## 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, 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

#### 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.

**Research students:** Johan Wright, Anna Clancy

**Postgraduate opportunities:** (classical and quantum, finite and affine) W-algebras

#### 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

#### Anna Clancy : PhD Student

**Research Topic:** Factorial Schur functions and Yang-Baxter algebras

**Supervisors:** Christian Korff, Daniele Valeri

## 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.