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.
The research project also deals with questions in "enumerative geometry", which in its original form concerns how many geometric objects of certain kinds exist on a given variety. This field has more recently been discovered to have strong links to both noncommutative geometry and to theoretical physics, and in the first year of the resarch project we have proved a certain conjecture within enumerative geometry.
This research project comprises basic research in algebraic geometry, which is the geometry of spaces that can be defined by particularly simple equations, so-called polynomial equations. The research is concerned with two closely related areas in algebraic geometry, namely non-commutative algebraic geometry and enumerative geometry.
Non-commutative algebraic geometry replaces the geometric objects studied in algebraic geometry with more abstract algebraic structures called "derived categories", or alternatively "non-commutative varieties". Our understanding of non-commutative algebraic geometry is much poorer than that of algebraic geometry, and in particular there is comparatively little understanding of which non-commutative varieties exist, and how to relate different non-commutative varieties to each other. In the non-commutative geometry part of this project, we attack these kinds of problems, exploiting recent ideas from theoretical physics along with the theory of homological projective duality. In particular, we aim to construct new examples of so-called "non-commutative K3 surfaces".
Enumerative geometry deals with the problem of counting geometric objects. A typical question would be how many algebraic curves in the plane exist with given properties. In modern enumerative geometry the focus has shifted to computing different kinds of "virtual number" of curves, which roughly speaking is a way of defining what the number of curves should be if it were finite, which it typically isn't. One such type of virtual number is called the Donaldson-Thomas invariants. The enumerative geometry part of the project will develop new techniques for computing and understanding these Donaldson-Thomas invariants, in technical terms for 3-dimensional "orbifolds" and 4-dimensional varieties.