Research

The Integrable Systems Group has a wide range of interests in integrable systems and related topics. Here we give a brief description of these, with further details available on individual members’ pages. Prospective postgraduate students may like to visit Postgraduate Opportunities, where some background reading matter can be found as well as a list of proposed PhD topics.

Integrable systems theory is concerned with models and systems, arising largely from, but not restricted to Physics which can, in some sense, be solved exactly. They take many forms, from ODEs and PDEs traditionally involved in classical (field) theories, to quantum mechanics and quantum field theory. This abundance of origins is matched by rich and fascinating mathematical structures making them worthwhile objects of study in their own right and allowing for the construction of their exact solutions. This fruitful interplay with areas of mathematics such as geometry and algebra led to many important and influential developments (e.g. twistor theory, quantum groups and Poisson-Lie groups). Topics of interest at Leeds include the following areas which overlap each other and have connections with other interests in the School of Mathematics

Research Topics and Expertise in our Group

  • Soliton theory, Lax pairs, inverse scattering transform, unified transform, integrable boundary conditions, integrable PDEs on graphs
  • Symmetries, infinite dimensional (Lie) algebras, algebraic theory of (partial) differential equations
  • Integrable and superintegrable Hamiltonian systems and Poisson algebras, classical Yang-Baxter equation, classical r-matrix
  • Classical and quantum many body problems, Calogero-Moser and Ruijsenaars-Schneider models
  • Classical and quantum integrable field theories
  • Integrable variational problems and Lagrangian multiform theory
  • Cherednik algebras, representations of quivers, non-commutative Poisson geometry
  • Special function theory and PainlevĂ© equations
  • Integrability of discrete systems, discrete differential geometry
  • The Laurent phenomenon and connections with quiver mutation, cluster algebras
  • Birational maps, set-theoretical Yang-Baxter equation, set-theoretical reflection equation
  • Quantum Yang-Baxter and reflection equations, quantum R matrix