Convexity of multi-valued momentum maps and the Gel'fand-Cetlin system

Given a group action on a symplectic manifold, there is a cohomological obstruction to the equivariance of the momentum map. In the first chapter of this thesis, we review the notions of Hamiltonian dynamical systems and that of momentum maps, and we give a recipe to compute the obstruction just mentioned.; In the second chapter we turn our attention to properties of the momentum map. A famous theorem of Atiyah, Guillemin and Sternberg states that, given a Hamiltonian torus action, the image of the momentum map is a convex polytope. We generalize this convexity theorem to the case when the action is not required to be Hamiltonian. The main goal of the second chapter is to prove that, given a symplectic torus action, the momentum map---defined on an appropriate covering of the manifold---has as its image the sum of a convex polytope contained in a rational subspace of t* with the orthogonal complement of such a subspace. This result is strictly related to another interesting theorem: the Hamiltonian nature of an action is stable under small perturbations of the symplectic structure. The technique developed allows us to extend the result to any compact group action and also to deduce that any symplectic n-dimensional torus action, with fixed points, on a 2n-dimensional symplectic manifold is Hamiltonian.; The third chapter's main goal is to prove a lemma used in Chapter 2. This lemma describes a property of closed Morse-Bott 1-forms. The two main statements of the chapter are: "if the form has no index 1 critical manifolds and has a minimum then it is exact" and "every leaf of a rational 1-form is homologically represented by at most 1 + |{lcub}index 1 crit manifolds{rcub}| - |{lcub}index 0 crit manifolds{rcub}| among its leaf-components."; The last chapter deals with dynamical systems with a 'weaker' type of symmetry. These dynamical systems are called completely integrable systems. We review the geometry that lies behind these systems, we recall the cohomological obstructions to existence of global action-angle coordinates, and we investigate in detail how the theory applies to a completely integrable system of algebraic origins: the Gelfand-Cetlin system. We prove that this system is built by nesting non-commutative integrable systems which present an obstruction to global action-angle coordinates and we show that this system is originated by a Lax pair.