Integrable systems

Integrable systems are a special class of dynamical systems with the property that they can be "solved" exactly. In the classical setting this means writing down exact solutions to the equations of motion, while quantum mechanically this means finding a basis of exact eigenstates of the Hamiltonian. Many interesting and important examples of integrable systems, both classical and quantum, are known to admit descriptions as Gaudin models.
In the case of quantum Gaudin models associated with finite-dimensional semisimple Lie algebras, the basis of eigenstates is usually obtained by the Bethe ansatz method. The latter is known to have deep connections with the geometric Langlands correspondence, through the work of B. Feigin, E. Frenkel and N. Reshetikhin. Another powerful and general approach to solving a finite-dimensional integrable system, developed initially by E. Sklyanin both at the classical and quantum levels, is the method of separation of variables. For the quantum Gaudin model, this has only been studied in detail in the case when the underlying Lie algebra is sl_2. The main aim of the project will be to develop the method of separation of variables for quantum Gaudin models associated with any finite-dimensional semisimple Lie algebra. The method could ultimately also be extended to the case of Gaudin models associated with affine Kac-Moody algebras, first at the classical level and then in the quantum case if time permits.