A multi-faceted exploration of the concept of space
Description
Inspired by physics and sheer mathematical curiosity, we take you on a ride exploring the ideas pushing the boundaries of the notion of Space! You get a taste and overview of the cutting-edge mathematical concepts employed in non-commutative algebraic geometry and its application in the physical theory of homological mirror symmetry.
Program
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.
Remarks
Successful completion of the summer school can be awarded with 3 credits according to the European Credit Transfer System (ECTS). Credits will be awarded by the University of Antwerp on the base of 100 % (active) participation during the course, group work and submission of exercises and a written summary on some part of the lectures.
