# 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 student:** Damian Wierzbicki

**Postgraduate opportunities:** Algebraic Combinatorics and Symbolic Computation, 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

W-algebras; classical and quantum integrable systems; ODE/IM correspondence; invariant measures for dynamical systems.

**Postgraduate opportunities:** (classical and quantum, finite and affine) W-algebras, (Poisson) vertex algebras and applications to integrable systems

#### 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 students:** Luke Ito, Dimitrios Gerontogiannis

## Postgraduates

## Postgraduate opportunities

### (classical and quantum, finite and affine) W-algebras (PhD)

**Supervisors:** Daniele Valeri

**Relevant research groups:** Integrable Systems and Mathematical Physics

A novel approach to W-algebras using Lax operators has been recently developed. This approach nicely shows the deep relation between W-algebras, integrable systems and Yangians. Further investigations of these relations deserves to be studied. In particular, the Lax type operator approach should be extended to quantum affine W-algebras and used to understand better their representations and relations with quantum integrable systems.

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

**Supervisors:** Christian Korff

**Relevant research groups:** Integrable Systems and Mathematical Physics, Algebra

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:** 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.

### (Poisson) vertex algebras and applications to integrable systems (PhD)

**Supervisors:** Daniele Valeri

**Relevant research groups:** Integrable Systems and Mathematical Physics

Infinite dimensional Hamiltonian systems and their integrability can be nicely studied within the framework of Poisson vertex algebras. This led to a new understanding and new results concerning the classification of Hamiltonion operators, cohomology of Poisson brackets and structure theory of W-algebras with applications to both classical and quantum, commutative and non-commutative, integrable systems. The goal of this project is to pursue this theory as far as possible from different perspectives: representation theory, geometry and mathematical physics.

### Algebraic Combinatorics and Symbolic Computation (PhD)

**Supervisors:** Christian Korff

**Relevant research groups:** Algebra, Integrable Systems and Mathematical Physics

This project will look at topics in algebraic combinatorics, for example the ring of symmetric functions, the Robinson-Schensted-Knuth correspondence, crystal graphs, and their implementation in symbolic computational software. An example can be found here:

http://demonstrations.wolfram.com/KirillovReshetikhinCrystals/

Emphasis of the project will be on the efficient design and writing of algorithms with the aim to provide data for open research problems in algebraic combinatorics.

The ideal candidate should have a dual interest in mathematics and computer science and, in particular, already possess some programming experience. As part of the project it is envisaged to spend some time with a private software company to explore a possible commercialisation of the resulting computer packages.