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In this paper we study noncommutative integrable systems on b-Poisson manifolds. One important source of examples (and motivation) of such systems comes from considering noncommutative systems on manifolds with boundary having the right asymptotics on the boundary. In this paper we describe this and other examples and prove an action-angle theorem for noncommutative integrable systems on a b-symplectic manifold in a neighborhood of a Liouville torus inside the critical set of the Poisson structure associated to the b-symplectic structure., Peer Reviewed, Postprint (author's final draft)

In this article we prove an action-angle theorem for b-integrable systems on b-Poisson manifolds improving the action-angle theorem contained in [14] for general Poisson manifolds in this setting. As an application, we prove a KAM-type theorem for b-Poisson manifolds. (C) 2015 Elsevier Masson SAS. All rights reserved., Postprint (published version)

In this paper we prove rigidity theorems for Poisson Lie group actions on Poisson manifolds. In particular, we prove that close infinitesimal momentum maps associated to Poisson Lie group actions are equivalent using a normal form theorem for SCI spaces. When the Poisson structure of the acted manifold is integrable, this yields rigidity also for lifted actions to the symplectic groupoid., Peer Reviewed, Postprint (updated version)

n this paper we generalize constructions of non-commutative in- tegrable systems to the context of weakly Hamiltonian actions on Poisson manifolds. In particular we prove that abelian weakly Hamiltonian actions on symplectic manifolds split into Hamiltonian and non-Hamiltonian factors, and explore generalizations in the Poisson setting., Peer Reviewed, Postprint (author's final draft)

In this paper we study non-commutative integrable systems on b-Poisson manifolds. One important source of examples (and motiva- tion) of such systems comes from considering non-commutative systems on manifolds with boundary having the right asymptotics on the boundary. In this paper we describe this and other examples and we prove an action-angle theorem for non-commutative integrable systems on a b-symplectic manifold in a neighbourhood of a Liouville torus inside the critical set of the Poisson structure associated to the b-symplectic structure, Preprint

In this paper we study noncommutative integrable systems on b-Poisson manifolds. One important source of examples (and motivation) of such systems comes from considering noncommutative systems on manifolds with boundary having the right asymptotics on the boundary. In this paper we describe this and other examples and prove an action-angle theorem for noncommutative integrable systems on a b-symplectic manifold in a neighborhood of a Liouville torus inside the critical set of the Poisson structure associated to the b-symplectic structure., Peer Reviewed, Postprint (author's final draft)

We present a collection of examples borrowed from celestial mechanics and projective dynamics. In these examples symplectic structures with singularities arise naturally from regularization transformations, Appell’s transformation or classical changes like McGehee coordinates, which end up blowing up the symplectic structure or lowering its rank at certain points. The resulting geometrical structures that model these examples are no longer symplectic but symplectic with singularities which are mainly of two types: bm-symplectic and m-folded symplectic structures. These examples comprise the three body problem as non-integrable exponent and some integrable reincarnations such as the two fixed-center problem. Given that the geometrical and dynamical properties of bm-symplectic manifolds and folded symplectic manifolds are well-understood (Guillemin et al., 2011, 2014, 2015; Kiesenhofer et al., 2016; Kiesenhofer, in press; Martinet, 1970; Cannas, 2011; Gualtieri and Li, 2014; Gualtier et al., 2014; Marcut and Osorno, 2014; Scott, 0000; Guillemin, 0000), we envisage that this new point of view in this collection of examples can shed some light on classical long-standing problems concerning the study of dynamical properties of these systems seen from the Poisson viewpoint, Peer Reviewed, Postprint (author's final draft)

We associate cotangent models to a neighbourhood of a Liouville torus in symplectic and Poisson manifolds focusing on b-Poisson/b-symplectic manifolds. The semilocal equivalence with such models uses the corresponding action-angle theorems in these settings: the theorem of Liouville–Mineur–Arnold for symplectic manifolds and an action-angle theorem for regular Liouville tori in Poisson manifolds (Laurent- Gengoux et al., IntMath Res Notices IMRN 8: 1839–1869, 2011). Our models comprise regular Liouville tori of Poisson manifolds but also consider the Liouville tori on the singular locus of a b-Poisson manifold. For this latter class of Poisson structures we define a twisted cotangent model. The equivalence with this twisted cotangent model is given by an action-angle theorem recently proved by the authors and Scott (Math. Pures Appl. (9) 105(1):66–85, 2016). This viewpoint of cotangent models provides a new machinery to construct examples of integrable systems, which are especially valuable in the b-symplectic case where not many sources of examples are known. At the end of the paper we introduce non-degenerate singularities as lifted cotangent models on b-symplectic manifolds and discuss some generalizations of these models to general Poisson manifolds., Peer Reviewed, Postprint (author's final draft)

In this note we prove that an analytic symplectic action of a semisimple Lie algebra can be locally linearized in Darboux coordinates. This result yields simultaneous analytic linearization for Hamiltonian vector fields in a neighbourhood of a common zero. We also provide an example of smooth non-linearizable Hamiltonian action with semisimple linear part. The smooth analogue only holds if the semisimple Lie algebra is of compact type. An analytic equivariant b-Darboux theorem for b-Poisson manifolds and an analytic equivariant Weinstein splitting theorem for general Poisson manifolds are also obtained in the Poisson setting, Preprint

A 2n-dimensional Poisson manifold (M,¿) is said to be bm-symplectic if it is symplectic on the complement of a hypersurface Z and has a simple Darboux canonical form at points of Z which we will describe below. In this paper we will discuss a desingularization procedure which, for m even, converts ¿ into a family of symplectic forms ¿¿ having the property that ¿¿ is equal to the bm-symplectic form dual to ¿ outside an ¿-neighborhood of Z and, in addition, converges to this form as ¿ tends to zero in a sense that will be made precise in the theorem below. We will then use this construction to show that a number of somewhat mysterious properties of bm-manifolds can be more clearly understood by viewing them as limits of analogous properties of the ¿¿'s. We will also prove versions of these results for m odd; however, in the odd case the family ¿¿ has to be replaced by a family of folded symplectic forms., Preprint

In this article we develop tools to compute the Geometric Quantization of a symplectic manifold with respect to a regular In this article we develop tools to compute the Geometric Quantization of a symplectic manifold with respect to a regular Lagrangian foliation via sheaf cohomology and obtain important new applications in the case of real polarizations. The starting point is the definition of representation spaces due to Kostant. Besides the classical examples of Gelfand-Cetlin systems due to Guillemin and Sternberg [13] very few examples of explicit computations of real polarizations are known. The computation of Geometric Quantization in [13] is based on a theorem due to Śniatycki for fibrations [32] which identifies the representation space with the set of Bohr-Sommerfeld leaves determined by the integral action coordinates. In this article we check that the associated sheaf cohomology apparatus of Geometric Quantization satisfies Mayer-Vietoris and Künneth formulae. As a consequence, a new short proof of this classical result for fibrations due to Śniatycki is obtained. We also compute Geometric Quantization with respect to any generic regular Lagrangian foliation on a 2-torus and the case of the irrational flow. In the way, we recover some classical results in the computation of foliated cohomology of these polarizations., Peer Reviewed, Preprint

In this article we prove an action-angle theorem for b-integrable systems on b-Poisson manifolds improving the action-angle theorem contained in [LMV11] for general Poisson manifolds in this setting. As an application, we prove a KAM-type theorem for b-Poisson manifolds., Postprint (published version)

There are no special prerequisites to follow this minicourse except for basic differential geometry. Poisson manifolds constitute a natural generalization of Symplectic manifolds. Many problems in mechanics turn out to be naturally formulated in the Poisson language. In contrast to symplectic manifolds where there are no local invariants (Darboux), Poisson manifolds do present local invariants. The aim of the minicourse is to present an introduction to Poisson Geometry and the study of normal forms (how the structures look like locally). We also plan to explain some of the problems considered in their semilocal and global study and introduce the study of symmetries (group actions) in these manifolds. We will end up the minicourse presenting an application of the study of group actions to integrable systems (stressing the particular case of b-Poisson manifolds)., Peer Reviewed, Postprint (published version)

The main goal of the thesis is to classify some particular Poisson structures, called b-Poisson, following results of Guillemin-Miranda-Pirés and Radko. To understand these results, this thesis contains some chapters of introduction to Poisson Geometry and Symplectic Geometry. There is also a new result extending the classification of Poisson structures to the non-orientable case. Moreover it is explained a way to obtain local classification of some Poisson structures that can lead to more original results.

Resum: Estudi de la transversalitat, una eina molt útil de la topologia diferencial tant en varietats (de manera geomètrica) i els espais de jets. També es fa un breu repàs a la geometria diferencial necessària. Per últim es mostra un aplicació a l'estudi d'equacions diferencials en el tor, una varietat molt coneguda. Summary: In this TFG we study transversality, a very useful tool in Differential Topology, which is applied to manifolds (in a geometric way) and to the space of jets. A summary of basic facts in differential geometry is also included. Finally, we include an application to the study of differential equations on the torus, a very well-known manifold.

Postprint (published version)

Hamiltonian actions constitute a central object of study in symplectic geometry. Special attention has been devoted to the toric case. Toric symplectic manifolds provide natural examples of integrable systems and every integrable system on a symplectic manifold is a toric manifold in a neighbourhood of a compact fiber (Arnold–Liouville). The classification of toric symplectic manifolds is given by Delzant’s theorem in terms of the image of the moment map (Delzant polytope)...., Postprint (published version)

n this paper we prove that close infinitesimal momentum maps associated to Poisson Lie actions are equivalent under some mild assumptions. We also obtain rigidity theorems for actual momentum maps (when the acting Lie group G is endowed with an arbitrary Poisson structure) combining a rigidity result for canonical Hamiltonian actions (\cite{MMZ}) and a linearization theorem(\cite{GW}). These results have applications to quantization of symmetries since these infinitesimal momentum maps appear as the classical limit of quantum momentum maps (\cite{BEN})., Peer Reviewed, Preprint

In [GMPS] we proved that the moment map image of a b-symplectic toric manifold is a convex b-polytope. In this paper we obtain convexity results for the more general case of non-toric hamiltonian torus actions on b-symplectic manifolds. The modular weights of the action on the connected components of the exceptional hypersurface play a fundamental role: either they are all zero and the moment map behaves as in classic symplectic one, or they are all nonzero and the moment map behaves as in the toric b-symplectic case studied in [GMPS]., Preprint

This article presents a Poincar e lemma for the Kostant complex, used to compute geometric quantisation, when the polarisation is given by a Lagrangian foliation de ned by an integrable system with nondegenerate singularities., Peer Reviewed, Postprint (published version)

2a prova

A Poisson manifold (M2n; ) is b-symplectic if Vn is transverse to the zero section. In this paper we apply techniques of Symplectic Topology to address global questions pertaining to b-symplectic manifolds. The main results provide constructions of: b-symplectic submanifolds a la Donaldson, b-symplectic structures on open manifolds by Gromov's h-principle, and of b-symplectic manifolds with a prescribed singular locus, by means of surgeries., Preprint

We study Hamiltonian actions on b-symplectic manifolds with a focus on the e ective case of half the dimension of the manifold. In particular, we prove a Delzant-type theorem that classi es these manifolds using polytopes that reside in a certain enlarged and decorated version of the dual of the Lie algebra of the torus. At the end of the paper we suggest further avenues of study, including an example of a toric action on a b 2-manifold and applications of our ideas to integrable systems on b-manifolds, Preprint

This paper presents a Poincaré lemma for the Kostant comple x, used to compute geometric quantisation, when the polarisat ion is given by a Lagrangian foliation defined by an integrable system wit h non-degenerate singularities., Preprint

A signi cant research activity has occurred in the area of convex semi- in nite optimization in the recent years. Many new theoretical, algorithm and computational contribution has been obtained . Despite these numerous con- tributions, there still exits a lack of representative convex semi-in nite test problems. Test problems are of major importance for researchers interested in the algorithmic development. This article is motivated by the scarcity of con- vex semi-in nite test problems and describes a procedure for generating convex semi-in nite families of test problems with optimal solution and optimal value known., Preprint

In this paper we prove a Poincar e lemma for forms tangent to a foliation with nondegenerate singularities given by an integrable system on a symplectic manifold. As a consequence, the Kostant complex in Geometric Quantization is a ne resolution of the sheaf of at sections when the polarization is spanned by the Hamiltonian vector elds of the rst integrals of this integrable system., Preprint

We study local normal forms for completely integrable systems on Poisson manifolds in the presence of additional symmetries. The symmetries that we consider are encoded in actions of compact Lie groups. The existence of Weinstein’s splitting theorem for the integrable system is also studied giving some examples in which such a splitting does not exist, i.e. when the integrable system is not, locally, a product of an integrable system on the symplectic leaf and an integrable system on a transversal. The problem of splitting for integrable systems with additional symmetries is also considered., Peer Reviewed, Postprint (published version)

In this article we develop tools to compute the Geometric Quantization of a symplectic manifold with respect to a regular Lagrangian foliation via sheaf cohomology. The starting point is the definition of representation spaces due to Kostant. We check that the associated sheaf cohomology apparatus satisfies Mayer-Vietoris and K\"unneth formulae. As a consequence, new proofs of classical results for fibrations are obtained. In the general case of Lagrangian foliations, we compute Geometric Quantization with respect to almost any generic regular Lagrangian foliation on a 2-torus., Preprint

This thesis shows an approach to geometric quantisation of integrable systems. It extends some results by Guillemin, Kostant, Rawnsley, Sniatycki and Sternberg in geometric quantisation, considering regular fibrations as real polarisations, to the singular setting: the real polarisations concerned here are given by integrable systems with nondegenerate singularities, and the definition of geometric quantisation used is the one suggested by Kostant (via higher cohomology groups). It also presents unifying proofs for results in geometric quantisation by exploring the existence of symplectic circle actions: the tools developed here highlight and unravel the role played by circle actions in known results in geometric quantisation. The originality of this thesis relies on the following aspects. Firstly, the use of symplectic circle actions to obtain results in geometric quantisation, and secondly, the nonexistence of Poincaré lemmata for foliated cohomology when the foliation has singularities. Previous results on circle actions, due to Rawnsley, could not be used when the circle action is not free, and it is not straightforward to adapt them to accommodate fixed points. After developing these techniques, the computation of geometric quantisation is performed in a series of situations, which includes: the cotangent bundle of the circle and products of it with any quantisable manifold, and neighbourhoods of nondegenerate singularities of integrable systems (hyperbolic singularities need special treatment, since there is no natural circle action). These computations imply that the Kostant complex is a fine resolution (for the sheaf of sections of the prequantum line bundle which are flat along the polarisation) when the real polarisations are given by integrable systems with nondegenerate singularities. It is important to mention that the proofs are original, since, contrary to expectations, there is no Poincaré Lemma when singularities are allowed for the foliated cohomol, Postprint (published version)

Prova L1

The famous Hirsch conjecture was formulated in 1957 in a conversation of Warren Hirsch with George Dantzig, the father of the simplex method. It asserts that given a polytope of dimension d and n facets, its combinatorial diameter is smaller than or equal to n–d, i.e. any two vertices of the polytope can be connected to each other by a path of at most n–d edges. The conjecture is related to the problem of complexity of the simplex method (it gives a lower bound). The conjecture has been open for 53 years and has attracted the interest of many mathematicians in discrete, combinatorial and computational geometry. The simplicity of its statement has also attracted mathematicians in other areas. On 10 May 2010, the entry about the Hirsch Conjecture in Wikipedia was updated, announcing that Francisco Santos had found a counterexample to the Hirsch conjecture. Later, on 14 June, Francisco Santos posted a preprint where the first counterexample was given. It was a polytope in dimension 43 and with 86 facets. The paper is now published in Annals of Mathematics [Sa4]. The HirschConjecture has been disproved., Peer Reviewed, Postprint (published version)

In this paper we study normal forms problems for integrable systems on Poisson manifolds in the presence of additional symmetries. The symmetries that we consider are encoded in actions of compact Lie groups. The equivariant normal forms are obtained at the local level. The existence of Weinstein’s splitting theorem for the integrable system is also studied giving some examples in which such a splitting cannot split. This splitting allows to decompose the integrable system locally as a product of an integrable system on the symplectic leaf and a symplectic leaf on the transversal. The problem of splitting for integrable systems with additional symmetries is also considered, Peer Reviewed, Preprint

Postprint (published version)

Let M2n be a Poisson manifold with Poisson bivector field . We say thatM is b-Poisson if the map n : M ! 2n(TM) intersects the zero section transversally on a codimension one submanifold Z M. This paper will be a systematic investigation of such Poisson manifolds. In particular, we will study in detail the structure of (M, ) in the neighbourhood of Z and using symplectic techniques define topological invariants which determine the structure up to isomorphism. We also investigate a variant of de Rham theory for these manifolds and its connection with Poisson cohomology, Preprint

Peer Reviewed, Postprint (published version)

The main purpose of this paper is to present in a uni¯ed approach to see di®erent results concerning group actions and integrable systems in symplectic, Poisson and contact manifolds. Rigidity problems for integrable systems in these manifolds will be explored from this perspective., Preprint