List of projects » Wheeld props, differential geometry and quantum field theory

# Wheeld props, differential geometry and quantum field theory

Project leader

Funding source

Swedish Research Council - Vetenskapsrådet (VR) |

Project Details

Start date: 01/01/2009

End date: 31/12/2012

Funding: 1843000 SEK

Description

The theory of operads and props has undergone an explosive development in recent years in the sense of both the number of research publications and the variety of questions which are effectively approached with the help of this theory. Operads and props can be seen almost everywhere nowadays --- in algebraic topology, in homological algebra, in classical differential geometry, in non-commutative geometry, in string topology, in deformation theory, in quantization theory, in theoretical computer sciences. The theory is now developing into a kind of universal language in which mathematicians from, historically, different branches of mathematics can effectively and fruitfully communicate with each other. Our project aims to develop Koszul duality for quadratic wheeled properads and apply the results to the homotopy theory of bialgebras, computation of homology of several important graph complexes, deformation quantization of unimodular Poisson structures, and homotopy theory of Batalin-Vilkovisky manifolds. Our plans include a study of the new quantum simplicial BF theory for unimodular Lie 1-bialgebras and subsequent application of the Batalin quantization machinery for computing homotopy transfer formulae of certain quadratic wheeled props which control Poisson and Nijenhuis geometries. In particular, we want to give a new properadic meaning to the famous Feynman integrals in the theoretical physics, and hence develop new methods for regularizing and computing them.