PhD projects
PhD Applicants
I am always happy to hear from prospective PhD students with an interest in mathematical physics or in areas related to my research interests. The School of Mathematics and Statistics, University of Glasgow offers every year a limited number of funded PhD places. Candidates are shortlisted starting January, so any applications for a PhD position should be made early December in the year before you wish to start your studies.
Further details on how to make an application and possible projects can be found on the School's website.
The pictures shows a 5-vertex model (osculating walkers) on a square lattice with periodic boundary conditions. The latter can be used to compute Gromov-Witten invariants for Grassmannians.
Quantum spin-chains and exactly solvable lattice models (PhD)
Supervisors: Christian Korff
Relevant research groups: Algebra, Integrable Systems and Mathematical Physics
Quantum spin-chains and 2-dimensional statistical lattice models, such as the Heisenberg spin-chain and the six and eight-vertex models remain an active area of research with many surprising connections to other areas of mathematics.
Some of the algebra underlying these models deals with quantum groups and Hecke algebras, the Temperley-Lieb algebra, the Virasoro algebra and Kac-Moody algebras. There are many unanswered questions ranging from very applied to more pure topics in representation theory and algebraic combinatorics. For example, recently these models have been applied in combinatorial representation theory to compute Gromov-Witten invariants (enumerative geometry) and fusion coefficients in conformal field theory (mathematical physics).
The six-vertex (or ice) model from statistical mechanics can be used as a combinatorial to compute symmetric functions given in terms of tableaux. The example shows an ice-configuration and its corresponding shifted tableau occurring in the description of Schur's Q-functions which are linked to spin-characters of the symmetric group and solutions of the BKP-hierarchy.
Symmetric functions & vertex operators (PhD)
Supervisors: Christian Korff
Relevant research groups: Algebra, Integrable Systems and Mathematical Physics
The ring of symmetric functions plays an important role in representation theory and integrable systems: solutions of so-called integrable hierarchies, infinite sets on nonlinear PDEs, are known to be particular symmetric functions. Examples include Schur functions which are solutions to the Kadomtsev-Petiashvili (KP) hierarchy and Schur's Q-functions which are solutions to the so-called BKP hierarchy. They can be interpreted as characters of Weyl modules of Lie algebras. In their construction so-called vertex operators come up, a term which originates from quantum field theory. There exist recent developments where some of these vertex operators have been constructed using lattice models and solutions of the Yang-Baxter equation. This project will explore this new connection and possible generalisations of the celebrated boson-fermion correspondence.
One of the simplest cluster algebras is probably the one for type A describing the triangulation of regular polygons (here the pentagon). Maps between triangulations (called "mutations") allow to define a system of recurrence relations which describe the UV limit of QFT.
Integrable quantum field theory and Y-systems (PhD)
Supervisors: Christian Korff
Relevant research groups: Algebra, Integrable Systems and Mathematical Physics
The mathematically rigorous and exact construction of a quantum field theory remains a tantalising challenge. In 1+1 dimensions exact results can be found by computing the scattering matrices of such theories using a set of functional relations. These theories exhibit beautiful mathematical structures related to Weyl groups and Coxeter geometry.
In the thermodynamic limit (volume and particle number tend to infinity while the density is kept fixed) the set of functional relations satisfied by the scattering matrices leads to so-called Y-systems which appear in cluster algebras introduced by Fomin and Zelevinsky and the proof of dilogarithm identities in number theory.