Non-Commutative Algebraic Geometry University of Antwerp

Overview

The intuitive notion of a “space” has seen a variety of incarnations in mathematics, in fields ranging from topology to geometry in all its flavours. 

The subject of Non-Commutative Algebraic Geometry (NCAG) pushes the boundaries of the concept of space by using algebraic and categorical models in contexts where classical point-set based representations fail.  

This summer school is aimed at gaining understanding of key concepts and examples that motivate NCAG and which see their application in Homological Mirror Symmetry (HMS), which has its origins in mathematical physics.  

The programme provides two parallel courses, one on Algebraic Models for Spaces (AMS) and one on HMS, which consist each of lectures by experts in the field and highly interactive problem sessions in groups tailored to a shared background knowledge. 

• In the AMS course, we furnish the background to model non-commutative spaces, working towards A-infinity algebras and quasi-categories as models for infinity-categories.  
• In the HMS course, we illustrate the idea of NCAG by Mirror Symmetry (MS) by starting from Classical Mirror Symmetry as an exchange of numerical data between “mirror” complex and symplectic space. From there, we build towards Kontsevich’ HMS conjecture. 

The programme also includes one or two afternoons of research talks by experts in the field.