# 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:** David Palazzo

**Postgraduate opportunities:** Algebraic Combinatorics and Symbolic Computation, Cluster Algebras and their implementation on the computer

#### 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 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

### 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.

### Cluster Algebras and their implementation on the computer (PhD)

**Supervisors:** Sira Gratz, Christian Korff

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

This project will look at cluster algebras and their implementation in symbolic computational software. An example can be found here:

http://demonstrations.wolfram.com/ClusterAlgebras/

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 cluster algebras as well as their visualisation, e.g. in the context of triangulations of surfaces.

The ideal candidate should have a dual interest in algebra 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.