Descrizione del progetto
Un quadro per la geometria algebrica non commutativa superiore e curva
La geometria algebrica non commutativa (NCAG, noncommutative algebraic geometry) studia le proprietà geometriche delle duplicità formali degli oggetti algebrici non commutativi. Quest’area di studio ha registrato una vasta gamma di applicazioni sia nel campo matematico che in quello della fisica teorica. La topologia algebrica è un’altra branca della matematica che si avvale di strumenti provenienti dall’algebra astratta per studiare gli spazi topologici. Il progetto FHiCuNCAG, finanziato dall’UE, intende fondere queste branche di studio sviluppando una teoria sui topos lineari superiori. La ricerca del progetto fornirà ulteriori indizi sul problema di lunga data della curvatura nella teoria della deformazione algebrica e istituirà un nuovo quadro per la NCAG, che integra oggetti curvi, traendo ispirazione dal reame delle categorie superiori.
Obiettivo
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.
Campo scientifico
Programma(i)
Argomento(i)
Meccanismo di finanziamento
ERC-COG - Consolidator GrantIstituzione ospitante
2000 Antwerpen
Belgio