Carnegie PhD Scholar awarded Robertson Medal 2022-23
Project Title: Donaldson-Thomas theory for 3-manifolds
In mathematics one of the ways to understand different shapes is to assign a mathematical object, called an invariant, to each shape. If two shapes are the same, then the object assigned to them must also be the same. One such invariant is constructed by considering loops inside the shape. This invariant and its higher dimensional analogues contain a lot of information about the shape but are very difficult to compute.
Donaldson-Thomas (DT) invariants is a theory in algebraic geometry connecting the field with physics. DT theory is an enumerative theory, meaning it provides a count of certain objects important in algebraic geometry. It is possible to interpret these counts as coming from certain algebraic structures, which provide new insights and are interesting on their own.
Recently, it has become feasible to connect DT invariants with 3-manifolds, which can be thought of as generalisations of shapes like the surface of a sphere or a donut to 3 dimensions. The project is therefore concerned with defining new invariants for 3-manifolds and the algebraic structures associated to them. Furthermore, I will explore connections to theoretical physics and how DT invariants can be used to calculate known, but difficult to compute invariants.
Awarded: Carnegie PhD Scholarship
Field: Mathematics and Statistics
University: University of Edinburgh