C^*_sing(X,Z) to the category derived binomial rings is fully faithful when restricted to a certain natural subcategory of spaces (e.g. simply-connected spaces of finite type). This gives a quite reasonable "algebraic model" for such a space, which really loses almost no information about it. The key step which makes the above fully-faithfulness statement work is the computation of free derived binomial rings LBin(Z[-n]): they turn out to match directly the singular cohomology of Eilenberg-Maclane spaces K(Z,n). If time permits I would also like to compare this with the results of joint work with Shizhang Li where we describe cohomology (e.g. de Rham, crystalline, prismatic or, say, cohomology of structure sheaf) of the higher classifying stacks K(G,n) where G is a commutative group scheme. While here binomial structure on cohomology does not typically appear, the deformed version, with LBin replaced by the divided power algebra LГ, turns out to provide a universal formula in a certain natural class of examples, however only as an E_n-algebra.