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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

The periodic orbit conjecture states that, on closed manifolds, the set of lengths of the orbits of a non-vanishing vector field all whose orbits are closed admits an upper bound. This conjecture is known to be false in general due to a counterexample by Sullivan. However, it is satisfied under the geometric condition of being geodesible. In this work, we use the recent characterization of Eulerisable flows (or more generally flows admitting a strongly adapted one-form) to prove that the conjecture remains true for this larger class of vector fields., 12 pages, overall improvements, new title and new section 4

Can every physical system simulate any Turing machine? This is a classical problem that is intimately connected with the undecidability of certain physical phenomena. Concerning fluid flows, Moore [C. Moore, Nonlinearity 4, 199 (1991)] asked if hydrodynamics is capable of performing computations. More recently, Tao launched a program 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 to quantum-field theories. To the best of our knowledge, the existence of undecidable particle paths of three-dimensional fluid flows has remained an elusive open problem since Moore¿s works in the early 1990s. In this article, we construct a Turing complete stationary Euler flow on a Riemannian S3 and speculate on its implications concerning Tao¿s approach to the blow-up problem in the Navier¿Stokes equations., Robert Cardona was supported by the Spanish Ministry of Economy and Competitiveness, through the María de Maeztu Program for Units of Excellence in R&D (MDM-2014-0445) via an FPI grant. R.C. and E.M. are partially supported by Grants MTM2015-69135-P/FEDER, the Spanish Ministry of Science and Innovation PID2019-103849GB-I00/AEI/10.13039/501100011033, and Agència de Gestió d’Ajuts Universitaris i de Recerca Grant 2017SGR932. E.M. is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2016. D.P.-S. is supported by MICINN Grant MTM PID2019-106715GB-C21 and MCIU Grant Europa Excelencia EUR2019-103821. F.P. is supported by MICINN/FEDER Grants MTM2016-79400-P and PID2019-108936GB-C21. This work was partially supported by ICMAT–Severo Ochoa Grant CEX2019-000904-S.

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., 13 pages, 1 figure

For thousands of years, humankind has utilized star tracking to measure both time and geographical location. In the modern technological era, the problem of telling one's orientation and position from images of the stars has newfound importance when related to satellite communication systems. For example, developments in laser-based communication systems promise huge gains in data rates; however, they tend to require a much finer pointing accuracy in order to hit and track their target as compared to radio frequency due to having a more focused emission pattern. As such, satellites with laser-based communications systems require the ability to obtain their attitude with a much higher degree of accuracy than traditional radio-based communication. For example, in [1], the Mars-to-Earth optical communications system studied requires a pointing accuracy on the order of 2-5 microradians, with an estimated update clock of several hundred Hertz. For contrast, the high-gain antenna of the Mars Reconnaissance Orbiter (MRO) had a pointing accuracy requirement of 2.08 milliradians that could update at 10Hz-10kHz [2] (note that MRO did not use star trackers, but rather used the Electra radio; one can study its performance in [3]).

The goal of Delay Tolerant Networking (DTN) is to take a collection of heterogeneous, disparate connections between satellites, space assets, ground stations, and ground infrastructure and bring it together into a cohesive, functioning overlay network. Depending on the systems being considered, one can find links with a one-way light time exceeding minutes (and hours), periodic links which can sometimes be predicted by orbital mechanics, and restrictions based on the variety of capabilities built into these systems. These characteristics preclude traditional network models and routing techniques and have classically led to either rigid routing tables or purely probabilistic models. As the deeper underlying structures remain unknown, development of more DTN-optimized algorithms has lacked the necessary foundation. In a continuation of previous work, the goal of this paper is to identify and study these fundamental structures that exist in delay tolerant networks (DTN), with a focus on space networks. The current routing methodology has been to use contact graph routing (CGR) algorithms. CGR models a series of known contacts as a static graph. For CGR to work, this graph must be globally consistent and must have an accurate picture of the network. Because this is a globally controlled structure, there is little room for flexibility in the event of changes to the network which would naturally occur as the network grows. As a response to the desire for flexibility as the network changes, we introduced the mathematical structure known as sheaves to DTNs last year. The tag-line for sheaves is that they are a mathematically precise way of gluing local data together into unique global data. Thus, sheaves lend extra power to traditional models (and routing algorithms) by taking additional information and merging it, in as consistent a manner as possible, with the representation itself. The clearest example of how Earth-bound networks exhibit behavior that is “sheafy” is link state routers, which build a local-to-global picture of their network by gluing local information together into a global network, exactly as a sheaf would do. For routing within delay tolerant networks to truly exploit this structure, a deeper structure than a graph is required. In this paper, we develop sheaves that can work over directed graphs such as temporal flow networks, we construct a sheaf representation for Dijkstra's algorithm, and we outline a construction for routing sheaves capable of modeling multicast scenarios. Finally, there is a section of future work suggesting follow-on research.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 We use a new geometrical approach to the universality of Euler flows. By proving flexibility results on embeddings for Reeb flows in contact topology, we deduce some new universal properties for Euler flows. As a byproduct, we deduce the Turing completeness of stationary Euler flows, answering an open question for steady solutions. The results contained in this article are an announcement and short version of [2], where the complete list of results and proofs can be found. This work was partially supported by the ICMAT–Severo Ochoa grant SEV-2015-0554

We construct non-vanishing steady solutions to the Euler equations (for some metric) with analytic Bernoulli function in each three-manifold where they can exist: graph manifolds. Using the theory of integrable systems, any admissible Morse-Bott function can be realized as the Bernoulli function of some non-vanishing steady Euler flow. This can be interpreted as an inverse problem to Arnold's structure theorem and yields as a corollary the topological classification of such solutions. Finally, we prove that the topological obstruction holds without the non-vanishing assumption: steady Euler flows with a Morse-Bott Bernoulli function only exist on graph three-manifolds., 29 pages, 4 figures. Minor correctios, final version

Using open books, we prove the existence of a non-vanishing steady solution to the Euler equations for some metric in every homotopy class of non-vanishing vector fields of any odd dimensional manifold. As a corollary, any such field can be realized in an invariant submanifold of a contact Reeb field on a sphere of high dimension. The constructed solutions are geodesible and hence of Beltrami type, and can be modified to obtain chaotic fluids. We characterize Beltrami fields in odd dimensions and show that there always exist volume-preserving Beltrami fields which are neither geodesible nor Euler flows for any metric. This contrasts with the three dimensional case, where every volume-preserving Beltrami field is a steady Euler flow for some metric. Finally, we construct a non-vanishing Beltrami field (which is not necessarily volume-preserving) without periodic orbits in every manifold of odd dimension greater than three., Comment: 21 pages, 5 figures. Minor corrections, final version to appear at Ergodic Theory and Dynamical Systems

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., 8 pages

Tichler proved (Tischler D. 1970 Topology 9, 153-154. (doi:10.1016/0040-9383(70)90037-6)) that a manifold admitting a smooth non-vanishing and closed oneform fibres over a circle. More generally, a manifold admitting k-independent closed one-form fibres over a torus Tk. In this article, we explain a version of this construction for manifolds with boundary using the techniques of b-calculus (Melrose R. 1993 The Atiyah Patodi Singer index theorem. Research Notes in Mathematics. Wellesley, MA: A. K. Peters; Guillemin V, Miranda E, Pires AR. 2014 Adv. Math. (N. Y.) 264, 864-896. (doi:10.1016/j.aim.2014.07.032)). 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 (Miranda E, Oms C. 2018 Contact structures with singularities. https://arxiv.org/abs/1806.05638) and what we call b-Beltrami fields is established, thus mimicking the classical correspondence between Beltrami fields and contact structures (see for instance Etnyre J, Ghrist R. 2000 Trans. Am. Math. Soc. 352, 5781-5794. (doi:10.1090/S0002-9947-00-02651-9)). These results provide a new technique to analyse the geometry of steady fluid flows on non-compact manifolds with cylindrical ends., Robert Cardona is supported by FPI-BGSMath doctoral grant. Eva Miranda is supported by the CatalanInstitution for Research and Advanced Studies via an ICREA Academia Prize 2016 and partially supported bythe grants reference number MTM2015-69135-P (MINECO/FEDER) and reference number 2017SGR932 (AGAUR).Daniel Peralta-Salas is supported by the ERC Starting Grant 335079, the MTM grant 2016-76702-P, and partiallysupported by the ICMAT–Severo Ochoa grant SEV-2015-0554. This material is based upon work supported by theNational Science Foundation under Grant No. DMS-1440140 while Eva Miranda was in residence at the MathematicalSciences Research Institute in Berkeley, California, during the Fall 2018 semester.

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