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In this paper, we study confoliations in dimensions higher than three mostly from the perspective of symplectic fillability. Our main result is that Massot-Niederkr\"uger-Wendl's bordered Legendrian open book, an object that obstructs the weak symplectic fillability of contact manifolds, admits a generalization for confoliations equipped with symplectic data. Applications include the non-fillability of the product of an overtwisted contact manifold and a class of symplectic manifolds, and the fact that Bourgeois contact structures associated with overtwisted contact manifolds admit no weak symplectic fillings for which the symplectic structure restricts at the boundary to a positive generator of the second cohomology of the torus factor. In addition, along the lines of the original $3$-dimensional work of Eliashberg and Thurston, we give a new definition of approximation and deformation of confoliations by contact structures and describe some natural examples., Comment: 55 pages. Comments are welcome

We push forward the study of higher dimensional stable Hamiltonian topology by establishing two non-density results. First, we prove that stable hypersurfaces are not $C^2$-dense in any isotopy class of embedded hypersurfaces on any ambient symplectic manifold of dimension $2n\geq 8$. Our second result is that on any manifold of dimension $2m+1\geq 5$, the set of non-degenerate stable Hamiltonian structures is not $C^2$-dense among stable Hamiltonian structures in any given stable homotopy class that satisfies a mild assumption. The latter generalizes a result by Cieliebak and Volkov to arbitrary dimensions., Comment: 26 pages

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

Ideal magnetohydrodynamic (MHD) equilibria on a Riemannian 3-manifold satisfy the stationary Euler equations for ideal fluids. A stationary solution $X$ admits a large set of ``adapted" metrics in $M$ for which $X$ solves the corresponding MHD equilibrium equations with the same pressure function. We prove different versions of the following statement: an MHD equilibrium with non-constant pressure on a compact three-manifold with or without boundary admits no continuous Killing symmetries for an open and dense set of adapted metrics. This contrasts with the classical conjecture of Grad which loosely states that an MHD equilibrium on a toroidal Euclidean domain in $\mathbb{R}^3$ with pressure function foliating the domain with nested toroidal surfaces must admit Euclidean symmetries., Comment: 38 pages. Version 2 has fixed typos, more references, and some clarifications. Results unchanged

We prove that on any closed Riemannian three-manifold $(M,g)$ the time-dependent Euler equations are non-mixing on the space of smooth volume-preserving vector fields endowed with the $C^1$-topology, for any fixed helicity and large enough energy, solving a problem posed by Khesin, Misiolek, and Shnirelman. To prove this, we introduce a new framework that assigns contact/symplectic geometry invariants to large sets of time-dependent solutions to the Euler equations on any 3-manifold with an arbitrary fixed metric. This greatly broadens the scope of contact topological methods in hydrodynamics, which so far have had applications only for stationary solutions and without fixing the ambient metric. We further use this framework to prove that spectral invariants obtained from Floer theory, concretely embedded contact homology, define new non-trivial continuous first integrals of the Euler equations in certain regions of the phase space endowed with the $C^{1,s}$-topology, producing countably many disjoint invariant open sets., Comment: 36 pages, 1 Figure. Minor corrections

Given an embedded stable hypersurface in a four-dimensional symplectic manifold, we prove that it is stable isotopic to a $C^0$-close stable hypersurface with the following property: $C^\infty$-nearby hypersurfaces are generically unstable. This shows that the stability property is neither open nor generic, independently of the isotopy class of hypersurfaces and ambient symplectic manifold. The proof combines tools from stable Hamiltonian topology with techniques in three-dimensional dynamics such as partial sections, integrability and KAM theory. On our way, we establish non-density properties of Reeb-like flows and a generic non-integrability theorem for cohomologous Hamiltonian structures and volume-preserving fields in dimension three., Comment: 47 pages, 3 figures. Minor corrections and overall improvement of the exposition

In this paper we introduce some new algebraic and geometric perspectives on networked space communications. Our main contribution is a novel definition of a time-varying graph (TVG), defined in terms of a matrix with values in subsets of the real line P(R). We leverage semi-ring properties of P(R) to model multi-hop communication in a TVG using matrix multiplication and a truncated Kleene star. This leads to novel statistics on the communication capacity of TVGs called lifetime curves, which we generate for large samples of randomly chosen STARLINK satellites, whose connectivity is modeled over day-long simulations. Determining when a large subsample of STARLINK is temporally strongly connected is further analyzed using novel metrics introduced here that are inspired by topological data analysis (TDA). To better model networking scenarios between the Earth and Mars, we introduce various semi-rings capable of modeling propagation delay as well as protocols common to Delay Tolerant Networking (DTN), such as store-and-forward. Finally, we illustrate the applicability of zigzag persistence for featurizing different space networks and demonstrate the efficacy of K-Nearest Neighbors (KNN) classification for distinguishing Earth-Mars and Earth-Moon satellite systems using time-varying topology alone., Comment: Figures updated and improved based on more exhaustive simulations. Conjecture 2.27 now has weak and strong variations

Dynamical systems and physical models defined on idealized continuous phase spaces are known to exhibit non-computable phenomena, examples include the wave equation, recurrent neural networks, or Julia sets in holomorphic dynamics. Inspired by the works of Moore and Siegelmann, we show that ideal fluids, modeled by the Euler equations, are capable of simulating poly-time Turing machines with polynomial advice on compact three-dimensional domains. This is precisely the complexity class $P/poly$ considered by Siegelmann in her study of analog recurrent neural networks. In addition, we introduce a new class of symbolic systems, related to countably piecewise linear transformations of the unit square, that is capable of simulating Turing machines with advice in real-time, contrary to previously known models., Comment: 28 pages. Minor corrections and new title. To appear in Revista Matem\'atica Iberoamericana

Given a class of embeddings into a contact or a symplectic manifold, we give a sufficient condition, that we call isocontact or isosymplectic realization, for this class to satisfy a general $h$-principle. The flexibility follows from the $h$-principles for isocontact and isosymplectic embeddings, it provides a framework for classical results, and we give two new applications. Our main result is that embeddings transverse to a contact structure satisfy a full $h$-principle in two cases: if the complement of the embedding is overtwisted, or when the intersection of the image of the formal derivative with the contact structure is strictly contained in a proper symplectic subbundle. We illustrate the general framework on symplectic manifolds by studying the universality of Hamiltonian dynamics on regular level sets via a class of embeddings., Comment: 33 pages. Minor corrections and expository improvements. To appear in Journal of Topology

A $b$-contact structure on a $b$-manifold $(M,Z)$ is a singular Jacobi structure on $M$ satisfying a transversality condition along the hypersurface $Z$. We show that, in three dimensions, $b$-contact structures with overtwisted three-dimensional leaves satisfy an existence $h$-principle that allows prescribing the induced singular foliation. We give a method to classify $b$-contact structures on a given $b$-manifold and use it to give a classification on $S^3$ with either a two-sphere or an unknotted torus as the critical surface. We also discuss generalizations to higher dimensions., Comment: 33 pages. Overall exposition improvement, minor corrections

Stable Hamiltonian structures generalize contact forms and define a volume-preserving vector field known as the Reeb vector field. We study two aspects of Reeb vector fields defined by stable Hamiltonian structures on 3-manifolds: on one hand, we classify all the examples with finitely many periodic orbits under a non-degeneracy condition; on the other, we give sufficient conditions for the existence of a supporting broken book decomposition and for the existence of a Birkhoff section., Comment: 53 pages, 4 figures. Overall improvement of the exposition and minor corrections

Let $f$ be a Morse function on a closed surface $\Sigma$ such that zero is a regular value and such that $f$ admits neither positive minima nor negative maxima. In this expository note, we show that $\Sigma\times \mathbb{R}$ admits an $\mathbb{R}$-invariant contact form $\alpha=fdt+\beta$ whose characteristic foliation along the zero section is (negative) weakly gradient-like with respect to $f$. The proof is self-contained and gives explicit constructions of any $\mathbb{R}$-invariant contact structure in $\Sigma \times \mathbb{R}$, up to isotopy. As an application, we give an alternative geometric proof of the homotopy classification of $\mathbb{R}$-invariant contact structures in terms of their dividing set., Comment: 15 pages, 5 figures. Expository note

Adapting a definition given by Bjerkevik and Lesnick for multiparameter persistence modules, we introduce an $\ell^p$-type extension of the interleaving distance on merge trees. We show that our distance is a metric, and that it upper-bounds the $p$-Wasserstein distance between the associated barcodes. For each $p\in[1,\infty]$, we prove that this distance is stable with respect to cellular sublevel filtrations and that it is the universal (i.e., largest) distance satisfying this stability property. In the $p=\infty$ case, this gives a novel proof of universality for the interleaving distance on merge trees., Comment: 20 pages

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

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., Comment: 12 pages, overall improvements, new title and new section 4

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 efforts to incorporate sheaves into networking, we seek to reinterpret pathfinding algorithms in terms of cellular sheaves, using Dijkstra's algorithm as an example. We construct sheaves on a graph with distinguished source and sink vertices, in which paths are represented by sections. The first sheaf is a very general construction that can be applied to other algorithms, while the second is created specifically to capture the decision making of Dijkstra's algorithm. In both cases, Dijkstra's algorithm can be described as a systematic process of extending local sections to global sections. We discuss the relationship between the two sheaves and summarize how other pathfinding algorithms can be interpreted in a similar way. While the sheaves presented here address paths and pathfinding algorithms, we suggest that future work could explore connections to other concepts from graph theory and other networking algorithms. This work was supported by the NASA Internship Project and SCaN Internship Project during the summer of 2020.

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

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., Comment: 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

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 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