Deformation theory of algebraic structures

Información del proyecto

DefAlgS

Identificador del acuerdo de subvención: 704589

DOI

10.3030/704589

Proyecto finalizado el 1 Septiembre 2016

Fecha de la firma de la CE 17 Febrero 2016

Fecha de inicio 1 Marzo 2016

Fecha de finalización 31 Octubre 2017

Financiado con arreglo a

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Coste total

€ 166 829,00

Aportación de la UE

€ 166 829,00

Coordinado por

KOBENHAVNS UNIVERSITET
Denmark

Objetivo

This project focuses on questions related to the deformation theory of algebraic structures in the very general setting of algebras over a prop (which includes various kinds of algebras and bialgebras). In particular, it aims to understand realizations of an algebraic structure at the (co)homology level in an algebraic structure up to homotopy at the (co)chain level. When this complex is of topological or geometric nature (e.g. the de Rham complex of a manifold), one expects to extract from these realizations new topological and geometric invariants. These realizations are not understood at present in various cases where bialgebra structures play a crucial role, especially with Poincaré duality, String topology and Deformation quantization. We use methods relying in particular on Quillen’s work on homotopical algebra, Lurie's higher category theory and Toen-Vezzosi’s derived algebraic geometry. We define and study the homotopy type of realization spaces of algebraic structures and develop a derived geometry approach to deformation theory of such structures. This machinery is set up to address three kinds of problems related to topology, geometry and mathematical physics. The first one is to use realization spaces of Poincaré duality structures and equivariant string topology to build new topological and geometric invariants of manifolds, solve an open problem of Sullivan about the realisation of known geometric invariants and understand the action of the Grothendieck-Teichmüller group on such structures. The second one is to use a derived geometry approach to these moduli spaces to compare deformation theories of bialgebras and algebras over the little disks operad and solve a longstanding formality conjecture of Kontsevich. The third one is to use deformation quantization of moduli stacks of algebraic structures up to homotopy to perform a far reaching generalization of Turaev's work to the construction of link invariants in higher dimensional manifolds.

Ámbito científico (EuroSciVoc)

• ciencias naturales › matemáticas › matemáticas aplicadas › física matemática
• ciencias naturales › matemáticas › matemáticas puras › topología
• ciencias naturales › matemáticas › matemáticas puras › geometría
• ciencias naturales › matemáticas › matemáticas puras › álgebra › geometría algebraica

Palabras clave

• operads
• props
• bialgebras
• moduli spaces
• deformation theory
• homotopical algebra
• higher category theory
• derived algebraic geometry
• Poincaré duality
• string topology
• deformation quantization

Programa(s)

• H2020-EU.1.3. - EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions PROGRAMA PRINCIPAL
  
• H2020-EU.1.3.2. - Nurturing excellence by means of cross-border and cross-sector mobility
  

Tema(s)

• MSCA-IF-2015-EF - Marie Skłodowska-Curie Individual Fellowships (IF-EF)
  

Régimen de financiación

MSCA-IF - Marie Skłodowska-Curie Individual Fellowships (IF)

Convocatoria de propuestas

H2020-MSCA-IF-2015

Coordinador

KOBENHAVNS UNIVERSITET

• Aportación neta de la UEn: € 166 829,00
• Dirección: NORREGADE 10 ; 1165 KOBENHAVN ; Dinamarca
• Región: Danmark › Hovedstaden › Byen København
• Tipo de actividad: Higher or Secondary Education Establishments
• Coste total: € 166 829,00