Project description
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.
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
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Keywords
Programme(s)
Funding Scheme
ERC-COG - Consolidator GrantHost institution
2000 Antwerpen
Belgium