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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.

We define the Bohr-Sommerfeld quantization via T-modules for a b-symplectic toric manifold and show that it coincides with the formal geometric quantization of [GMW18b]. In particular, we prove that its dimension is given by a signed count of the integral points in the moment polytope of the toric action on the manifold. We define the Bohr-Sommerfeld quantization via T-modules for a b-symplectic toric manifold and show that it coincides with the formal geometric quantization of [GMW18b]. In particular, we prove that its dimension is given by a signed count of the integral points in the moment polytope of the toric action on the manifold. Finançat pel projecte ICREA ACADEMIA 2016-05 i ICREA ACADEMIA 2021

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 Poincare’s orbits going to infinity in the (restricted) three-body problem ´ (see [29] and [7]), we investigate the generic existence of heteroclinic-like orbits in a neighbourhood of the critical set of a b-contact form. This is done by using a singular counterpart [4] of Etnyre– Ghrist’s contact/Beltrami correspondence [11], and genericity results concerning eigenfunctions of the Laplacian established by Uhlenbeck [33]. Specifically, we analyze the b-Beltrami vector fields on b-manifolds of dimension 3 and prove that for a generic asymptotically exact b-metric they exhibit escape orbits. We also show that a generic asymptotically symmetric b-Beltrami vector field on an asymptotically flat b-manifold has a generalized singular periodic orbit and at least 4 escape orbits. Generalized singular periodic orbits are trajectories of the vector field whose a- and ¿-limit sets intersect the critical surface. These results are a first step towards proving the singular Weinstein conjecture. E. M. is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2016. Eva Miranda and Cedric Oms are supported by the grants reference number MTM2015-69135-P (MINECO/FEDER) ánd reference number 2017SGR932 (AGAUR) and the project PID2019-103849GB-I00 / AEI / 10.13039/501100011033. C. O. has been supported by an FNR-AFR PhD predoctoral grant (project GLADYSS) until October 2nd, 2020 and by a SECTI-Postdoctoral grant financed by Eva Miranda’s ICREA Academia immediately after. D. P.-S. is supported by the grants MTM PID2019-106715GB-C21 (MICINN) and Europa Excelencia EUR2019- 103821 (MCIU). This work is supported in part by the ICMAT–Severo Ochoa grant SEV-2015-0554 and the CSIC grant 20205CEX001.

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.

Les teories gauge descriuen la dinàmica d'una partícula clàssica amb graus interns de llibertat. L'espai de configuracions és un fibrat associat a un fibrat principal $\pi\colon P \longrightarrow M$ amb grup d'estructura $G$, i les equacions de moviment reben el nom d'equacions de Wong. Les teories Yang-Mills, les quals són la base del model estàndard de la física, són una teoria gauge amb grup d'estructura $G = \operatorname{U}(1) \times \operatorname{SU}(2) \times \operatorname{SU}(3)$. Weinstein va provar que l'espai de fase natural d'una teoria gauge es pot descriure mitjançant una reducció simplèctica i que les equacions de Wong són hamiltonianes. Respecte de l'espai de fase en una teoria de Yang-Mills, Montgomery va observar que l'espai de fase de Weinstein és una fulla simplèctica d'una varietat de Poisson i va generalitzar l'isomorfisme de Weinstein amb l'espai de fase de Sternberg a varietats de Poisson. Aquest resultat rep el nom d'acoblament mínim. En aquest treball estenem els resultats de Weinstein i Montgomery a $E$-varietats, on el fibrat tangent se substitueix per un subfibrat anomenat fibrat $E$-tangent. Les varietats amb vora, les varietats amb cantonades i les varietats foliades són exemples d'$E$-varietats. Fent servir resultats coneguts, hem provat que els formalismes de les teories clàssiques de gauge i de Yang-Mills, proposades per Weinstein i Montgomery, respectivament, es poden aplicar quan l'espai base $M$ és una $E$-varietat. Es pot aplicar aquest resultat, per exemple, a l'estudi del model estàndard clàssic en un espai-temps compactificat de Penrose. Las teorías gauge describen la dinámica de una partícula clásica con grados internos de libertad. El espacio de configuraciones es un fibrado asociado a un fibrado principal $\pi\colon P \longrightarrow M$ con grupo de estructura $G$, y las ecuaciones de movimiento reciben el nombre de ecuaciones de Wong. Las teorías Yang-Mills, las cuales son la base del modelo estándar de la física, son una teoría gauge con grupo de estructura $G = \operatorname{U}(1) \times \operatorname{SU}(2) \times \operatorname{SU}(3)$. Weinstein demostró que el espacio de fase natural en una teoría gauge se puede obtener por medio de una reducción simpléctica y que las ecuaciones de Wong son hamiltonianas. Respecto al espacio de fases en una teoría de Yang-Mills, Montgomery observó que el espacio de fases de Weinstein es una hoja simpléctica de un espacio de Poisson y generalizó el isomorfismo de Weinstein con el espacio de fases de Sternberg a variedades de Poisson. Este resultado se conoce como el procedimiento de acoplamiento mínimo. En este trabajo extendemos los resultados conocidos para teorías de gauge clásicas a $E$-variedades, en las que el fibrado tangente se reemplaza por un subfibrado integrable llamado fibrado $E$-tangente. Las variedades con borde, las variedades con esquinas y las variedades foliadas son ejemplos de $E$-variedades. Usando resultados conocidos en la literatura, hemos demostrado que los formalismos de las teorías de gauge y de Yang-Mills clásicas, propuestos por Weinstein y Montgomery, respectivamente, son válidos cuando el espacio base $M$ es una $E$-variedad. Este resultado puede aplicarse, por ejemplo, para el estudio del modelo estándar clásico en un espacio-tiempo compactificado de Penrose. Gauge theories describe the dynamics of a classical particle with internal degrees of freedom. The natural configuration space is associated to a $G$-principal bundle $\pi\colon P \longrightarrow M$ and the equations of motion are called Wong's equations. Yang-Mills theories, which are the basis of the standard model of physics, are a gauge theory with structure group $G = \operatorname{U}(1) \times \operatorname{SU}(2) \times \operatorname{SU}(3)$. Weinstein observed that the natural phase space of gauge theories can be obtained by a symplectic reduction, and that Wong's equations are Hamiltonian. For the natural phase space of Yang-Mills theories, Montgomery observed that the natural phase space of Weinstein is a symplectic leaf of a Poisson manifold, and generalized Weinstein's isomorphism with Sternberg's phase space to Poisson manifolds. This result is called the minimal coupling procedure. In this work, we extend this setting to classical gauge theories over $E$-manifolds, in which the tangent bundle is replace by an integrable subbundle, called the $E$-tangent bundle. Manifolds with boundary, manifolds with corners, and foliated manifolds are examples of $E$-manifolds. Using standard results in the literature, we have proved that the formulation of classical gauge and Yang-Mills theories by Weinstein and Montgomery, respectively, hold when the base manifold is taken to be an $E$-manifold. This result can be applied, for example, to study the classical standard model in a Penrose compactified space-time.

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.

En aquesta tesi estudiarem la relació entre les varietats simplèctiques tòriques i els sistemes integrables. Per això, presentem en primer lloc tots els preliminars necessaris de geometria simplèctica. Després presentem el teorema de Delzant i alguns exemples. Finalment, s'estudia i es presenta una demostració alternativa del teorema que relaciona les varietats amb els sistemes integrables: el teorema de Arnold-Liouville.

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.

En aquest treball es presenten les nocions de geometria simplèctica, de Poisson i b-simplèctica. En cada cas està demostrat un teorema de Arnold-Liouville per a sistemes integrables. Desprès de demostrar una part d'aquests teoremes usant eines diferents de topologia diferencial, ens centre en les varietats fold-simplèctiques. Per aquestes definim la noció de sistema integrable i demostrem un teorema de Arnold-Liouville. Finalment, analitzem l'estructura singular simplèctica de un exemple de Mecànica Celest i l'usem per demostrar un teorema d'existència d'òrbites de col.lisió-ejecció d'una manera més simple.

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.