The associativity condition of quantum cohomology provides links with integrable systems through the WDVV equations, Frobenius manifolds and noncommutative Schur polynomials.
Integrable Systems and Mathematical Physics
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Quantum cohomology
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Dynkin diagrams
Dynkin diagrams encode the structure of Weyl groups and Lie algebras. Their representation theory is used in many areas, e.g. quantum many body systems and Frobenius manifolds.
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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.
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A deformation of a singularity
The space of deformations of a singularity carries the structure of a Frobenius manifold.
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Soliton theory
Solitons are special solutions to nonlinear partial differential equations but they also parametrize surfaces such as the one depicted above.
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Ultradiscrete integrable systems
Ultradiscrete integrable systems are cellular automata having soliton solutions. In the figure, three solitons with different speeds interact and each emerges unchanged apart from a phase shift.
Integrable systems is a branch of mathematics which has a long history but first came to prominance in the mid 1960's with the (mathematical) discovery of the soliton by Kruskal and Zabusky in connection with a dispersive shallow water wave problem. Since then this field has come to embrace many different aspects of mathematical physics. The group is working on a range of different problems in this area. Details of these are given below.
Dr Chris Athorne Senior lecturer
Geometric representation theory; algebraic curves;soliton theory
Member of other research groups: Geometry and Topology
Research student: Sophie Rachel Norman
Dr Mikhail Feigin Lecturer
Quantum integrable systems; Hadamard's problem; WDVV equations; random matrices
Member of other research groups: Geometry and Topology, Algebra
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
Member of other research groups: Algebra
Research student: Mary Clark
Postgraduate opportunities: Integrable quantum field theory and Y-systems, Quantum spin-chains, exactly solvable lattice models and representation theory
Dr Jonathan J C Nimmo Reader
Discrete and ultradiscrete integrable systems; Darboux transformations; quasideterminants
Research student: Ying Shi
Prof Ian A B Strachan Professor of Mathematical Physics
Geometry and integrable systems; Frobenius manifolds; twistor theory and self-duality
Member of other research groups: Geometry and Topology
Research student: Richard Stedman
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: Andrew Hawkins, Gabriel Tornetta
Mary Clark PhD Student
Research Topic: Quantum Integrable Systems
Supervisor: Christian Korff
Sophie Rachel Norman PhD Student
Research Topic: Geometry of higher genus Weierstrass P-functions
Supervisor: Chris Athorne
Ying Shi PhD Student
Research Topic: Inverse Scattering For Discrete Integrable Systems
Supervisor: Jonathan J C Nimmo
Richard Stedman PhD Student
Research Topic: Frobenius Manifolds
Supervisor: Ian A B Strachan
Quantum spin-chains, exactly solvable lattice models and representation theory (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 model 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 and Hecke algebras, the Temperley-Lieb algebra, the Virasoro algebra and Kac-Moody algebras. Recently these models have been applied in combinatorial representation theory to compute Gromov-Witten invariants and fusion coefficients in conformal field theory.
The sl(n)-WZNW fusion ring: a combinatorial construction and a realisation as quotient of quantum cohomology. with Catharina Stroppel. Adv Math 225, 1 (2010) 200-268; arXiv:0909.2347
Integrable quantum field theory and Y-systems (PhD)
Supervisors: Christian Korff
Relevant research groups: Integrable Systems and Mathematical Physics, Algebra
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.