School of Mathematics & Statistics

Information about postgraduate research opportunities and how to apply can be found on the Postgraduate Research Study page. Below is a selection of projects that could be undertaken with our group.

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).

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.

Cherednik Algebras and related topics (PhD)

Supervisors: Misha Feigin

Relevant research groups: Algebra, Integrable Systems and Mathematical Physics

The project is aimed at clarifying certain questions related to Cherednik algebras. These questions may include study of homomorphisms between rational Cherednik algebras for particular Coxeter groups and special multiplicity parameters, defining and studying of new partial spherical Cherednik algebras and their representations related to quasi-invariant polynomials, study of differential operators on quasi-invariants related to non-Coxeter arrangements. Relations with quantum integrable systems of Calogero-Moser type may be explored as well. Some other possible topics may include study of quasi-invariants for non-Coxeter arrangements in relation to theory of free arrangements of hyperplanes.

qDT invariants and deformations of hyperKahler geometry (PhD)

Supervisor: Ian Strachan

Relevant research groups: Geometry and Topology, Integrable Systems and Mathematical Physics

The project seeks to understand and exploit the integrable structure behind quantum Donaldson-Thomas invariants in terms of deformation of hyperKahler geometry and quantum-Riemann-Hilbert problems.

Almost-duality for arbitrary genus Hurwitz spaces (PhD)

Supervisor: Ian Strachan

Relevant research groups: Geometry and Topology, Integrable Systems and Mathematical Physics

The space of rational functions (interpreted as the space of holomorphic maps from the Riemann sphere to itself) may be endowed with the structure of a Frobenius manifolds, and hence there also exists an almost-dual Frobenius manifold structure. The class of examples include Coxeter and Extended-Affine-Weyl orbit group spaces. This extends to spaces of holomorphic maps between the torus and the sphere, where one can proved stronger results than just existence results. The project will seek to extend this to the explicit study of the space of holomorphic maps from an arbitrary genus Riemann surface to the Riemann sphere.

Ultra-discrete Integrable Systems (PhD)

Supervisor: Claire Gilson

Relevant research groups: Integrable Systems and Mathematical Physics

An ultra-discrete equation is an equation where not only are the independent variables x and t discrete but the dependent variable are also discrete and often take on the values 0 or 1.  There are many interesting results known about these systems.  They are usually obtained by the process of ultra-discretization, this is a limiting process which takes you from a continuous integrable system to an ultra-discrete system via a discrete system.  This project investigates the world of ultra-discrete systems and soliton cellular automata both periodic and non periodic.  Looking in particular at how one can build different systems and extract the conserved quantities of these systems.

Non-commutative Integrable Systems (PhD)

Supervisor: Claire Gilson

Relevant research groups: Integrable Systems and Mathematical Physics

Integrable Systems are a very special class of differential equations with exact (soliton) solutions.  There are several well known integrable systems, most of which are commutative (i.e the order in which the dependent variables are written down in the equation doesn't matter).  Most of these equations have multiple soliton solutions that can be written down in terms of determinants such as Wronskians or Grammians.   There are many fewer non-commutative integrable systems that are known,  these have solutions in terms of quasi-determinants (Gelfand and Retakh, vol25, p91, Functional Analysis and its Applications).  This project aims to investigate the properties of known and new non-commutative systems.

Regular seminars relevant to the group are held as part of the Applied Mathematics seminar series. You can find a full list of the seminars within the school on the main seminars page, where you can subscribe to their respective calendars.