Mathematics Colloquia and Seminars

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Given a (quantum) completely integrable system we shall investigate the corresponding eigenvalue problem for the generic values of the parameters. We shall show that the corresponding differential Galois group is always a reductive algebraic group. This result would imply among the rest that an integrable system is algebraically integrable (i.e. the corresponding eigenvalue problem is explicitly solvable) if and only if the differential Galois group is commutative for generic eigenvalues. We shall apply the above criterion of algebraic integrability to prove a conjecture by Chalykh and Veselov which predicts that the Calogero-Moser system with an integrable parameter is algebraically integrable.