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 Matthew England Lecturer
Abelian functions; algebraic curves; symbolic computation
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 systems; representation theoretic and combinatorial aspects
Member of other research groups: Algebra
Dr Jonathan J C Nimmo Reader
Discrete and ultradiscrete integrable systems; Darboux transformations; quasideterminants
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: Ewan Kenneth Morrison
Ewan Kenneth Morrison PhD Student
Research Topic: Modular Frobenius Manifolds
Supervisor: Ian A B Strachan
Sophie Rachel Norman PhD Student
Research Topic: Geometry of higher genus Weierstrass P-functions
Supervisor: Chris Athorne