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We provide a close analysis of the connections between pseudogroups, groupoids, and toposes. This analysis provides a topos perspective on both the localic germ groupoid of a pseudogroup defined by Resende, and the topological groupoid of a pseudogroup defined by Lawson and Lenz. In particular, we show how to analyze the topos of a pseudogroup using sheaf theory, leading to an examination of pseudogroup torsors. Consequently we obtain a concrete description of the category of points of an étendue. [ABSTRACT FROM AUTHOR]

We take advantage of the correspondence between pseudogroups and inverse quantal frames, and of the recent description of Morita equivalence for inverse quantal frames in terms of biprincipal bisheaves, to define Morita equivalence for pseudogroups and to investigate its applications. In particular, two pseudogroups are Morita equivalent if and only if their corresponding localic étale groupoids are. We explore the clear analogies between our definition of Morita equivalence for pseudogroups and the usual notion of strong Morita equivalence for C ⁎ -algebras and these lead to a number of concrete results. [ABSTRACT FROM AUTHOR]

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 consider finitely generated groups of real-analytic circle diffeomorphisms. We show that if such a group admits an exceptional minimal set (i.e., a minimal invariant Cantor set), then its Lebesgue measure is zero; moreover, there are only finitely many orbits of connected components of its complement. For the case of minimal actions, we show that if the underlying group is (algebraically) free, then the action is ergodic with respect to the Lebesgue measure. This provides first answers to questions due to É. Ghys, G. Hector and D. Sullivan. [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]

We prove that "generalized Lie algebroid," a geometric object which appeared recently in the literature, is a misconception. [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 prove a global algebraic version of the Lie-Tresse theorem which states that the algebra of differential invariants of an algebraic pseudogroup action on a differential equation is generated by a finite number of rational-polynomial differential invariants and invariant derivations. [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 classical theory of invariant means, which plays an important rôle in the theory of paradoxical decompositions, is based upon what are usually termed 'pseudogroups'. Such pseudogroups are in fact concrete examples of the Boolean inverse monoids which give rise to étale topological groupoids under non-commutative Stone duality. We accordingly initiate the theory of invariant means on arbitrary Boolean inverse monoids. Our main theorem is a characterization of when just such a Boolean inverse monoid admits an invariant mean. This generalizes the classical Tarski alternative proved, for example, by de la Harpe and Skandalis, but using different methods. [ABSTRACT FROM AUTHOR]

We study integrable non-degenerate Monge-Ampère equations of Hirota type in 4D and demonstrate that their symmetry algebras have a distinguished graded structure, uniquely determining those equations. This knowledge is used to deform these heavenly type equations into new integrable PDEs of the second-order with large symmetry pseudogroups. We classify the symmetric deformations obtained in this way and discuss self-dual hyper-Hermitian geometry of their solutions, thus encoding integrability via the twistor theory. [ABSTRACT FROM AUTHOR]

We study the Hopf *-algebra structures on the Hopf algebra H(1, q) over ℂ. It is shown that H(1, q) is a Hopf *-algebra if and only if | q| = 1 or q is a real number. Then the Hopf *-algebra structures on H(1, q) are classified up to the equivalence of Hopf *-algebra structures. [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.

We classify the affine connections on compact orientable surfaces for which the pseudogroup of local isometries acts transitively. We prove that such a connection is either torsion-free and flat, the Levi-Civita connection of a Riemannian metric of constant curvature or the quotient of a translation-invariant connection in the plane. This refines previous results by Opozda. [ABSTRACT FROM AUTHOR]

The action of the pseudogroup of point transformations on the set of smooth functions on the contact 1-jet space Jℝ is considered. Differential invariants of this action are found, and the problem of local point equivalence of regular smooth functions is solved. [ABSTRACT FROM AUTHOR]

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]

We show that for a given finitely generated group, its Bernoulli shift space can be equivariantly embedded as a subset of a space of pointed trees with Gromov–Hausdorff metric and natural partial action of a free group. Since the latter can be realized as a transverse space of a foliated space with leaves Riemannian manifolds, this embedding allows us to obtain a suspension of such Bernoulli shift. By a similar argument, we show that the space of pointed trees is universal for compactly generated expansive pseudogroups of transformations. [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]

Inverse semigroups are the algebraic counterparts of pseudogroups of transformations. The algebraic counterparts of atlases in differential geometry are what Wagner termed 'generalized heaps'. These are sets equipped with a ternary operation satisfying certain axioms. We prove that there is a bijective correspondence between generalized heaps and the equivalence bimodules, defined by Steinberg. Such equivalence bimodules are used to define the Morita equivalence of inverse semigroups. This paper therefore shows that the Morita equivalence of inverse semigroups is determined by Wagner's generalized heaps. [ABSTRACT FROM AUTHOR]

class of non-Dirac-hermitian many-particle quantum systems admitting entirely real spectra and unitary time-evolution is presented. These quantum models are isospectral with Dirac-hermitian systems and are exactly solvable. The general method involves a realization of the basic canonical commutation relations defining the quantum system in terms of operators those are hermitian with respect to a pre-determined positive definite metric in the Hilbert space. Appropriate combinations of these operators result in a large number of pseudo-hermitian quantum systems admitting entirely real spectra and unitary time evolution. Examples of a pseudo-hermitian rational Calogero model and XXZ spin-chain are considered. [ABSTRACT FROM AUTHOR]

The author first introduces the notion of affine structures on a ringed space and then obtains several related properties. Affine structures on a ringed space, arising from complex analytical spaces of algebraic schemes, behave like differential structures on a smooth manifold. As one does for differential manifolds, pseudogroups of affine transformations are used to define affine atlases on a ringed space. An atlas on a space is said to be an affine structure if it is maximal. An affine structure is said to be admissible if there is a sheaf on the underlying space such that they are coincide on all affine charts, which are in deed affine open sets of a scheme. In a rigour manner, a scheme is defined to be a ringed space with a specified affine structure if the affine structures make a contribution to the cases such as analytical spaces of algebraic schemes. Particularly, by the whole of affine structures on a space, two necessary and sufficient conditions, that two spaces are homeomorphic and that two schemes are isomorphic, coming from the main theorems of the paper, are obtained respectively. A conclusion is drawn that the whole of affine structures on a space and a scheme, as local data, encode and reflect the global properties of the space and the scheme, respectively. [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]

We consider the Lie algebra that corresponds to the Lie pseudogroup of all conformal transformations on the plane. This conformal Lie algebra is canonically represented as the Lie algebra of holomorphic vector fields in ℝ2≃ℂ. We describe all representations of $\mathfrak{g}$ via vector fields in J0ℝ2=ℝ3( x, y, u), which project to the canonical representation, and find their algebra of scalar differential invariants. [ABSTRACT FROM AUTHOR]

A modification of the standard product used in local field theory by means of an associative deformed product is proposed. We present a class of deformed products, one for every spin S = 0, 1/2, 1, that induces a nonlocal theory, displaying different form for different fields. This type of deformed product is naturally supersymmetric and it has an intriguing duality. [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]

The inverse problem of the Calculus of Variations for Lagrangians and Euler–Lagrange equations invariant under a pseudogroup $\mathcal{P}$ of local transformations of the base manifold is considered. Exploiting some ideas of Krupka, a theorem is proved showing that, if the configuration space consists of sections of tensor bundles or of local maps of a manifold into another, then such inverse problem is solvable whenever a certain cohomology class of $\mathcal{P}$-invariant forms on the configuration space is vanishing. In addition, for a few pseudogroups, the cohomology groups considered in the main result are explicitly determined in terms of the de Rham cohomology of the configuration space. [ABSTRACT FROM AUTHOR]

We study the equivalence problem of submanifolds with respect to a transitive pseudogroup action. The corresponding differential invariants are determined via formal theory and lead to the notions of l-variants and l-covariants, even in the case of non-integrable pseudogroup. Their calculation is based on the cohomological machinery: we introduce a complex for covariants, define their cohomology and prove the finiteness theorem. This implies the well-known Lie–Tresse theorem about differential invariants. We also generalize this theorem to the case of pseudogroup action on differential equations. [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]

Anear-identitynilpotent pseudogroup of orderm= 1 is a familyf1, . . . ,fn : (-1, 1) ? R ofC2 functions for which:for some small positive real number ?<1/10m+1 and commutators of the functionsfi of order at leastmequal the identity. We present a classification of near-identity nilpotent pseudogroups: our results are similar to those of Plante, Thurston, Farb and Franks. As an application, we classify certain foliations of nilpotent manifolds. [ABSTRACT FROM AUTHOR]