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In 1991, Moore [20] raised a question about whether hydrodynamics is capable of performing computations. Similarly, in 2016, Tao [25] asked whether a mechanical system, including a fluid flow, can simulate a universal Turing machine. In this expository article, we review the construction in [8] of a "Fluid computer" in dimension 3 that combines techniques in symbolic dynamics with the connection between steady Euler flows and contact geometry unveiled by Etnyre and Ghrist. In addition, we argue that the metric that renders the vector field Beltrami cannot be critical in the Chern-Hamilton sense [9]. We also sketch the completely different construction for the Euclidean metric in $\mathbb R^3$ as given in [7]. These results reveal the existence of undecidable fluid particle paths. We conclude the article with a list of open problems., Comment: 11 pages, 3 figures

We explore the relationship between Turing completeness and topological entropy of dynamical systems. We first prove that a natural class of Turing machines that we call "regular Turing machines" (which includes most of the examples of universal Turing machines) has positive topological entropy. We deduce that any Turing complete dynamics with a continuous encoding that simulates a universal machine in this class is chaotic. This applies to our previous constructions of Turing complete area-preserving diffeomorphisms of the disk and 3D stationary Euler flows. The article concludes with an appendix written by Ville Salo that introduces a method to construct universal Turing machines that are not regular and have zero topological entropy., Comment: 24 pages, 3 figures, includes an appendix by Ville Salo

We establish analogs of the Darboux, Moser and Weinstein theorems for prequantum systems. We show that two prequantum systems on a manifold with vanishing first cohomology, with symplectic forms defining the same cohomology class and homotopic to each other within that class, differ only by a symplectomorphism and a gauge transformation. As an application, we show that the Bohr-Sommerfeld quantization of prequantum system on a manifold with trivial first cohomology is independent of the choice of the connection., Comment: 5 pages

In this article, we study the dynamical properties of Reeb vector fields on b-contact manifolds. We show that in dimension 3, the number of so-called singular periodic orbits can be prescribed. These constructions illuminate some key properties of escape orbits and singular periodic orbits, which play a central role in formulating singular counterparts to the Weinstein conjecture and the Hamiltonian Seifert conjecture. In fact, we prove that the above-mentioned constructions lead to counterexamples of these conjectures as stated in [23]. Our construction shows that there are b-contact manifolds with no singular periodic orbit and no regular periodic orbit away from Z. We do not know whether there are constructions with no generalized escape orbits whose $\alpha$ and $\omega$-limits both lie on Z (a generalized singular periodic orbit). This is the content of the generalized Weinstein conjecture., Comment: 22 pages, 11 figures, overall improvement of the paper, formulated the generalized Weinstein conjecture

We prove that the correspondence between Reeb and Beltrami vector fields can be made equivariant whenever additional symmetries of the underlying geometric structures are considered. As a corollary of this correspondence, we show that energy levels above the maximum of the potential energy of mechanical Hamiltonian systems can be viewed as stationary fluid flows, though the metric is not prescribed. In particular, we showcase the emblematic example of the $n$-body problem and focus on the Kepler problem. We explicitly construct a compatible Riemannian metric that makes the Kepler problem of celestial mechanics a stationary fluid flow (of Beltrami type) on a suitable manifold, the Kepler-Euler flow., Comment: 16 pages, 3 figures. Overall improvements of the paper. Final section added

In this article, we introduce $b$-semitoric systems as a generalization of semitoric systems, specifically tailored for $b$-symplectic manifolds. The objective of this article is to furnish a collection of examples and investigate the distinctive characteristics of these systems. A $b$-semitoric system is a 4-dimensional $b$-integrable system that satisfies certain conditions: one of its momentum map components is proper and generates an effective global $S^1$-action, and all singular points are non-degenerate and devoid of hyperbolic components. To illustrate this concept, we provide five examples of $b$-semitoric systems by modifying the coupled spin oscillator and the coupled angular momenta, and we also classify their singular points. Additionally, we describe the dynamics of these systems through the image of their respective momentum maps., Comment: 39 pages, 8 figures

In this short note, we prove that singular Reeb vector fields associated with generic $b$-contact forms have either (at least) $2N$ or an infinite number of escape orbits, where $N$ denotes the number of connected components of the critical set., Comment: 16 pages, for Alain Chenciner on his 2N-birthday. The article has undergone substantial improvement incorporating stronger results and introducing a novel count of escape orbits with multiplicity

This monograph explores classification and perturbation problems for integrable systems on a class of Poisson manifolds called $b^m$-Poisson manifolds. Even if the class of $b^m$-Poisson manifolds is not ample enough to represent general Poisson manifolds, this investigation can be seen as a first step for the study of perturbation theory for general Poisson manifolds. We prove an action-angle coordinate and a KAM theorem for $b^m$-Poisson manifolds which improves the one obtained for $b$-Poisson manifolds for $m=1$ in [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. We also obtain a new KAM theorem for symplectic manifolds where the perturbation keeps track of a distinguished hypersurface. In several problems in celestial mechanics, this distinguished hypersurface can be the line at infinity or can represent the collision set., Comment: 117 pages, 7 figures, overall improvement, new section added

In this article, we study the Hamiltonian dynamics on singular symplectic manifolds and prove the Arnold conjecture for a large class of $b^m$-symplectic manifolds. Novel techniques are introduced to associate smooth symplectic forms to the original singular symplectic structure, under some mild conditions. These techniques yield the validity of the Arnold conjecture for singular symplectic manifolds across multiple scenarios. More precisely, we prove a lower bound on the number 1-periodic Hamiltonian orbits for $b^{2m}$-symplectic manifolds depending only on the topology of the manifold. Moreover, for $b^m$-symplectic surfaces, we improve the lower bound depending on the topology of the pair $(M,Z)$. We then venture into the study of Floer homology to this singular realm and we conclude with a list of open questions., Comment: overall improvement of the article, new results included, 50 pages, 7 figures, typos corrected, final version to appear at the Journal of Fixed Point Theory and Applications

Manifolds with boundary, with corners, $b$-manifolds and foliations model configuration spaces for particles moving under constraints and can be described as $E$-manifolds. $E$-manifolds were introduced in [NT01] and investigated in depth in [MS20]. In this article we explore their physical facets by extending gauge theories to the $E$-category. Singularities in the configuration space of a classical particle can be described in several new scenarios unveiling their Hamiltonian aspects on an $E$-symplectic manifold. Following the scheme inaugurated in [Wei78], we show the existence of a universal model for a particle interacting with an $E$-gauge field. In addition, we generalize the description of phase spaces in Yang-Mills theory as Poisson manifolds and their minimal coupling procedure, as shown in [Mon86], for base manifolds endowed with an $E$-structure. In particular, the reduction at coadjoint orbits and the shifting trick are extended to this framework. We show that Wong's equations, which describe the interaction of a particle with a Yang-Mills field, become Hamiltonian in the $E$-setting. We formulate the electromagnetic gauge in a Minkowski space relating it to the proper time foliation and we see that our main theorem describes the minimal coupling in physical models such as the compactified black hole., Comment: 38 pages, 6 figures

In this article we analyze several mathematical models with singularities where the classical cotangent model is replaced by a $b$-cotangent model. We provide physical interpretations of the singular symplectic geometry underlying in $b$-cotangent bundles featuring two models: the canonical (or non-twisted) model and the twisted one. The first one models systems on manifolds with boundary and the twisted model represents Hamiltonian systems where the singularity of the system is in the fiber of the bundle. The twisted cotangent model includes (for linear potentials) the case of fluids with dissipation. We relate the complexity of the fluids in terms of the Reynolds number and the (non)-existence of cotangent lift dynamics. We also discuss more general physical interpretations of the twisted and non-twisted $b$-symplectic models. These models offer a Hamiltonian formulation for systems which are dissipative, extending the horizons of Hamiltonian dynamics and opening a new approach to study non-conservative systems., Comment: 19 pages, 7 figures. This paper has been improved with respect to the previous version

The investigation of symmetries of b-symplectic manifolds and folded-symplectic manifolds is well-understood when the group under consideration is a torus (see, for instance, [GMPS,GLPR, GMW18a] for b-symplectic manifolds and [CGP, CM] for folded symplectic manifolds). However, reduction theory has not been set in this realm in full generality. This is fundamental, among other reasons, to advance in the "quantization commutes with reduction" programme for these manifolds initiated in [GMW18b, GMW21]. In this article, we fill in this gap and investigate the Marsden-Weinstein reduction theory under general symmetries for general $b^m$-symplectic manifolds and other singular symplectic manifolds, including certain folded symplectic manifolds. In this new framework, the set of admissible Hamiltonian functions is larger than the category of smooth functions as it takes the singularities of the differential forms into account. The quasi-Hamiltonian set-up is also considered and brand-new constructions of (singular) quasi-Hamiltonian spaces are obtained via a reduction procedure and the fusion product., Comment: 35 pages, 10 figures, minor changes

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., Comment: 19 pages, 4 figures

In this article, we pursue our investigation of the connections between the theory of computation and hydrodynamics. We prove the existence of stationary solutions of the Euler equations in Euclidean space, of Beltrami type, that can simulate a universal Turing machine. In particular, these solutions possess undecidable trajectories. Heretofore, the known Turing complete constructions of steady Euler flows in dimension 3 or higher were not associated to a prescribed metric. Our solutions do not have finite energy, and their construction makes crucial use of the non-compactness of $\mathbb R^3$, however they can be employed to show that an arbitrary tape-bounded Turing machine can be robustly simulated by a Beltrami flow on $\mathbb T^3$ (with the standard flat metric). This shows that there exist steady solutions to the Euler equations on the flat torus exhibiting dynamical phenomena of (robust) computational complexity as high as desired. We also quantify the energetic cost for a Beltrami field on $\mathbb T^3$ to simulate a tape-bounded Turing machine, thus providing additional support for the space-bounded Church-Turing thesis. Another implication of our construction is that a Gaussian random Beltrami field on Euclidean space exhibits arbitrarily high computational complexity with probability $1$. Finally, our proof also yields Turing complete flows and diffeomorphisms on $\mathbb{S}^2$ with zero topological entropy, thus disclosing a certain degree of independence within different hierarchies of complexity., Comment: overall improvement of the article, proofs revised, 37 pages, 3 figures, final version to appear at J. Math. Pures Appl

The dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by the Euler equations. In recent papers [5, 6, 7, 8] several unknown facets of the Euler flows have been discovered, including universality properties of the stationary solutions to the Euler equations. The study of these universality features was suggested by Tao as a novel way to address the problem of global existence for Euler and Navier-Stokes [28]. Universality of the Euler equations was proved in [7] for stationary solutions using a contact mirror which reflects a Beltrami flow as a Reeb vector field. This contact mirror permits the use of advanced geometric techniques in fluid dynamics. On the other hand, motivated by Tao's approach relating Turing machines to Navier-Stokes equations, a Turing complete stationary Euler solution on a Riemannian $3$-dimensional sphere was constructed in [8]. Since the Turing completeness of a vector field can be characterized in terms of the halting problem, which is known to be undecidable [30], a striking consequence of this fact is that a Turing complete Euler flow exhibits undecidable particle paths [8]. In this article, we give a panoramic overview of this fascinating subject, and go one step further in investigating the undecidability of different dynamical properties of Turing complete flows. In particular, we show that variations of [8] allow us to construct a stationary Euler flow of Beltrami type (and, via the contact mirror, a Reeb vector field) for which it is undecidable to determine whether its orbits through an explicit set of points are periodic., Comment: 24 pages, 1 figure

In this article we construct a compact Riemannian manifold of high dimension on which the time dependent Euler equations are Turing complete. More precisely, the halting of any Turing machine with a given input is equivalent to a certain global solution of the Euler equations entering a certain open set in the space of divergence-free vector fields. In particular, this implies the undecidability of whether a solution to the Euler equations with an initial datum will reach a certain open set or not in the space of divergence-free fields. This result goes one step further in Tao's programme to study the blow-up problem for the Euler and Navier-Stokes equations using fluid computers. As a remarkable spin-off, our method of proof allows us to give a counterexample to a conjecture of Moore dating back to 1998 on the non-existence of analytic maps on compact manifolds that are Turing complete., Comment: 13 pages, 1 figure

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 previous cotangent models 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 previous models 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. 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., Comment: 26 pages, 5 figures. This paper has been completely rewritten with respect to the previous version

Can every physical system simulate any Turing machine? This is a classical problem which is intimately connected with the undecidability of certain physical phenomena. Concerning fluid flows, Moore asked in [15] if hydrodynamics is capable of performing computations. More recently, Tao launched a programme based on the Turing completeness of the Euler equations to address the blow up problem in the Navier-Stokes equations. In this direction, the undecidability of some physical systems has been studied in recent years, from the quantum gap problem [7] to quantum field theories [11]. To the best of our knowledge, the existence of undecidable particle paths of 3D fluid flows has remained an elusive open problem since Moore's works in the early 1990's. In this article we construct a Turing complete stationary Euler flow on a Riemannian $S^3$ and speculate on its implications concerning Tao's approach to the blow up problem in the Navier-Stokes equations., Comment: minor changes, 16 pages, 2 figures

Motivated by the group of Galilean transformations and the subgroup of Galilean transformations which fix time zero, we introduce the notion of a $b$-Lie group as a pair $(G,H)$ where $G$ is a Lie group and $H$ is a codimension-one Lie subgroup. Such a notion allows us to give a theoretical framework for transformations of space-time where the initial time can be seen as a boundary. In this theoretical framework, we develop the basics of the theory and study the associated canonical $b$-symplectic structure on the $b$-cotangent bundle $^b {T}^\ast G$ together with its reduction theory. Namely, we extend the minimal coupling procedure to $^bT^*G/H$ and prove that the Poisson reduction under the cotangent lifted action of $H$ by left translations can be described in terms of the Lie Poisson structure on $\mathfrak{h}^\ast$ (where $\mathfrak{h}$ is the Lie algebra of $H$) and the canonical $b$-symplectic structure on $^b {T}^\ast(G/H)$, where $G/H$ is viewed as a one-dimensional $b$-manifold having as critical hypersurface (in the sense of $b$-manifolds) the identity element., Comment: this article has been completely rewritten; 11 pages, accepted for publication at Journal of Geometry and Physics. arXiv admin note: text overlap with arXiv:1811.11894

Motivated by Poincar\'e's orbits going to infinity in the (restricted) three-body (see [26] and [6]), 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 the singular counterpart [3] of Etnyre--Ghrist's contact/Beltrami correspondence [9], and genericity results concerning eigenfunctions of the Laplacian established by Uhlenbeck [29]. 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 $\alpha$- and $\omega$-limit sets intersect the critical surface. These results are a first step towards proving the singular Weinstein conjecture., Comment: 18 pages, 2 figures, minor changes

In this article we consider integrable systems on manifolds endowed with singular symplectic structures of order one. These structures are symplectic away from an hypersurface where the symplectic volume goes either to infinity or to zero in a transversal way (singularity of order one) resulting either in 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 [KM] and [KMS] 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 $b$-symplectic forms in [KM]. Global constructions of integrable systems are provided and obstructions for 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 $Z$., Comment: minor changes, 32 pages, 4 figures

In this article 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 sytems with degenerate singularities which are invariant under a torus action., Comment: 17 pages, 1 figure

In this article, we investigate Reeb dynamics on $b^m$-contact manifolds, previously introduced in [MiO], 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 $\mathbb R^+$-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., Comment: minor changes

The dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by the Euler equations. Recently, Tao launched a programme to address the global existence problem for the Euler and Navier Stokes equations based on the concept of universality. Inspired by this proposal, in this article we prove that the stationary Euler equations exhibit several universality features. More precisely, we show that any non-autonomous flow on a compact manifold can be extended to a smooth stationary solution of the Euler equations on some Riemannian manifold of possibly higher dimension. The solutions we construct are of Beltrami type, and being stationary they exist for all time. Using this result, we establish the Turing completeness of the steady Euler flows,i.e., there exist solutions that encode a universal Turing machine and, in particular, these solutions have undecidable trajectories. Our proofs deepen the correspondence between contact topology and hydrodynamics, which is key to establish the universality of the Reeb flows and their Beltrami counterparts. An essential ingredient in the proofs, of interest in itself, is a novel flexibility theorem for embeddings in Reeb dynamics in terms of an h-principle in contact geometry, which unveils the flexible behavior of the steady Euler flows. These results can be viewed as lending support to the intuition that solutions to the Euler equations can be extremely complicated in nature., Comment: 27 pages, 3 figures, minor changes, accepted for publication at Advances in Mathematics

Tichler proved that a manifold admitting a smooth closed one-form fibers over a circle. More generally a manifold admitting $k$ independent closed one-forms fibers over a torus $T^k$. In this article we explain a version of this construction for manifolds with boundary using the techniques of $b$-calculus. We explore new applications of this idea to Fluid Dynamics and more concretely in the study of stationary solutions of the Euler equations. In the study of Euler flows on manifolds, two dichotomic situations appear. For the first one, in which the Bernoulli function is not constant, we provide a new proof of Arnold's structure theorem and describe $b$-symplectic structures on some of the singular sets of the Bernoulli function. When the Bernoulli function is constant, a correspondence between contact structures with singularities and what we call $b$-Beltrami fields is established, thus mimicking the classical correspondence between Beltrami fields and contact structures. These results provide a new technique to analyze the geometry of steady fluid flows on non-compact manifolds with cylindrical ends., Comment: 18 pages

Moser proved in 1965 in his seminal paper that two volume forms on a compact manifold can be conjugated by a diffeomorphism, that is to say they are equivalent, if and only if their associated cohomology classes in the top cohomology group of a manifold coincide. In particular, this yields a classification of compact symplectic surfaces in terms of De Rham cohomology. In this paper we generalize these results for volume forms admitting transversal zeroes. In this case there is also a cohomology capturing the classification: the relative cohomology with respect to the set of critical hypersurface. We compare this classification scheme with the classification of Poisson structures on surfaces which are symplectic away from a hypersurface where they fulfill a transversality assumption ($b$-Poisson structures). We do this using the desingularization technique introduced by Guillemin-Miranda-Weitsman and extend it to $b^m$-Nambu structures., Comment: 10 pages, 2 figures

In this article, motivated by the study of symplectic structures on manifolds with boundary and the systematic study of $b$-symplectic manifolds started in [12], we prove a slice theorem for Lie group actions on $b$-symplectic manifolds., Comment: 18 pages, 2 figure, final form accepted at Bulletin of the London Mathematical Society

In this article we introduce and analyze in detail singular contact structures, with an emphasis on $b^m$-contact structures, which are tangent to a given smooth hypersurface $Z$ and satisfy certain transversality conditions. These singular contact structures are determined by the kernel of non-smooth differential forms, called $b^m$-contact forms, having an associated critical hypersurface $Z$. We provide several constructions, prove local normal forms, and study the induced structure on the critical hypersurface. The topology of manifolds endowed with such singular contact forms are related to smooth contact structures via desingularization. The problem of existence of $b^m$-contact structures on a given manifold is also tackled in this paper. We prove that a connected component of a convex hypersurface of a contact manifold can be realized as a connected component of the critical set of a $b^m$-contact structure. In particular, given an almost contact manifold $M$ with a hypersurface $Z$, this yields the existence of a $b^{2k}$-contact structure on $M$ realizing $Z$ as a critical set. As a consequence of the desingularization techniques in [GMW], we prove the existence of folded contact forms on any almost contact manifold., Comment: 28 pages, new high dimensional construction added

In this paper, we show that every singular fiber of the Gelfand--Cetlin system on coadjoint orbits of unitary groups is a smooth isotropic submanifold which is diffeomorphic to a $2$-stage quotient of a compact Lie group by free actions of two other compact Lie groups. In many cases, these singular fibers can be shown to be homogeneous spaces or even diffeomorphic to compact Lie groups. We also give a combinatorial formula for computing the dimensions of all singular fibers, and give a detailed description of these singular fibers in many cases, including the so-called (multi-)diamond singularities. These (multi-)diamond singular fibers are degenerate for the Gelfand--Cetlin system, but they are Lagrangian submanifolds diffeomorphic to direct products of special unitary groups and tori. Our methods of study are based on different ideas involving complex ellipsoids, Lie groupoids, and also general ideas coming from the theory of singularities of integrable Hamiltonian systems., Comment: version 2, 32 pages, new corrected proof of a result on the isotropic character of the singular fibers

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 applications of this machinery (including the study of Poisson cohomology), and study specific examples of a few of them in depth., Comment: 19 pages

We study the formal geometric quantization of $b^m$-symplectic manifolds equipped with Hamiltonian actions of a torus $T$ with nonzero leading modular weight. The resulting virtual $T$-modules are finite dimensional when $m$ is odd, as in [GMW2]; when $m$ is even, these virtual modules are not finite dimensional, and we compute the asymptotics of the representations for large weight., Comment: 7 pages

We prove a convexity theorem for the image of the moment map of a Hamiltonian torus action on a $b^m$-symplectic manifold., Comment: 6 pages, 1 figure

In the first part of this paper we revisit a classical topological theorem by Tischler (1970) and deduce a topological result about compact manifolds admitting a set of independent closed forms proving that the manifold is a fibration over a torus. As an application we reprove the Liouville theorem for integrable systems asserting that the invariant sets or compact connected fibers of a regular integrable system are tori. We give a new proof of this theorem (including the non-commutative version) for symplectic and more generally Poisson manifolds., Comment: 8 pages

We prove that for regular Poisson manifolds, the zeroth homology group is isomorphic to the top foliated cohomology group and we give some applications. In particular, we show that for regular unimodular Poisson manifolds top Poisson and foliated cohomology groups are isomorphic. Inspired by the symplectic setting, we define what is a perfect Poisson manifold. We use these Poisson homology computations to provide families of perfect Poisson manifolds., Comment: 8 pages

In this paper we analyze in detail a collection of motivating examples to consider $b^m$-symplectic forms and folded-type symplectic structures. In particular, we provide models in Celestial Mechanics for every $b^m$-symplectic structure. At the end of the paper, we introduce the odd-dimensional analogue to $b$-symplectic manifolds: $b$-contact manifolds., Comment: 14 pages, 1 figure