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In this article we give a universal model for geometric quantization associated to a real polarization given by an integrable system with non-degenerate singularities. This universal model goes one step further than the cotangent models in [KM17] by both considering singular orbits and adding to the cotangent models a model for the prequantum line bundle. These singularities are generic in the sense that are given by Morse-type functions and include elliptic, hyperbolic and focus-focus singularities. Examples of systems admitting such singularities are toric, semitoric and almost toric manifolds, as well as physical systems such as the coupling of harmonic oscillators, the spherical pendulum or the reduction of the Euler’s equations of the rigid body on T *(SO(3)) to a sphere. Our geometric quantization formulation coincides with the models given in [HM10] and [MPS20] away from the singularities and corrects former models for hyperbolic and focus-focus singularities cancelling out the infinite dimensional contributions obtained by former approaches. The geometric quantization models provided here match the classical physical methods for mechanical systems such as the spherical pendulum as presented in [CS16]. Our cotangent models obey a local-to-global principle and can be glued to determine the geometric quantization of the global systems even if the global symplectic classification of the systems is not known in general.

In this article, we consider integrable systems on manifolds endowed with symplectic structures with singularities of order one. These structures are symplectic away from a hypersurface where the symplectic volume goes either to infinity or to zero transversally, yielding either a b-symplectic form or a folded symplectic form. The hypersurface where the form degenerates is called critical set. We give a new impulse to the investigation of the existence of action-angle coordinates for these structures initiated in [36] and [37] by proving an action-angle theorem for folded symplectic integrable systems. Contrary to expectations, the action-angle coordinate theorem for folded symplectic manifolds cannot be presented as a cotangent lift as done for symplectic and bsymplectic forms in [36]. Global constructions of integrable systems are provided and obstructions for the global existence of action-angle coordinates are investigated in both scenarios. The new topological obstructions found emanate from the topology of the critical set Z of the singular symplectic manifold. The existence of these obstructions in turn implies the existence of singularities for the integrable system on Z. Robert Cardona acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the María de Maeztu Programme for Units of Excellence in R&D (MDM-2014-0445). Eva Miranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2016. Both authors are supported by the grants reference number 2017SGR932 (AGAUR) and PID2019-103849GB-I00 / AEI / 10.13039/501100011033.

In this article we generalize a theorem by Palais on the rigidity ofcompact group actions to cotangent lifts. We use this result to prove rigidity forintegrable systems on symplectic manifolds including sytems with degeneratesingularities which are invariant under a torus action.

In this article, we investigate Reeb dynamics onbm-contact manifolds, previously introduced in [MO], which are contact away from a hypersurface $Z$ but satisfy certain transversality conditions on $Z$. In this article, we investigate Reeb dynamics on $b^m$-contact manifolds, previously introduced in \cite{MO}, which are contact away from a hypersurface $Z$ but satisfy certain transversality conditions on $Z$. The study of these contact structures is motivated by that of contact manifolds with boundary. The search of periodic Reeb orbits on those manifolds thereby starts with a generalization of the well-known Weinstein conjecture. Contrary to the initial expectations, examples of compact $b^m$-contact manifolds without periodic Reeb orbits outside $Z$ are provided. Furthermore, we prove that in dimension $3$, there are always infinitely many periodic orbits on the critical set if it is compact. We prove that traps for the $b^m$-Reeb flow exist in any dimension. This investigation goes hand-in-hand with the Weinstein conjecture on non-compact manifolds having compact ends of convex type. In particular, we extend Hofer's arguments to open overtwisted contact manifolds that are $\R^+$-invariant in the open ends, obtaining as a corollary the existence of periodic $b^m$-Reeb orbits away from the critical set. The study of $b^m$-Reeb dynamics is motivated by well-known problems in fluid dynamics and celestial mechanics, where those geometric structures naturally appear. In particular, we prove that the dynamics on positive energy level-sets in the restricted planar circular three body problem are described by the Reeb vector field of a $b^3$-contact form that admits an infinite number of periodic orbits at the critical set. Eva Miranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA AcademiaPrize 2016. C ́edric Oms is supported by an AFR-Ph.D. grant of FNR - Luxembourg National Research Fund. Eva Mi-randa and C ́edric Oms are partially supported by the grants reference number MTM2015-69135-P (MINECO/FEDER)and reference number 2017SGR932 (AGAUR). Eva Miranda was supported by aChaire d’Excellenceof theFondationSciences Math ́ematiques de Pariswhen this project started and this work has been supported by a public grant overseenby the French National Research Agency (ANR) as part of the“Investissements d’Avenir”program (reference: ANR-10-LABX-0098). This material is based upon work supported by the National Science Foundation under Grant No.DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, Califor-nia, during the Fall 2018 semester.

Motivated by the study of symplectic Lie algebroids, we study a describe a type of algebroid (called an E-tangent bundle) which is particularly well-suited to study of singular differential forms and their cohomology. This setting generalizes the study of b-symplectic manifolds, foliated manifolds, and a wide class of Poisson manifolds. We generalize Moser's theorem to this setting, and use it to construct symplectomorphisms between singular symplectic forms. We give appli- cations of this machinery (including the study of Poisson cohomology), and study specific examples of a few of them in depth.

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

We present a collection of examples borrowed from celes- tial mechanics and projective dynamics. In these examples symplectic structures with singularities arise naturally from regularization trans- formations, Appell's transformation or classical changes like McGehee coordinates, which end up blowing up the symplectic structure or lower- ing its rank at certain points. The resulting geometrical structures that model these examples are no longer symplectic but symplectic with sin- gularities which are mainly of two types: b m -symplectic and m -folded symplectic structures. These examples comprise the three body prob- lem as non-integrable exponent and some integrable reincarnations such as the two xed-center problem. Given that the geometrical and dy- namical properties of b m -symplectic manifolds and folded symplectic manifolds are well-understood [GMP, GMP2, GMPS, KMS, Ma, CGP, GL, GLPR, MO, S, GMW], 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.

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.

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

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.

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.

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

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.

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

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.

In this short note we give a complete characterization of a certain class of compact corank one Poisson manifolds, those equipped with a closed one-form defining the symplectic foliation and a closed two-form extending the symplectic form on each leaf. If such a manifold has a compact leaf, then all the leaves are compact, and furthermore the manifold is a mapping torus of a compact leaf. These manifolds and their regular Poisson structures admit an extension as the critical hypersurface of a b-Poisson manifold as we will see in [GMP].

We present a normal form theorem for singular integrable systems on contact manifolds

This paper is about the rigidity of compact group actions in the Poisson context. The main result is that Hamiltonian actions of compact semisimple type are rigid. We prove it via a Nash-Moser normal form theorem for closed subgroups of SCI-type. This Nash-Moser normal form has other applications to stability results that we will explore in a future paper. We also review some classical rigidity results for differentiable actions of compact Lie groups and export it to the case of symplectic actions of compact Lie groups on symplectic manifolds.

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].

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

We prove that the singular Lagrangian foliation of a 2-degree of freedom integrable Hamiltonian system, is symplectically equivalent to the linearized foliation in a neighbourhood of a non-degenerate singular orbit.

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)....

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.

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.

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).

This thesis explores classification and perturbation problems for group actions on a class of Poisson manifolds called $b^m$-Poisson manifolds. $b^m$-Poisson manifolds are manifolds which are symplectic away from a hypersurface along which they satisfy some transversality properties. They often model problems on symplectic manifolds with boundary such as the study of their deformation quantization and celestial mechanics. One of the interesting properties of $b^m$-Poisson manifolds is that their study can be achieved considering the language of $b^m$-forms. That is to say, we can work with forms which are symplectic away from the critical set and admit a smooth extension as a \emph{form over a Lie algebroid} generalizing De Rham forms as form over the standard Lie algebroid of the tangent bundle of the manifold. To consider $b^m$-forms the standard tangent bundle is replaced by the $b^m$-tangent bundle. This thesis starts with the equivariant classification of $b^m$-Poisson structures investigating, in particular, the analogue of Moser's classification theorem for symplectic surfaces and their equivariant analogues. The classification invariants in the case of surfaces are encoded in a cohomology called $b^m$-cohomology which has been deeply studied by \cite{Scott16}. Mazzeo-Melrose type formula for $b^m$-cohomology decomposes it in two pieces which can be read off the De Rham cohomology of both the ambient manifold $M$ and the critical hypersurface. As an outcome of this identification, the Poisson classification of these manifolds is given by the De Rham cohomology of the manifold and the hypersurface. This classification is extended to the equivariant setting if we assume that the singular forms are preserved by the group action of a compact Lie group. These techniques can be extended to the classification of $b^m$-Nambu structures which are also considered in this thesis. Group actions re-appear in the last chapters as integrable systems on these manifolds turn out to have associated Hamiltonian actions of tori in a neighbourhood of a Liouville torus. We use this Hamiltonian group action to prove existence of action-angle coordinates in a neighborhood of a Liouville torus. The action-angle coordinate theorem that we prove gives a semilocal normal form in the neighbourhood of a Liouville torus for the $b^m$-symplectic structure which depends on the modular weight of the connected component of the critical set in which the Liouville torus is lying and the modular weights of the associated toric action. This action-angle theorem allows us to identify a neighborhood of the Liouville torus with the $b^m$-cotangent lift of the action of a torus acting by translations on itself. We end up this thesis proving a KAM theorem for $b^m$-Poisson manifolds which clearly refines and improves the one obtained for $b$-Poisson manifolds in \cite{KMS16}. As an outcome of this result together with the extension of the desingularization techniques of Guillemin-Miranda-Weitsman to the realm of integrable systems, we obtain a KAM theorem for folded symplectic manifolds where KAM theory has never been considered before. In the way, we also obtain a brand new KAM theorem for symplectic manifolds where the perturbation keeps track of a distinguished hypersurface. In celestial mechanics this distinguished hypersurface can be the line at infinity or the collision set. Esta tesis doctoral explora problemas de clasificación y perturbación para acciones de grupo en una clase particular de variedades de Poisson llamadas variedades de $b^m$-Poisson. Las variedades de $b^m$-Poisson son variedades que son simplécticas fuera de una hipersuperficie en la cual satisfacen ciertas propiedades de transversalidad. A menudo modelan problemas en variedades simplécticas con borde tales como el estudio de la \textcolor{black}{cuantización por deformación} o problemas de mecánica celeste. Una de las propiedades interesantes de las variedades $b^m$-Poisson es que se pueden estudiar usando el lenguage de $b^m$-formas. Es decir, que podemos trabajar con formas que son simplécticas lejos un conjunto crítico y que admiten una extensión suave como forma sobre un algebroide de Lie generalizando formas de De Rham como formas sobre el algebroide de Lie del fibrado tangente de la variedad. Para considerar $b^m$-formas el fibrado tangente estándar debe reemplazarse por el fibrado $b^m$-tangente. Esta tesis empieza con una clasificación equivariante de estructuras $b^m$-Poisson, investigando, en particular, el análogo del teorema de clasificación de Moser para superficies simplécticas y sus análogos equivariantes. La clasificación de invariantes en el caso de superfícies estan codificados en una cohomologia llamada $b^m$-cohomologia que ha sido estudiada en profundiad por \cite{Scott16}. Una fórmula del tipo de Mazzeo-Melrose para la $b^m$-cohomologia descompone en dos partes que pueden interpretarse como las cohomologias de De Rham tanto de la variedad ambiente $M$ como de la hypersuperficie crítica. Como consecuencia de esta identificación, la clasificación de Poisson de estas variedades viene dada por la cohomologia de De Rham de la variedad y de la hipersuperficie. Esta clasificación se extiende al contexto equivariante si asumimos que las formas singulares son preservadas por la acción de un grupo de Lie compacto. Estas técnicas pueden ser extendidas a la clasificación de structuras de $b^m$-Nambu que se consideran también en esta tesis. Las acciones de grupo reaparecen en los últimos capitulos ya que los sistemas integrables en estas variedades resulta que tienen asociadas acciones Hamiltonianas de toros en un entorno de un toro de Liouville. Usamos estas acciones Hamiltonianas de grupos para demostrar la existencia de coordenadas accion-ángulo en un entorno de un toro de Liouville. El teorema de acción-ángulo que demostramos da un teorema de formas normales semilocales en un entorno del toro de Liouville para la forma $b^m$-simpléctica que depende tanto del peso modular de la componente conexa de la hipersuperfície donde se encuentra el toro de Liouville como los pesos asociados a la acción tórica. Este teorema de accion-ángulo nos permite identificar un entorno del toro de Liouville como el $b^m$-cotangent lift de la acción de un toro actuano por translaciones sobre sí mismo. Acabamos la tesis demostrando un teorema KAM para variedades de $b^m$-Poisson que claramente refina y mejora el teorema obtenido para variedades de $b$-Poisson en \cite{KMS16}. Como consecuencia de este resultado junto con la extension de las técnicas de desingularización de Guillemin-Miranda-Weitsman en el ambiente de los sistemas integrables, obtenemos un teorema KAM para variedad folded-simplécticas donde la teoria KAM nunca ha sido considerada con anterioridad. En el camino, también obtenemos un nuevo teorema KAM para variedades simplécticas dónde la perturbación permite seguir con detalle una hipersuperície concreta. En mecánica celeste esta hipersuperfície puede ser interpretada cómo la línea al infinito o el conjunto de colisión.

In this master thesis we generalize a theorem by Palais on the rigidity of compact group actions to cotangent lifts. We use this result to prove rigidity for integrable systems on symplectic manifolds including systems with degenerate singularities which are invariant under a torus action. We also prove the $b$-symplectic analogue of the rigidity results. We illustrate the three basic types of singularities of integrable systems through three models from classical mechanics and we give them as cotangent lifts. Finally we review the focus-focus singularity and the saddle-focus singularity.

The study of b-symplectic manifolds was initiated in 2012 by the works of Victor Guillemin, Eva Miranda and Ana Rita Pires (Adv. Math. 264 (2014), 864¿896). These manifolds, which can be understood as symplectic manifolds with singularities, have since then become the object of intense study. In the language of Poisson tensors, a b-symplectic manifold is a manifold $M^{2n}$ with a Poisson tensor $\Pi$ such that $\Pi^n$ vanishes transversally to the zero section of the bundle $¿^{2n}TM$. This thesis contributes important results about the dynamics of b-symplectic manifolds. After reviewing the general theory of Poisson and $b$-symplectic manifolds we present the definitions of integrable systems on these manifolds. The main results of this thesis are the action-angle coordinate theorems for commutative and non-commutative b-integrable systems [KMS, KM2], which state the existence of invariant "Liouville" tori on the singular set of the b-symplectic manifold, in perfect analogy to the symplectic case. We go on to present a cotangent model of this result, identifying a neighborhood of such a Liouville torus with a certain cotangent lift of a torus action [KM1]. These models also allow us to construct examples of b-integrable systems using torus actions as a starting point. The existence of action-angle coordinates motivates us to explore an analogue of the classical stability result for symplectic manifolds known as KAM theory. We prove a result that shows stability of a large number of invariant tori under certain perturbations. Finally, we present several examples of singular symplectic structures that arise naturally as the result of non-canonical transformations used in regularization of singularities in celestial mechanics [DKM]., El estudio de variedades b-simplécticas se inició en 2012 con el trabajo de Victor Guillemin, Eva Miranda y Ana Rita Pires (Adv. Math. 264 (2014), 864-896). Estas variedades que pueden ser interpretadas como variedades simplécticas con singularidades se han convertido en un campo de investigación muy activo. Desde el punto de vista de estructuras de Poisson, una variedad b-sympléctica se define como una variedad de dimensión par (orientada) $M^{2n}$ con una estructura de Poisson $\Pi$ tal que $\Pi^n$ se anula transversalmente como sección del fibrado $¿^{2n}TM$. Esta tesis contribuye resultados importantes en geometría y dinámica de estas variedades b-simplécticas. Tras explicar las nociones básicas sobre variedades de Poisson y variedades b-simplécticas introducimos las definiciones de sistemas integrables en estas variedades. En el caso de variedades b-simplécticas los sistemas integrables que investigamos son llamados "sistemas b-integrables". Los resultados principales de esta tesis son teoremas de coordenadas acción-ángulo para sistemas b-integrables en el caso conmutativo y no conmutativo [KMS, KM2]. Este teorema demuestra la existencia de toros invariantes, llamados toros de Liouville, en el conjunto singular de la variedad b-simpléctica, y su estructura b-simpléctica en un entorno del toro. Posteriormente presentamos un modelo cotangente que nos permite de identificar un entorno de un toro Liouville con un tipo de lift cotangente de una acción de un toro [KM1]. Estos modelos también nos permiten la construcción de ejemplos de sistemas b-integrables utilizando acciones de un toro como punto de partida. El teorema de coordenadas acción-ángulo es la motivación para explorar la estibilidad de sistemas b-integrables de manera analoga al resultado clásico de la teoria KAM para variedades simplécticas. Nuestro teorema KAM demuestra la existencia de un "gran número" de toros Liouville que son invariantes bajo pequeñas perturbaciones de cierta forma. Para terminar, presentamos varios ejemplos de estructuras simplécticas singulares que aparecen como resultado de transformaciones non-canónicas en la regularización de singularidades en mecánica celeste [DKM].

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 cohomology associated to foliations induced by integrable systems. This nontrivial result turns out to be interesting in its own right, but only the aspects related to geometric quantisation are presented in the thesis, e.g. the need for a new proof that the Kostant complex is a fine resolution for the sheaf of flat sections. The thesis also provides a different proof of a theorem, firstly proved by Guillemin and Sternberg, that shows that the set of regular Bohr-Sommerfeld fibres is discrete -it not only bares the role played by circle actions, it also excludes the compactness assumption from the theorem. The exploitation of circle actions culminate in an alternative proof for the theorems of Sniatycki and Hamilton. It is an original and unifying proof: the argument works for both situations, Lagrangian fibre bundles and locally toric manifolds. In addition, this approach casts some light on a conjecture about the contributions coming from focus-focus type of singularities. It actually proves that, in degree zero, there is no contribution to geometric quantisation coming from focus-focus fibres for compact 4-dimensional almost toric manifolds.

Morse homology studies the topology of smooth manifolds by examining the critical points of a real-valued function defined on the manifold, and connecting them with the negative gradient of the function. Rather surprisingly, the resulting homology is proved to be independent of the choice of the real-valued function and metric defining the negative gradient. This leads to a topological lower bound on the number of critical points. In the 1980s, the construction of Morse homology served as a prototype to define a homology spanned by $1$-periodic Hamiltonian diffeomorphisms on symplectic manifolds. The resulting homology, introduced by Andreas Floer, spectacularly revolutionized the area of symplectic topology and leaded to a proof of the famous Arnold conjecture. Floer theory still is the subject of a lot of active and exciting research and is nowadays an essential technique in symplectic topology.