Description du projet
Un cadre pour la géométrie algébrique non commutative supérieure et courbe
La géométrie algébrique non commutative (GANC) étudie les propriétés géométriques des duals formels d’objets algébriques non commutatifs. Ce domaine a été le théâtre de la découverte d’un large éventail d’applications aussi bien en mathématiques qu’en physique théorique. La topologie algébrique est une autre branche des mathématiques qui utilise des outils de l’algèbre générale pour étudier les espaces topologiques. Le projet FHiCuNCAG, financé par l’UE, vise à fusionner ces branches en développant une théorie des topologies linéaires supérieures. Les recherches menées dans le cadre du projet permettront de mieux appréhender le problème de la courbure, identifié de longue date, dans la théorie de la déformation algébrique. FHiCuNCAG établira un nouveau cadre pour la GANC, qui tient compte des objets courbes, s’inspirant du domaine des catégories supérieures.
Objectif
With this research programme, inspired by open problems within noncommutative algebraic geometry (NCAG) as well as by actual developments in algebraic topology, it is our aim to lay out new foundations for NCAG. On the one hand, the categorical approach to geometry put forth in NCAG has seen a wide range of applications both in mathematics and in theoretical physics. On the other hand, algebraic topology has received a vast impetus from the development of higher topos theory by Lurie and others. The current project is aimed at cross-fertilisation between the two subjects, in particular through the development of “higher linear topos theory”. We will approach the higher structure on Hochschild type complexes from two angles. Firstly, focusing on intrinsic incarnations of spaces as large categories, we will use the tensor products developed jointly with Ramos González and Shoikhet to obtain a “large version” of the Deligne conjecture. Secondly, focusing on concrete representations, we will develop new operadic techniques in order to endow complexes like the Gerstenhaber-Schack complex for prestacks (due to Dinh Van-Lowen) and the deformation complexes for monoidal categories and pasting diagrams (due to Shrestha and Yetter) with new combinatorial structure. In another direction, we will move from Hochschild cohomology of abelian categories (in the sense of Lowen-Van den Bergh) to Mac Lane cohomology for exact categories (in the sense of Kaledin-Lowen), extending the scope of NCAG to “non-linear deformations”. One of the mysteries in algebraic deformation theory is the curvature problem: in the process of deformation we are brought to the boundaries of NCAG territory through the introduction of a curvature component which disables the standard approaches to cohomology. Eventually, it is our goal to set up a new framework for NCAG which incorporates curved objects, drawing inspiration from the realm of higher categories.
Champ scientifique
Programme(s)
Régime de financement
ERC-COG - Consolidator GrantInstitution d’accueil
2000 Antwerpen
Belgique