# Josh Cork

**Email:**mmjsc@leeds.ac.uk**Thesis title:****Supervisor:**Dr Derek Harland

#### Profile

MMath graduate from Durham University, and current PhD research student at the University of Leeds, with supervisor Derek Harland.

#### Research interests

Broadly, my research interests incline towards understanding various aspects of pure mathematics found in physics. In particular, I am interested in physical systems with topological and differential geometric structure. Various physical field theories possess topological soliton configurations -- particle-like solutions to equations of motion which are stable under continuous deformations. Two prominent examples, which I am interested in, are the Yang-Mills, and Yang-Mills-Higgs gauge theories, whose topological solitons are generally called instantons and monopoles respectively. Both instantons and monopoles have a rich geometry, providing examples of hyperkaehler manifolds, and links to geometric invariant theories, alongside important physical interpretations.

Currently, my research has focused on instantons for which one direction is periodic; these are also known as calorons. Alongside many fascinating mathematical properties, calorons play an interesting role in physics, especially in lattice gauge theories, and are arguably more physically relevant than monopoles or instantons. Importantly, calorons are explicitly related to the aforementioned topological solitons, but are significantly less well-understood. One very special property of calorons is the existence of a discrete isometry known as the rotation map - this is an action on the moduli space induced by the outer automorphism of a loop group which `rotates' the Dynkin diagram.

Recently I have been considering calorons which are invariant under isometry groups which contain the rotation map, as a means of semi-explicit construction. I am also currently working on understanding relationships between calorons and critical points of a lower dimensional field theory, called skyrmions.