## 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" or see below for examples of possible projects, and click here for information on how to apply.

# Information about

## Staff

#### Dr Chris Athorne: **Senior lecturer**

Geometric representation theory; algebraic curves; soliton theory

**Member of other research groups:** Geometry and Topology

**Dr Panupong Cheewaphutthisakun: EPSRC Research Associate**

Toroidal quantum groups, quantum Knizhnik-Zamolodchikov equations, Nekrasov factors

#### Prof. Misha Feigin: Professor of Mathematics

Quantum integrable systems; Frobenius manifolds; Cherednik algebras

**Member of other research groups:** Geometry and Topology, Algebra

**Research students:** , Leo Kaminski, Martin Vrabec, Johan Wright

**Dr Francesco Giglio: Lecturer in Applied Mathematics**

C-integrable PDEs; nonlinear waves; statistical thermodynamics

**Member of other research groups**: Theory and Modelling of Liquid Crystals

#### Dr Claire R Gilson: **Senior lecturer**

Integrable systems: in particular discrete, ultradiscrete and non-commutative integrable systems; quasi-determinants

**Research students**: Chen Shu

#### Prof. Christian Korff: Professor of Mathematical Physics

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: **Leo Kaminski, Andre Bedell, Alessandro, Alessandro Proserpio

#### Prof Joachim Zacharias: **Professor**

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

## Postgraduates

#### Simone Castellan: **PhD Student**

**Research Topic:** Short star products in (Poisson) vertex algebras

**Member of other research groups:** Geometry and Topology, Algebra

**Supervisors:** Gwyn Bellamy

#### Anna Clancy: **PhD Student**

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

**Supervisors:** Christian Korff, Christina Cobbold

#### Leo Kaminski: **PhD Student**

**Research Topic:** Special solutions of WDVV equations

**Member of other research groups:** Geometry and Topology, Algebra

**Supervisor:** Misha Feigin, Ian Strachan

**Johan Wright: ****PhD Student**

**Research Topic:** Frobenius manifolds

**Member of other research groups:** Geometry and Topology, Algebra

**Supervisor:** Misha Feigin

#### Martin Vrabec: **PhD Student**

**Research Topic:** Relativistic integrable systems and related structures

**Member of other research groups:** Geometry and Topology, Algebra

**Supervisor:** Misha Feigin

## Integrable Systems and Mathematical Physics example research projects

**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 groups 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.

#### Cherednik Algebras and related topics (PhD)

**Supervisors**: Misha Feigin

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

The project is aimed at clarifying certain questions related to Cherednik algebras. These questions may include study of homomorphisms between rational Cherednik algebras for particular Coxeter groups and special multiplicity parameters, defining and studying of new partial spherical Cherednik algebras and their representations related to quasi-invariant polynomials, study of differential operators on quasi-invariants related to non-Coxeter arrangements. Relations with quantum integrable systems of Calogero-Moser type may be explored as well. Some other possible topics may include study of quasi-invariants for non-Coxeter arrangements in relation to theory of free arrangements of hyperplanes.

#### qDT invariants and deformations of hyperKahler geometry (PhD)

**Supervisor**: Ian Strachan

**Relevant research groups**: Geometry and Topology, Integrable Systems and Mathematical Physics

The project seeks to understand and exploit the integrable structure behind quantum Donaldson-Thomas invariants in terms of deformation of hyperKahler geometry and quantum-Riemann-Hilbert problems.

#### Almost-duality for arbitrary genus Hurwitz spaces (PhD)

**Supervisor**: Ian Strachan

**Relevant research groups**: Geometry and Topology, Integrable Systems and Mathematical Physics

The space of rational functions (interpreted as the space of holomorphic maps from the Riemann sphere to itself) may be endowed with the structure of a Frobenius manifolds, and hence there also exists an almost-dual Frobenius manifold structure. The class of examples include Coxeter and Extended-Affine-Weyl orbit group spaces. This extends to spaces of holomorphic maps between the torus and the sphere, where one can proved stronger results than just existence results. The project will seek to extend this to the explicit study of the space of holomorphic maps from an arbitrary genus Riemann surface to the Riemann sphere.