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Einstein–Weyl structures on a three-dimensional manifold M are given by a system E of PDEs on sections of a bundle over M . This system is invariant under the Lie pseudogroup G of local diffeomorphisms on M . Two Einstein–Weyl structures are locally equivalent if there exists a local diffeomorphism taking one to the other. Our goal is to describe the quotient equation E ∕ G whose solutions correspond to nonequivalent Einstein–Weyl structures. The approach uses symmetries of the Manakov–Santini integrable system and the action of the corresponding Lie pseudogroup. [ABSTRACT FROM AUTHOR]

We study the projections of an arbitrary stably Gelfand quantale Q and show that each projection determines a pseudogroup S ⊂ Q (and a corresponding localic étale groupoid G ) together with a map of involutive quantales p : Q → L ∨ ( S ) [ = O ( G ) ] . As an application we obtain a simplified axiomatization of inverse quantal frames (= quantales of étale groupoids) whereby such a quantale is shown to be the same as a unital stably Gelfand quantal frame Q whose partial units cover Q . [ABSTRACT FROM AUTHOR]

We compute the quotient of the self-duality equation for conformal metrics by the action of the diffeomorphism group. We also determine Hilbert polynomial, counting the number of independent scalar differential invariants depending on the jet-order, and the corresponding Poincaré function. We describe the field of rational differential invariants separating generic orbits of the diffeomorphism pseudogroup action, resolving the local recognition problem for self-dual conformal structures. [ABSTRACT FROM AUTHOR]

The work establishes a relation between deformed cohomologies of symmetry pseudo-groups and coverings of differential equations. Examples include the potential Khokhlov–Zabolotskaya equation and the Boyer–Finley equation. [ABSTRACT FROM AUTHOR]

The goal and the main result of the paper is to provide a complete description of the field of rational differential invariants of one class of second order ordinary differential equations with scalar control parameter with respect to Lie pseudo-group of local feedback transformations. In particular, considered class describes behavior of conservative mechanical systems. We construct the class of rational differential invariants that separate regular orbits. It is well known that differential invariants form algebra with respect to the operation of addition and multiplication (Alekseevskij et al. 1991) [20] . In our case, constructed rational differential operators form a field (in algebraic sense). Rational differential invariants were studied by Rosenlicht (1956, 1963) [25,26] , Kruglikov and Lychagin (2011) [24] . [ABSTRACT FROM AUTHOR]

We compute the Hilbert polynomial and the Poincaré function counting the number of fixed jet-order differential invariants of conformal metric structures modulo local diffeomorphisms, and we describe the field of rational differential invariants separating generic orbits of the diffeomorphism pseudogroup action. This resolves the local recognition problem for conformal structures. [ABSTRACT FROM AUTHOR]

A permutoid is a set of partial permutations that contains the identity and is such that partial compositions, when defined, have at most one extension in the set. In 2004 Peter Cameron conjectured that there can exist no algorithm that determines whether or not a permutoid based on a finite set can be completed to a finite permutation group. In this note we prove Cameron's conjecture by relating it to our recent work on the profinite triviality problem for finitely presented groups. We also prove that the existence problem for finite developments of rigid pseudogroups is unsolvable. In an appendix, Steinberg recasts these results in terms of inverse semigroups. [ABSTRACT FROM AUTHOR]

We construct the differential invariants of Lie symmetry pseudogroups of the (2+1)-dimensional breaking soliton equation and analyze the structure of the induced differential invariant algebra. Their syzygies and recurrence relations are classified. In addition, a moving frame and the invariantization of the breaking soliton equation are also presented. The algorithms are based on the method of equivariant moving frames. [ABSTRACT FROM AUTHOR]

We examine the connections between complexity of a pseudogroup, its equicontinuity, the mixing property and entropy. We prove that the entropy of a pseudogroup can be (under some additional assumptions) computed using a continuous and dynamically generating pseudometric. [ABSTRACT FROM AUTHOR]

The article presents a mathematical analysis on general problems of contact and point classifications for functions defined on the n-jet space Jn&#211D; for an arbitrary n≿2. It describes the actions of contact and point pseudogroups on the n-jet space in the Lie alegra of the contact pseudogroup contact. It uses the methods of the theory of differential invariants in solving the problem.

In this paper we suggest an approach to the study of the action of point pseudogroup in (contact) 1-jet space J 1 R n . This approach is based on the following idea. We replace the canonic projection π 1 , 0 : J 1 R n → J 0 ( R n ) from 1-jet space to 0-jet space with projective action on the fiber by some bundle, which is called symplectization , with the linear action on the fiber. This makes it possible to apply methods of studying the linear actions of algebraic groups and to find the set of independent differential invariants. Finally, we rewrite the action of point pseudogroup in terms of these invariants and classify the orbits of point pseudogroup action. We also give two examples of application of this method. In the first example we obtain classification of contact vector fields in J 1 R n , and in the second we obtain classification of symbols of linear differential operators. [ABSTRACT FROM AUTHOR]

A pseudogroup is a complete infinitely distributive inverse monoid. Such inverse monoids bear the same relationship to classical pseudogroups of transformations as frames do to topological spaces. The goal of this paper is to develop the theory of pseudogroups motivated by applications to group theory, -algebras and aperiodic tilings. Our starting point is an adjunction between a category of pseudogroups and a category of étale groupoids from which we are able to set up a duality between spatial pseudogroups and sober étale groupoids. As a corollary to this duality, we deduce a non-commutative version of Stone duality involving what we call boolean inverse semigroups and boolean étale groupoids, as well as a generalization of this duality to distributive inverse semigroups. Non-commutative Stone duality has important applications in the theory of -algebras: it is the basis for the construction of Cuntz and Cuntz–Krieger algebras and in the case of the Cuntz algebras it can also be used to construct the Thompson groups. We then define coverages on inverse semigroups and the resulting presentations of pseudogroups. As applications, we show that Paterson’s universal groupoid is an example of a booleanization, and reconcile Exel’s recent work on the theory of tight maps with the work of the second author. [ABSTRACT FROM AUTHOR]

We show that generically a pseudogroup generated by holomorphic diffeomorphisms defined about $0\in \mathbb{C} $ is free in the sense of pseudogroups even if the class of conjugacy of the generators is fixed. This result has a number of consequences on the topology of leaves for a (singular) holomorphic foliation defined on a neighborhood of an invariant curve. In particular, in the classical and simplest case arising from local nilpotent foliations possessing a unique separatrix which is given by a cusp of the form $\{ {y}^{2} - {x}^{2n+ 1} = 0\} $, our results allow us to settle the problem of showing that a generic foliation possesses only countably many non-simply connected leaves. [ABSTRACT FROM AUTHOR]

We prove that a homogeneous matchbox manifold is homeomorphic to a McCord solenoid, thereby proving a strong version of a conjecture of Fokkink and Oversteegen, which is a general form of a conjecture of Bing. A key step in the proof shows that if the foliation of a matchbox manifold has equicontinuous dynamics, then it is minimal. Moreover, we then show that a matchbox manifold with equicontinuous dynamics is homeomorphic to a weak solenoid. A result of Effros is used to conclude that a homogeneous matchbox manifold has equicontinuous dynamics, and the main theorem is a consequence. The proofs of these results combine techniques from the theory of foliations and pseudogroups, along with methods from topological dynamics and coding theory for pseudogroup actions. These techniques and results provide a framework for the study of matchbox manifolds in general, and exceptional minimal sets of smooth foliations. [ABSTRACT FROM AUTHOR]

We prove that the category of boolean inverse monoids is dually equivalent to the category of boolean groupoids. This generalizes the classical Stone duality between boolean algebras and boolean spaces. As an instance of this duality, we show that the boolean inverse monoid Cn associated with the Cuntz groupoid Gn is the strong orthogonal completion of the polycyclic (or Cuntz) monoid Pn. The group of units of Cn is the Thompson group Vn, 1. [ABSTRACT FROM AUTHOR]

According to J. Feldman and C. Moore's well-known theorem on Cartan subalgebras, a variant of the group measure space construction gives an equivalence of categories between twisted countable standard measured equivalence relations and Cartan pairs, i.e., a von Neumann algebra (on a separable Hilbert space) together with a Cartan subalgebra. A. Kumjian gave a C*-algebraic analogue of this theorem in the early eighties. After a short survey of maximal abelian self-adjoint subalgebras in operator algebras, I present a natural definition of a Cartan subalgebra in a C*-algebra and an extension of Kumjian's theorem which covers graph algebras and some foliation algebras. [ABSTRACT FROM AUTHOR]

Abstract: A singular foliation on a complete Riemannian manifold is said to be Riemannian if each geodesic that is perpendicular at one point to a leaf remains perpendicular to every leaf it meets. The singular foliation is said to admit sections if each regular point is contained in a totally geodesic complete immersed submanifold that meets every leaf orthogonally and whose dimension is the codimension of the regular leaves. A typical example of such a singular foliation is the partition by orbits of a polar action, e.g. the orbits of the adjoint action of a compact Lie group on itself. We prove that a singular Riemannian foliation with compact leaves that admits sections on a simply connected space has no exceptional leaves, i.e., each regular leaf has trivial normal holonomy. We also prove that there exists a convex fundamental domain in each section of the foliation and in particular that the space of leaves is a convex Coxeter orbifold. [Copyright &y& Elsevier]

In [Topology Appl., forthcoming paper], we have proved that for a 2-antilocal H-space there exists a strictly commutative simplicial pseudogroup structure on its minimal simplicial model. Here, we show that for a p-antilocal H-space there exists a p-variable simplicial pseudogroup structure on its minimal simplicial model which has strictly cyclic symmetry. [Copyright &y& Elsevier]