About the project
This project works towards new theoretical understanding in the mathematical field of algebraic geometry. The main goal is to construct and analyze new examples of non-commutative algebraic varieties.
In algebraic geometry one studies algebraic varieties. These are geometric objects that are defined by means of algebraic equations in several unknowns. In school you learn about the simplest examples of such objects, for example, the equation x + y = 1 defines a line, while x*x + y*y = 1 defines a circle.
By adding more unknowns and making the equations more complicated, one can produce algebraic varieties with complicated geometry. The properties of an algebraic variety can be described by means of so-called invariants: Some of these are easy to illustrate, such as the invariant of dimension (a line has dimension 1, the surface of a ball dimension 2) and the invariant of curvature (the surface of a ball curves positively, while the wide end of a trumpet curves negatively).
The derived category of an algebraic variety is, by comparison, a very abstract and complicated invariant. Loosely explained, the derived category is a structure that describes all the objects (technically: the "coherent sheaves") that exist in the algebraic variety, as well as the relationships between them. The study of the derived category and structures of this type is often called non-commutative algebraic geometry.
A particularly interesting class of objects in non-commutative algebraic geometry are the so-called non-commutative K3 surfaces. We would like to have ways of constructing more examples of these and tools to analyse them better. The non-commutative perspective turns out to be important in theoretical physics as well, and a key goal of this project is to use a phenomenon from physics (the title's "gauge duality") to construct and analyse new examples of non-commutative K3 surfaces.
Financing
Research council of Norway, Independent projects - project number 302277. Total budget approximately NOK 10,6 million.