# Integrable Systems and Mathematical Physics

# 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, Georgios Antoniou

#### 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:** David Palazzo, Lucia Rotheray

**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

#### 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 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 staff:** Joan Bosa

**Research students:** Luke Hamblin, Dimitrios Gerontogiannis

## Postgraduates

#### Maali Alkadhem PhD Student

**Supervisor:** Mikhail Feigin

#### Georgios Antoniou PhD Student

**Supervisors:** Ian A B Strachan, Mikhail Feigin

#### David Palazzo PhD Student

**Research Topic:** Quantum integrable systems

**Supervisor:** Christian Korff

## Postgraduate opportunities

### Quantum spin-chains, exactly solvable lattice models and representation theory (PhD)

**Supervisors:** Christian Korff

**Relevant research groups:** 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. 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:** 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.