Classical and Quantum Integrable Systems
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Outline: Introduction to symplectic geometry, including the notion of symmetry of a Hamiltonian system and the concept of a moment map; Poisson geometry; symplectic leaves; co-adjoint orbits; Lie-bi algebras; Poisson-Lie groups; classical r-matrices; the quantum Yang-Baxter algebra; the Algebraic Bethe Ansatz; quantum groups and quantum R-matrices; quantum integrability and combinatorics.
This course will be of particular interest to PhD students in mathematical physics, geometry, algebra (particularly representation theory and Hopf algebras) and topology. For mathematical physics students it will provide the mathematical foundations of integrable systems which are used widely in theoretical physics, particularly in condensed matter physics and string theory. For students of geometry, algebra and topology it will introduce an interdisciplinary area of research which has a long track record of initiating new and fruitful directions in mathematics (for example the study of quantum groups and their application to knot theory and geometric representation theory).