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

  • Personal Website
  • Publications

    • 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, Leo Kaminski, Georgios Antoniou, Johan Wright

  • Personal Website
  • Publications

    • Dr Claire R Gilson : Senior lecturer

    Discrete and ultradiscrete integrable systems; quasideterminants

  • Publications

    • 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

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  • Publications

    • 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

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  • Publications

    • Dr Daniele Valeri : Lecturer

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

    Research students: Simone Castellan, Johan Wright, Anna Clancy

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    • 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

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  • Publications


    • Postgraduates

    Maali Alkadhem : PhD Student

    Supervisor: Mikhail Feigin

    • Georgios Antoniou : PhD Student

    Supervisors: Ian A B Strachan, Mikhail Feigin

    • Simone Castellan : PhD Student

    Research Topic: Short star products in (Poisson) vertex algebras
    Member of other research groups: Geometry and Topology, Algebra
    Supervisors: Daniele Valeri, Gwyn Bellamy

    • Anna Clancy : PhD Student

    Research Topic: Factorial Schur functions and Yang-Baxter algebras
    Supervisors: Christian Korff, Daniele Valeri

    • Leo Kaminski : PhD Student

    Research Topic: Special solutions of WDVV equations
    Member of other research groups: Geometry and Topology, Algebra
    Supervisor: Mikhail Feigin



    • Quantum cohomology

    The associativity condition of quantum cohomology provides links with integrable systems through Frobenius manifolds and Yang-Baxter algebras

    The associativity condition of quantum cohomology provides links with integrable systems through Frobenius manifolds and Yang-Baxter algebras

    Dynkin diagrams

    Dynkin diagrams encode the structure of Weyl groups and Lie algebras. Their representation theory is used in many areas of mathematical physics

    Dynkin diagrams encode the structure of Weyl groups and Lie algebras. Their representation theory is used in many areas of mathematical physics

    Statistical models

    The integrable six-vertex model is used to describe ferroelectrics such as ice as well as to count alternating sign matrices in combinatorics

    The integrable six-vertex model is used to describe ferroelectrics such as ice as well as to count alternating sign matrices in combinatorics

    A deformation of a singularity

    The space of deformations of a singularity carries the structure of a Frobenius manifold

    The space of deformations of a singularity carries the structure of a Frobenius manifold

    Soliton theory

    Solitons are special solutions to nonlinear partial differential equations but they also parametrize surfaces such as the one depicted above

    Solitons are special solutions to nonlinear partial differential equations but they also parametrize surfaces such as the one depicted above

    Ultradiscrete integrable systems

    Discrete time evolution of "soliton solutions" in a periodic cellular automaton: three strings of balls of length 1, 2 and 3 interact and emerge unchanged

    Discrete time evolution of "soliton solutions" in a periodic cellular automaton: three strings of balls of length 1, 2 and 3 interact and emerge unchanged