Descripción del proyecto
Marco para una geometría algebraica no conmutativa mejor y más precisa
La geometría algebraica no conmutativa (NCAG, por sus siglas en inglés) estudia las propiedades geométricas de los duales formales en objetos algebraicos no conmutativos. Este campo del saber tiene una amplia gama de aplicaciones, tanto en matemáticas como en física teórica. La topología algebraica es otra rama de las matemáticas que utiliza herramientas del álgebra abstracta para estudiar espacios topológicos. El objetivo del proyecto financiado con fondos europeos FHiCuNCAG es combinar estas ramas mediante el desarrollo de una teoría de topos lineales superiores. La investigación del proyecto brindará más información sobre el antiguo problema de la curvatura en la teoría de la deformación algebraica y establecerá un nuevo marco para la NCAG, que incorpora objetos curvos, inspirándose en el reino de las categorías superiores.
Objetivo
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.
Ámbito científico
Programa(s)
Régimen de financiación
ERC-COG - Consolidator GrantInstitución de acogida
2000 Antwerpen
Bélgica