Project description DEENESFRITPL Framework for higher and curved noncommutative algebraic geometry Noncommutative algebraic geometry (NCAG) studies the geometric properties of formal duals of non-commutative algebraic objects. The field has seen a wide range of applications both in mathematics and in theoretical physics. Algebraic topology is another branch of mathematics that uses tools from abstract algebra to study topological spaces. The EU-funded project FHiCuNCAG aims to fuse these branches by developing a higher linear topos theory. Project research will shed further insight into the long-standing curvature problem in the algebraic deformation theory. It will set up a new framework for NCAG, which incorporates curved objects, drawing inspiration from the realm of higher categories. Show the project objective Hide the project objective Objective 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. Fields of science natural sciencesmathematicspure mathematicstopologyalgebraic topologynatural sciencesmathematicspure mathematicsgeometrynatural sciencesmathematicspure mathematicsalgebraalgebraic geometrynatural sciencesphysical sciencestheoretical physics Programme(s) H2020-EU.1.1. - EXCELLENT SCIENCE - European Research Council (ERC) Main Programme Topic(s) ERC-2018-COG - ERC Consolidator Grant Call for proposal ERC-2018-COG See other projects for this call Funding Scheme ERC-COG - Consolidator Grant Host institution UNIVERSITEIT ANTWERPEN Net EU contribution € 1 171 360,00 Address PRINSSTRAAT 13 2000 Antwerpen Belgium See on map Region Vlaams Gewest Prov. Antwerpen Arr. Antwerpen Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Total cost € 1 171 360,00 Beneficiaries (1) Sort alphabetically Sort by Net EU contribution Expand all Collapse all UNIVERSITEIT ANTWERPEN Belgium Net EU contribution € 1 171 360,00 Address PRINSSTRAAT 13 2000 Antwerpen See on map Region Vlaams Gewest Prov. Antwerpen Arr. Antwerpen Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Total cost € 1 171 360,00
