Integrable Systems and Mathematical Physics
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.
Staff
Dr Chris Athorne : Senior lecturer
Geometric representation theory; algebraic curves;soliton theory
Member of other research groups: Geometry and Topology
Dr Mikhail Feigin : Senior lecturer
Quantum integrable systems; Hadamard's problem; WDVV equations; random matrices
Member of other research groups: Geometry and Topology, Algebra
Research students: Maali Alkadhem, Leo Kaminski, Georgios Antoniou, Johan Wright
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
Research students: Anna Clancy, Damian Wierzbicki
Postgraduate opportunities: Integrable quantum field theory and Y-systems, Quantum spin-chains and exactly solvable lattice models
Prof Ian A B Strachan : Professor of Mathematical Physics
Geometry and integrable systems; Frobenius manifolds; Bi-Hamiltonian structures, twistor theory and self-duality
Member of other research groups: Geometry and Topology
Research student: Georgios Antoniou
Dr Daniele Valeri : Lecturer
Lie algebras, vertex algebras and W-algebras, Yangians; classical and quantum integrable systems; ODE/IM correspondence; invariant measures for dynamical systems.
Research students: Simone Castellan, Johan Wright, Anna Clancy
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: Dimitrios Gerontogiannis
Postgraduates
Maali Alkadhem : PhD Student
Supervisor: Mikhail Feigin
Georgios Antoniou : PhD Student
Supervisors: Ian A B Strachan, Mikhail Feigin
Simone Castellan : PhD Student
Research Topic: Short star products in (Poisson) vertex algebras
Member of other research groups: Geometry and Topology, Algebra
Supervisors: Daniele Valeri, Gwyn Bellamy
Anna Clancy : PhD Student
Research Topic: Factorial Schur functions and Yang-Baxter algebras
Supervisors: Christian Korff, Daniele Valeri
Leo Kaminski : PhD Student
Research Topic: Special solutions of WDVV equations
Member of other research groups: Geometry and Topology, Algebra
Supervisor: Mikhail Feigin
Postgraduate opportunities
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 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.
