School of Mathematics and Statistics

Integrable Systems and Mathematical Physics

  • Quantum cohomology

    Forbenius manifolds

    The associativity condition of quantum cohomology provides links with integrable systems through Frobenius manifolds and Yang-Baxter algebras.

  • Dynkin diagrams

    Dynkin diagram

    Dynkin diagrams encode the structure of Weyl groups and Lie algebras. Their representation theory is used in many areas of mathematical physics.

  • Statistical models

    Vertex model

    The integrable six-vertex model is used to describe ferroelectrics such as ice as well as to count alternating sign matrices in combinatorics.

  • A deformation of a singularity

    Singularity

    The space of deformations of a singularity carries the structure of a Frobenius manifold.

  • Soliton theory

    Breather

    Solitons are special solutions to nonlinear partial differential equations but they also parametrize surfaces such as the one depicted above.

  • Ultradiscrete integrable systems

    Cellular automata

    Discrete time evolution of "soliton solutions" in a periodic cellular automaton: three strings of balls of length 1, 2 and 3 interact and emerge unchanged.

Integrable systems is a branch of mathematics which first came to prominence in the mid 1960's with the (mathematical) discovery of the soliton by Kruskal and Zabusky while studying dispersive shallow water waves. Broadly speaking the focus is on systems for which - despite their nonlinear behaviour - exact solutions can be derived. Today the field has come to embrace many different aspects of mathematical physics and is at the cross-roads with other modern branches of pure and applied mathematics.

Our group is one of the largest in the UK, is part of an LMS network on quantum and classical integrability and hosts its own series of international conferences, ISLAND. Staff members have a diverse range of interests including topics in algebra and geometry; details of these are given below. We welcome applications by prospective PhD students; please click on "Postgraduate Opportunities" below for examples of possible projects

Dr Chris Athorne Senior lecturer

Geometric representation theory; algebraic curves;soliton theory

Member of other research groups: Geometry and Topology
Postgraduate opportunities: An equivariant approach to higher genus Weierstrass P-functions

Dr Mikhail Feigin Lecturer

Quantum integrable systems; Hadamard's problem; WDVV equations; random matrices

Member of other research groups: Geometry and Topology, Algebra

Dr Claire R Gilson Senior lecturer

Discrete and ultradiscrete integrable systems; quasideterminants

Dr Christian Korff Research Fellow of the Royal Society/Reader

Quantum integrable models; exactly solvable lattice models; low-dimensional QFT

Member of other research groups: Algebra
Research student: Mary Clark
Postgraduate opportunities: Integrable quantum field theory and Y-systems, Quantum spin-chains, exactly solvable lattice models and representation theory

Dr Jonathan J C Nimmo Reader

Discrete and ultradiscrete integrable systems; Darboux transformations; quasideterminants

Research student: Ying Shi

Prof Ian A B Strachan Professor of Mathematical Physics

Geometry and integrable systems; Frobenius manifolds; twistor theory and self-duality

Member of other research groups: Geometry and Topology
Research student: Richard Stedman

Dr Joachim Zacharias Reader

C*-algebras, their classification and amenability properties; special examples of C*-algebras; K-theory and non commutative topology, noncommutative dynamical systems, geometric group theory with applications to C*-algebras.

Member of other research groups: Geometry and Topology, Analysis, Algebra
Research student: Gabriel Tornetta

Mary Clark PhD Student

Research Topic: Quantum Integrable Systems
Supervisor: Christian Korff

Ying Shi PhD Student

Research Topic: Inverse Scattering For Discrete Integrable Systems
Supervisor: Jonathan J C Nimmo

Richard Stedman PhD Student

Research Topic: Frobenius Manifolds
Supervisor: Ian A B Strachan

Quantum spin-chains, exactly solvable lattice models and representation theory (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 model 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 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. 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 sl(n)-WZNW fusion ring: a combinatorial construction and a realisation as quotient of quantum cohomology. with Catharina Stroppel. Adv Math 225, 1 (2010) 200-268; arXiv:0909.2347

 

Integrable quantum field theory and Y-systems (PhD)

Supervisors: Christian Korff
Relevant research groups: Integrable Systems and Mathematical Physics, Algebra

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.

 

An equivariant approach to higher genus Weierstrass P-functions (PhD)

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

To obtain information about P-functions associated with planar curves of genus 4 and more using representation theory within an equivariant framework. Methods developed by the supervisor considerably simplify singularity based methods and illuminate the geometric structures of differential relations between these multi-periodic, Abelian functions. The functions themselves are of use in the theory of Integrability.