Project Details
Projekt
Constructive derived equivalences and equivariant vector bundles
Applicant
Professor Dr. Mohamed Barakat
Subject Area
Mathematics
Term
from 2013 to 2017
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 239466887
Computer algebra is facing new challenges as mathematicians are inventing new and more abstract tools to answer difficult problems and connect apparently remote fields of mathematics. With the invention of new high level programming languages these tools now start to find their way to computer algebra. The software project homalg emerged from the need to make the abstract notions of category theory and homological algebra accessible to the machine. They both provide extremely powerful organizing tools which translate to transparent algorithms able to organize many different computations and keep track of an enormous amount of data. The homalg project was designed to combine abstraction and efficiency, two traditionally conflicting goals. Written in GAP4 it is already interacting with almost all computer algebra systems of the priority program SPP1489, especially with SINGULAR and polymake. Abstract ABELian categories are already formalized in homalg. Driven by many concrete applications we will render derived categories and derived equivalences constructive. The ability of derived equivalences to relate two completely different ABELian categories is an invitation for computer algebra to change worlds; abandoning data structures with an unnecessary overhead in favor of more compact representations which speed up computations considerably. We aim at various tilting equivalences between derived categories of coherent sheaves on toric varieties and derived categories of modules over finite dimensional noncommutative algebras. Our goal is to setup a framework making such abstract equivalences computational exploitable. The main application of the proposed project is the search for low-rank vector bundles equivariant under a finite group action.
DFG Programme
Priority Programmes
