chow s theorem and universal holonomic quantum computation a theorem from control theory relating the lie algebra generated by vector fields on a manifold to the controllability of the dynamical system is shown to apply to holonomic quantum computation conditions for deriving the holonomy algebra are presented by taking covariant derivatives of the curvature associated with a non abelian gauge connection when applied to the optical holonomic computer these conditions determine that the holonomy group of the two qubit interaction model contains su 2 su 2 in particular a universal two qubit logic gate is attainable for this model

differential geometry palatini gravity and reduction the present article deals with a formulation of the so called vacuum palatini gravity as a general variational principle in order to accomplish this goal some geometrical tools related to the geometry of the bundle of connections of the frame bundle lm are used a generalization of lagrange poincare reduction scheme to these types of variational problems allows us to relate it with the einstein hilbert variational problem relations with some other variational problems for gravity found in the literature are discussed

a geometric setting for internal motions of the quantum three body system quantum mechanics for internal motions of the three body system is set up on the basis of the complex vector bundle theory the three body system is called a triatomic molecule in the born oppenheimer approximation the internal states of the molecule are described as cross sections in the complex vector bundle assigned by an eigenvalue of the square of the total angular momentum operator this bundle is equipped with a linear connection which is a natural consequence of a geometric interpretation of the so called eckart condition the coupling of the internal motion with the rotation is understood naturally in terms of this connection the internal hamiltonian operator is obtained which includes the internal motion rotation coupling and a centrifugal potential the complex vector bundle for the triatomic molecule proves to be a trivial bundle though the geometric setting for the internal motion is independent of whether the bundle is trivial or not

quantization of poisson manifolds from the integrability of the modular function we discuss a framework for quantizing a poisson manifold via the quantization of its symplectic groupoid combining the tools of geometric quantization with the results of renault s theory of groupoid c algebras this setting allows very singular polarizations in particular we consider the case when the modular function is multiplicatively integrable i e when the space of leaves of the polarization inherits a groupoid structure if suitable regularity conditions are satisfied then one can define the quantum algebra as the convolution algebra of the subgroupoid of leaves satisfying the bohr sommerfeld conditions

dirac lie systems and schwarzian equations abstract a lie system is a system of differential equations admitting a superposition rule i e a function describing its general solution in terms of any generic set of particular solutions and some constants following ideas going back to dirac s description of constrained systems we introduce and analyze a particular class of lie systems on dirac manifolds called dirac lie systems which are associated with dirac lie hamiltonians our results enable us to investigate constants of the motion superposition rules and other general properties of such systems in a more effective way several concepts of the theory of lie systems are adapted to this dirac setting and new applications of dirac geometry in differential equations are presented as an application we analyze solutions of several types of schwarzian equations but our methods can be applied also to other classes of differential equations important for physics

integration of holomorphic lie algebroids we prove that a holomorphic lie algebroid is integrable if and only if its underlying real lie algebroid is integrable thus the integrability criteria of crainic fernandes theorem 4 1 in crainic fernandes in ann math 2 157 2003 do also apply in the holomorphic context without any modification as a consequence we prove that a holomorphic poisson manifold is integrable if and only if its real part or imaginary part is integrable as a real poisson manifold

local models and integrability of certain almost kahler 4 manifolds we classify up to a local isometry all non kahler almost kahler 4 manifolds for which the fundamental 2 form is an eigenform of the weyl tensor and whose ricci tensor is invariant with respect to the almost complex structure equivalently such almost kahler 4 manifolds satisfy the third curvature condition of a gray we use our local classification to show that in the compact case the third curvature condition of gray is equivalent to the integrability of the corresponding almost complex structure

geometric phase geometric distance and length of the curve in quantum evolution the geometric phase and the geometric distance function are intimately related via length of the curve a concept the author introduces for any parametric evolution of the quantum system he offers an interpretation of the nonadiabatic berry phase as the integral over a difference between the differentials of two geometric quantities which enables him to say that the geometric phase is just half the integral of the contracted length of the curve that the system traverses during a cyclic excursion

on dual pairs in dirac geometry in this note we discuss weak dual pairs in dirac geometry we show that this notion appears naturally when studying the problem of pushing forward a dirac structure along a surjective submersion and we prove a dirac theoretic version of libermann s theorem from poisson geometry our main result is an explicit construction of self dual pairs for dirac structures this theorem not only recovers the global construction of symplectic realizations from crainic and m rcu j symplectic geom 9 4 435 444 2011 but allows for a more conceptual understanding of it yielding a simpler and more natural proof as an application of the main theorem we present a different approach to the recent normal form theorem around dirac transversals from bursztyn et al j fur die reine und angewandte mathematik crelles j doi 10 1515 crelle 2017 0014 2017

on the local time of random processes in random scenery random walks in random scenery are processes defined by z n sum k 1 n xi x 1 x k where basically x k k ge 1 and xi y y in mathbb z are two independent sequences of i i d random variables we assume here that x 1 is zz valued centered and with finite moments of all orders we also assume that xi 0 is zz valued centered and square integrable in this case h kesten and f spitzer proved that n 3 4 z nt t ge 0 converges in distribution as n to infty toward some self similar process delta t t ge 0 called brownian motion in random scenery in a previous paper we established that mathbb p z n 0 behaves asymptotically like a constant times n 3 4 as n to infty we extend here this local limit theorem we give a precise asymptotic result for the probability for z to return to zero simultaneously at several times as a byproduct of our computations we show that delta admits a bi continuous version of its local time process which is locally holder continuous of order 1 4 delta and 1 6 delta respectively in the time and space variables for any delta 0 in particular this gives a new proof of the fact previously obtained by khoshnevisan that the level sets of delta have hausdorff dimension a s equal to 1 4 we also get the convergence of every moment of the normalized local time of z toward its continuous counterpart

banach lie groupoids and generalized inversion abstract we study a few basic properties of banach lie groupoids and algebroids adapting some classical results on finite dimensional lie groupoids as an illustration of the general theory we show that the notion of locally transitive banach lie groupoid sheds fresh light on earlier research on some infinite dimensional manifolds associated with banach algebras

linear dynamical systems coherent state manifolds flows and matrix riccati equation the classical motion and the quantum evolution determined by linear hamiltonians on coherent state manifolds of compact and noncompact hermitian symmetric space structure are considered a matrix riccati equation with opposite signs of the quadratic terms in the compact and noncompact cases determines the classical motion and the quantum evolution the geometric meaning of equations of motion as flow on a complex grassmann manifold and his noncompact dual is emphasized possibilities of generalizing the results to flag manifolds are pointed out

strong invariance for local times let y 1 y 2 be a sequence of i i d random variables with distribution p y 1 k p k k 1 2 e y 1 0 e y 1 2 2 put t n y 1 y n and n x n k 0 k n t k x extending the result of revesz 1981 it is shown that for appropriate skorohod construction we have 1 mathop text sup limits x in mathbb z text l x n sigma 2 text sigma text 2 n x n o n 1 mathord left vphantom 1 4 varepsilon right kern nulldelimiterspace 4 varepsilon text a text s text provided all moments e y 1 m m 0 exists where l is the local time of a wiener process certain rate of convergence is given also under weaker conditions and for l x n 2 2 n x n too when x is fixed

optimal control for holonomic and nonholonomic mechanical systems with symmetry and lagrangian reduction in this paper we establish necessary conditions for optimal control using the ideas of lagrangian reduction in the sense of reduction under a symmetry group the techniques developed here are designed for lagrangian mechanical control systems with symmetry the benefit of such an approach is that it makes use of the special structure of the system especially its symmetry structure and thus it leads rather directly to the desired conclusions for such systems lagrangian reduction can do in one step what one can alternatively do by applying the pontryagin maximum principle followed by an application of poisson reduction the idea of using lagrangian reduction in the sense of symmetry reduction was also obtained by bloch and crouch proc 33rd cdc ieee 1994 pp 2584 2590 in a somewhat different context and the general idea is closely related to those in montgomery comm math phys 128 1990 pp 565 592 and vershik and gershkovich dynamical systems vii v arnold and s p novikov eds springer verlag 1994 here we develop this idea further and apply it to some known examples such as optimal control on lie groups and principal bundles such as the ball and plate problem and reorientation examples with zero angular momentum such as the satellite with moveable masses however one of our main goals is to extend the method to the case of nonholonomic systems with a nontrivial momentum equation in the context of the work of bloch krishnaprasad marsden and murray arch rational mech anal 1996 to appear the snakeboard is used to illustrate the method

pseudodifferential operators on manifolds with foliated boundaries abstract let x be a smooth compact manifold with boundary for smooth foliations on the boundary of x admitting a resolution in terms of a fibration we construct a pseudodifferential calculus generalizing the fibred cusp calculus of mazzeo and melrose in particular we introduce certain symbols leading to a simple description of the fredholm operators inside the calculus when the leaves of the fibration resolving the foliation are compact we also obtain an index formula for fredholm perturbations of dirac type operators along the way we obtain a formula for the adiabatic limit of the eta invariant for invertible perturbations of dirac type operators a result of independent interest generalizing the well known formula of bismut and cheeger

control aspects of holonomic quantum computation a unifying framework for the control of quantum systems with non abelian holonomy is presented it is shown that from a control theoretic point of view holonomic quantum computation can be treated as a control system evolving on a principal fiber bundle an extension of methods developed for these classical systems may be applied to quantum holonomic systems to obtain insight into the control properties of such systems and to construct control algorithms for two established examples of the computing paradigm

integrability of lie brackets in this paper we present the solution to a longstanding problem of differential geometry lie s third theorem for lie algebroids we show that the integrability problem is controlled by two computable obstructions as applications we derive explain and improve the known integrability results we establish integrability by local lie groupoids we clarify the smoothness of the poisson sigma model for poisson manifolds and we describe other geometrical applications

the range of stable random walks limit theorems are proved for the range of d dimensional random walks in the domain of attraction of a stable process of index p3 the range rn is the number of distinct sites of zd visited by the random walk before time n our results depend on the value of the ratio 3 d the most interesting results are obtained for 2 3 1 which can only occur if d 1 we prove the convergence in distribution of rn e rn toward some constant times the lebesgue measure of the range of the limiting stable process some of our results require regularity assumptions on the characteristic function of x

classical adiabatic angles and quantal adiabatic phase a semiclassical connection is established between quantal and classical properties of a system whose hamiltonian is slowly cycled by varying its parameters round a circuit the quantal property is a geometrical phase shift gamma n associated with an eigenstate with quantum numbers n nl the classical property is a shift delta theta l i in the lth angle variable for motion round a phase space torus with actions i il the connection is delta theta l delta gamma delta nl two applications are worked out in detail the generalised harmonic oscillator with quadratic hamiltonian whose parameters are the coefficients of q2 qp and p2 and the rotated rotator consisting of a particle sliding freely round a non circular hoop slowly turned round once in its own plane

reduction of poisson nijenhuis lie algebroids to symplectic nijenhuis lie algebroids with a nondegenerate nijenhuis tensor we show how to reduce under certain regularity conditions a poisson nijenhuis lie algebroid to a symplectic nijenhuis lie algebroid with a nondegenerate nijenhuis tensor we generalize the work done by magri and morosi for the reduction of poisson nijenhuis manifolds the choice of the more general framework of lie algebroids is motivated by the geometrical study of some reduced bi hamiltonian systems an explicit example of the reduction of a poisson nijenhuis lie algebroid is also provided

rigidity of hamiltonian actions on poisson manifolds abstract this paper is about the rigidity of compact group actions in the poisson context the main result is that hamiltonian actions of compact semisimple type are rigid we prove it via a nash moser normal form theorem for closed subgroups of sci type this nash moser normal form has other applications to stability results that we will explore in a future paper we also review some classical rigidity results for differentiable actions of compact lie groups and export it to the case of symplectic actions of compact lie groups on symplectic manifolds

geometric structures as deformed infinitesimal symmetries a general model for geometric structures on differentiable manifolds is obtained by deforming infinitesimal symmetries specifically this model consists of a lie algebroid equipped with an affine connection compatible with the lie algebroid structure the curvature of this connection vanishes precisely when the structure is locally symmetric this model generalizes cartan geometries a substantial class to the intransitive case simple examples are surveyed and corresponding local obstructions to symmetry are identified these examples include foliations riemannian structures infinitesimal g structures symplectic and poisson structures

remarks on generalized lie algebroids and related concepts we prove that generalized lie algebroid a geometric object which appeared recently in the literature is a misconception

nonholonomic constraints a new viewpoint the purpose of this paper is to show that at least for lagrangians of mechanical type nonholonomic euler lagrange equations for a nonholonomic linear constraint d may be viewed as nonconstrained euler lagrange equations but on a new generally not lie algebroid structure on d the proposed novel formalism allows us to treat in a unified way a variety of situations in nonholonomic mechanics and gives rise to a version of noether theorem producing actual first integrals in the case of symmetries the purpose of this paper is to show that at least for lagrangians of mechanical type nonholonomic euler lagrange equations for a nonholonomic linear constraint d may be viewed as nonconstrained euler lagrange equations but on a new generally not lie algebroid structure on d the proposed novel formalism allows us to treat in a unified way a variety of situations in nonholonomic mechanics and gives rise to a version of noether theorem producing actual first integrals in the case of symmetries

pontryagin maximum principle on almost lie algebroids the fundamental theorem of the theory of optimal control the pontryagin maximum principle pmp is extended to the setting of almost lie al algebroids geometrical objects generalizing lie algebroids this formulation of the pmp yields in particular a scheme comprising reductions of optimal control problems similar to the reduction for the rigid body in analytical mechanics on the other hand in the presented approach the reduced and unreduced pmps are parts of the same universal formalism the framework is based on a very general concept of homotopy of measurable paths and the geometry of al algebroids

groupoids and an index theorem for conical pseudo manifolds we define an analytical index map and a topological index map for conical pseudomanifolds these constructions generalize the analogous constructions used by atiyah and singer in the proof of their topological index theorem for a smooth compact manifold m a main ingredient is a non commutative algebra that plays in our setting the role of c 0 t m we prove a thom isomorphism between non commutative algebras which gives a new example of wrong way functoriality in k theory we then give a new proof of the atiyah singer index theorem using deformation groupoids and show how it generalizes to conical pseudomanifolds we thus prove a topological index theorem for conical pseudomanifolds

a normal form theorem around symplectic leaves we prove the poisson geometric version of the local reeb stability from foliation theory and of the slice theorem from equivariant geometry which is also a generalization of conn s linearization theorem

a central limit theorem for two dimensional random walks in random sceneries let s n n in bold n be a recurrent random walk on bold z 2 s 0 0 and let xi alpha alpha in bold z 2 be i i d bold r valued centered random variables it is shown that sum n i 1 xi s i sqrt n log n satisfies a central limit theorem a functional version is also presented

on integrability of infinitesimal actions we use foliations and connections on principal lie groupoid bundles to prove various integrability results for lie algebroids in particular we show under quite general assumptions that the semi direct product associated to an infinitesimal action of one integrable lie algebroid on another is integrable this generalizes recent results of dazord and nistor

explicitly solvable non linear optimal control problems the method of collective hamiltonians is applied to the study of optimal control problems on manifolds this approach enables us to single out exactly solvable situations various examples are considered necessary results from hamiltonian formalism are given including an industrial technique for the construction of completely integrable systems

k symplectic formalism on lie algebroids in this paper we introduce a geometric description of lagrangian and hamiltonian classical field theories on lie algebroids in the framework of k symplectic geometry we discuss the relation between the lagrangian and hamiltonian descriptions through a convenient notion of legendre transformation the theory is a natural generalization of the standard one in addition other interesting examples are studied in particular systems with symmetry and poisson sigma models

algebraic constructions in the category of lie algebroids the aim of this paper is to give the first general and abstract treatment of the algebraic properties of lie algebroids the concept of lie algebroid was introduced in 1967 by pradines ls as the basic infinitesimal invariant of a differentiable groupoid the construction of the lie algebroid of a differentiable groupoid is a direct generalization of the construction of the lie algebra of a lie group and 183 described a full lie theory for differentiable groupoids and lie algebroids encompassing many phenomena in the foundations of differential geometry a more detailed account and further references are given in 13 however at this stage the algebraic properties of lie algebroids were not pursued in 13 iii section 2 iv section 11 one of us gave a fairly detailed account of the abstract algebra of lie algebroids over a fixed base lie algebroids are vector bundles with several additional structures and a category of lie algebroids on a given base and morphisms which preserve that base has properties similar to those of the category of lie algebras using this algebra it was demonstrated in 13 that the lie theory of a category of locally trivial differentiable lie groupoids over a fixed base and the corresponding category of transitive lie algebroids over that base is coextensive with the standard theory of connections in principal bundles we now propose to show that the lie functor from the category of all differentiable groupoids over arbitrary bases and arbitrary smooth morphisms to the category of all lie algebroids preserves the basic algebraic constructions known to be possible in the category of differentiable groupoids the main work is to give abstract formulations of the corresponding lie algebroid concepts the groupoid results with which we

reduced classical field theories k cosymplectic formalism on lie algebroids in this paper we introduce a geometric description of lagrangian and hamiltonian classical field theories on lie algebroids in the framework of k cosymplectic geometry we discuss the relation between lagrangian and hamiltonian descriptions through a convenient notion of legendre transformation the theory is a natural generalization of the standard one in addition other interesting examples are studied mainly on reduction of classical field theories

isoholonomic problems and some applications we study the problem of finding the shortest loops with a given holonomy we show that the solutions are the trajectories of particles in yang mills potentials theorem 4 or equivalently the projections of kaluza klein geodesics theorem 2 applications to quantum mechanics berry s phase sect 3 and the optimal control of deformable bodies sect 6 are touched upon

homology of algebras of families of pseudodifferential operators abstract we compute the hochschild cyclic and periodic cyclic homology groups of algebras of families of laurent complete symbols on manifolds with corners we show in particular that the spectral sequence associated with hochschild homology degenerates at e 2 and converges to hochschild homology as a byproduct we identify the space of residue traces on fibrations by manifolds with corners in the process we prove some structural results about algebras of complete symbols on manifolds with corners

classical field theory on lie algebroids variational aspects the variational formalism for classical field theories is extended to the setting of lie algebroids given a lagrangian function we study the problem of finding critical points of the action functional when we restrict the fields to being morphisms of lie algebroids in addition to the standard case our formalism includes as particular examples the case of systems with symmetry covariant euler poincare and lagrange poincare cases sigma models or chern simons theories

representations up to homotopy of lie algebroids we introduce and study the notion of representation up to homotopy of a lie algebroid paying special attention to examples we use representations up to homotopy to define the adjoint representation of a lie algebroid and show that the resulting cohomology controls the deformations of the structure the weil algebra of a lie algebroid is defined and shown to coincide with kalkman s brst model for equivariant cohomology in the case of group actions the relation of this algebra with the integration of poisson and dirac structures is explained in arias abad crainic ann inst fourier

lagrangian submanifolds and dynamics on lie algebroids in some previous papers a geometric description of lagrangian mechanics on lie algebroids has been developed in this topical review we give a hamiltonian description of mechanics on lie algebroids in addition we introduce the notion of a lagrangian submanifold of a symplectic lie algebroid and we prove that the lagrangian hamiltonian dynamics on lie algebroids may be described in terms of lagrangian submanifolds of symplectic lie algebroids the lagrangian hamiltonian formalism on lie algebroids permits us to deal with lagrangian hamiltonian functions not defined necessarily on tangent cotangent bundles thus we may apply our results to the projection of lagrangian hamiltonian functions which are invariant under the action of a symmetry lie group as a consequence we obtain that lagrange poincare hamilton poincare equations are the euler lagrange hamilton equations associated with the corresponding atiyah algebroid moreover we prove that lagrange poincare hamilton poincare equations are the local equations defining certain lagrangian submanifolds of symplectic atiyah algebroids

variational calculus with constraints on general algebroids variational calculus on a vector bundle e equipped with a structure of a general algebroid is developed together with the corresponding analogs of euler lagrange equations constrained systems are introduced in the variational and geometrical settings the constrained euler lagrange equations are derived for analogs of holonomic vakonomic and nonholonomic constraints this general model covers the majority of first order lagrangian systems which are present in the literature and reduces to the standard variational calculus and the euler lagrange equations in classical mechanics for e tm

oscillating about coplanarity in the 4 body problem for the newtonian 4 body problem in space we prove that any zero angular momentum bounded solution suffers infinitely many coplanar instants that is times at which all 4 bodies lie in the same plane this result generalizes a known result for collinear instants syzygies in the zero angular momentum planar 3 body problem and extends to the d 1 body problem in d space the proof begins by identifying the translation reduced configuration space with real d times d matrices the degeneration locus set of coplanar configurations when d 3 with the set of matrices having determinant zero and the mass metric with the frobenius standard euclidean norm let s denote the signed distance from a matrix to the hypersurface of matrices with determinant zero the proof hinges on establishing a harmonic oscillator type ode for s along solutions bounds on inter body distances then yield an explicit lower bound omega for the frequency of this oscillator guaranteeing a degeneration within every time interval of length pi omega the non negativity of the curvature of oriented shape space the quotient of the translation reduced configuration space by the rotation group plays a crucial role in the proof

connections in poisson geometry i holonomy and invariants we discuss contravariant connections on poisson manifolds for vector bun dles the corresponding operational notion of a contravariant derivative had been introduced by i vaisman we show that these connections play an important role in the study of global properties of poisson manifolds and we use them to define poisson holonomy and new invariants of poisson manifolds

exact solutions of the isoholonomic problem and the optimal control problem in holonomic quantum computation the isoholonomic problem in a homogeneous bundle is formulated and solved exactly the problem takes a form of a boundary value problem of a variational equation the solution is applied to the optimal control problem in holonomic quantum computer we provide a prescription to construct an optimal controller for an arbitrary unitary gate and apply it to a k dimensional unitary gate which operates on an n dimensional hilbert space with n 2k our construction is applied to several important unitary gates such as the hadamard gate the cnot gate and the two qubit discrete fourier transformation gate controllers for these gates are explicitly constructed

the mechanics and control for multi particle systems this paper deals with the mechanics and control for multi particle systems from a geometric point of view the centre of mass system is viewed as a principal fibre bundle with structure group so 3 the base space of which is called the internal or shape space a natural connection and a natural riemannian metric are both defined on the centre of mass system the equations of motion for the multi particle system are derived in the lagrangian formalism adapted to the bundle structure and then reduced with the conserved total angular momentum in contrast with this the control problem is studied with non holonomic constraints i e with the vanishing total angular momentum and equations of motion are determined for an optimally controlled multi particle system the resultant equations derived in each of the mechanical and control systems are to be compared

deformation quantization and the baum connes conjecture alternative titles of this paper would have been index theory without index or the baum connes conjecture without baum

symmetry reduction of optimal control systems and principal connections this paper explores the role of symmetries and reduction in nonlinear control and optimal control systems the focus of the paper is to give a geometric framework of symmetry reduction of optimal control systems as well as to show how to obtain explicit expressions of the reduced system by exploiting the geometry in particular we show how to obtain a principal connection to be used in the reduction for various choices of symmetry groups as opposed to assuming such a principal connection is given or choosing a particular symmetry group to simplify the setting our result synthesizes some previous works on symmetry reduction of nonlinear control and optimal control systems affine and kinematic optimal control systems are of particular interest we explicitly work out the details for such systems and also show a few examples of symmetry reduction of kinematic optimal control problems

reduction in principal bundles covariant lagrange poincare equations a general reduction theory for field theoretical lagrangians on principal fiber bundles is presented the reduced variational problem as well as the associated equations are formulated the link between the solutions of the original problem and that of the reduced problem is discussed and an obstruction to the reconstruction of the solutions is isolated the important case of semidirect products is discussed in detail and several concrete examples are presented

diffusion in layered random flows polymers electrons in random potentials and spin depolarization in random fields several related models are studied in a common framework we first reconsider the model of matheron and de marsilly for anomalous tracer dispersion in a stratified porous medium in each horizontal layer the flow velocity is constant parallel to the layer and depends randomly on the vertical coordinate z this model is mapped onto ad 1 localization problem in a random potential and equivalently onto ad 1 polymer at larget theaveraged distribution of horizontal displacementsx takes the scaling form p x t z 0 at 5 4q bxt 3 4 whereq y is independent of the details of the model q y a andb are obtained exactly for a large class of models from the lifschitz tails of the localization problem we find in the regionx t3 4 i e y thatq y y exp c y 4 3 we also obtain exactly ind 1 the scaling functions for the local and total average magnetization of spins diffusing in a random magnetic field by mapping onto a polymer problem as well as the average local concentration for diffusion in the presence of random sources and sinks these mappings are then used to study higher dimensional extensions of these models

a geometric approach to conn s linearization theorem we give a soft geometric proof of the classical result due to conn stating that a poisson structure is linearizable around a singular point zero at which the isotropy lie algebra is compact and semisimple

reduction theory and the lagrange routh equations reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of euler jacobi lagrange hamilton routh poincare and others the modern vision of mechanics includes besides the traditional mechanics of particles and rigid bodies field theories such as electromagnetism fluid mechanics plasma physics solid mechanics as well as quantum mechanics and relativistic theories including gravity symmetries in these theories vary from obvious translational and rotational symmetries to less obvious particle relabeling symmetries in fluids and plasmas to subtle symmetries underlying integrable systems reduction theory concerns the removal of symmetries and their associated conservation laws variational principles along with symplectic and poisson geometry provide fundamental tools for this endeavor reduction theory has been extremely useful in a wide variety of areas from a deeper understanding of many physical theories including new variational and poisson structures to stability theory integrable systems as well as geometric phases this paper surveys progress in selected topics in reduction theory especially those of the last few decades as well as presenting new results on non abelian routh reduction we develop the geometry of the associated lagrange routh equations in some detail the paper puts the new results in the general context of reduction theory and discusses some future directions

on some 4 dimensional compact einstein almost kahler manifolds soit m m j une variete a 4 dimensions compacte d einstein presque de kahler dont la courbure scalaire est non negative alors m est une variete de kahler

integration of twisted dirac brackets given a lie groupoid g over a manifold m we show that multiplicative 2 forms on g relatively closed with respect to a closed 3 form phi on m correspond to maps from the lie algebroid of g into t m satisfying an algebraic condition and a differential condition with respect to the phi twisted courant bracket this correspondence describes as a special case the global objects associated to phi twisted dirac structures as applications we relate our results to equivariant cohomology and foliation theory and we give a new description of quasi hamiltonian spaces and group valued momentum maps

control and stabilization of nonholonomic dynamic systems a class of inherently nonlinear control problems has been identified the nonlinear features arising directly from physical assumptions about constraints on the motion of a mechanical system models are presented for mechanical systems with nonholonomic constraints represented both by differential algebraic equations and by reduced state equations control issues for this class of systems are studied and a number of fundamental results are derived although a single equilibrium solution cannot be asymptotically stabilized using continuous state feedback a general procedure for constructing a piecewise analytic state feedback which achieves the desired result is suggested

symplectic 4 manifolds with hermitian weyl tensor it is proved that any compact almost kihler einstein 4 manifold whose fundamental form is a root of the weyl tensor is necessarily kihler

quantal phase factors accompanying adiabatic changes a quantal system in an eigenstate slowly transported round a circuit c by varying parameters r in its hamiltonian r will acquire a geometrical phase factor exp i c in addition to the familiar dynamical phase factor an explicit general formula for c is derived in terms of the spectrum and eigenstates of r over a surface spanning c if c lies near a degeneracy of c takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy as an illustration c is calculated for spinning particles in slowly changing magnetic fields although the sign reversal of spinors on rotation is a special case the effect is predicted to occur for bosons as well as fermions and a method for observing it is proposed it is shown that the aharonov bohm effect can be interpreted as a geometrical phase factor

lie algebroids holonomy and characteristic classes abstract we extend the notion of connection in order to study singular geometric structures namely we consider a notion of connection on a lie algebroid which is a natural extension of the usual concept of a covariant connection it allows us to define holonomy of the orbit foliation of a lie algebroid and prove a stability theorem we also introduce secondary or exotic characteristic classes thus providing invariants which generalize the modular class of a lie algebroid

vector bundles over lie groupoids and algebroids abstract we study vb groupoids and vb algebroids which are vector bundles in the realm of lie groupoids and lie algebroids through a suitable reformulation of their definitions we elucidate the lie theory relating these objects i e their relation via differentiation and integration we also show how to extend our techniques to describe the more general lie theory underlying double lie algebroids and la groupoids

a local limit theorem for random walks in random scenery and on randomly oriented lattices random walks in random scenery are processes defined by z n sum k 1 n xi x 1 x k where x k k ge 1 and xi y y in mathbb z are two independent sequences of i i d random variables we assume here that their distributions belong to the normal domain of attraction of stable laws with index alpha in 0 2 and beta in 0 2 respectively these processes were first studied by h kesten and f spitzer who proved the convergence in distribution when alpha neq 1 and as n to infty of n delta z n for some suitable delta 0 depending on alpha and beta here we are interested in the convergence as n to infty of n delta mathbb p z n lfloor n delta x rfloor when x in rr is fixed we also consider the case of random walks on randomly oriented lattices for which we obtain similar results

reduction of some classical non holonomic systems with symmetry two types of nonholonomic systems with symmetry are treated i the configuration space is a total space of a g principal bundle and the constraints are given by a connection ii the configuration space is g itself and the constraints are given by left invariant forms the proofs are based on the method of quasicoordinates in passing a derivation of the maurer cartan equations for lie groups is obtained simple examples are given to illustrate the algorithmical character of the main results

gravity from lie algebroid morphisms inspired by the poisson sigma model and its relation to 2d gravity we consider models governing morphisms from t to any lie algebroid e where is regarded as a d dimensional spacetime manifold we address the question of minimal conditions to be placed on a bilinear expression in the 1 form fields sij x aiaj so as to permit an interpretation as a metric on this becomes a simple compatibility condition of the e tensor s with the chosen lie algebroid structure on e for the standard lie algebroid e tm the additional structure is identified with a riemannian foliation of m in the poisson case e t m with a sub riemannian structure which is poisson invariant with respect to its annihilator bundle for integrable image of s this means that the induced riemannian leaves should be invariant with respect to all hamiltonian vector fields of functions which are locally constant on this foliation this provides a huge class of new gravity models in d dimensions embedding known 2d and 3d models as particular examples

holonomy reduced dynamics of triatomic molecules whereas it is easy to reduce the translational symmetry of a molecular system using e g jacobi coordinates the situation is much more involved for rotational symmetry in this paper we address the latter problem using holonomy reduction we suggest that the configuration space may be considered as the reduced holonomy bundle with a connection induced by the mechanical connection using the fact that for the special case of the three body problem the holonomy group is so 2 as opposed to so 3 like in systems with more than three bodies we obtain a holonomy reduced configuration space of topology the dynamics then takes place on the cotangent bundle over the holonomy reduced configuration space on this phase space there is an s1 symmetry action coming from the conserved reduced angular momentum which can be reduced using the standard symplectic reduction method using a theorem by arnold it follows that the resulting symmetry reduced phase space is again a natural mechanical phase space i e a cotangent bundle this is different from what is obtained from the usual approach where symplectic reduction is used from the outset this difference is discussed in some detail and a connection between the reduced dynamics of a triatomic molecule and the motion of a charged particle in a magnetic field is established

multiplicative forms and spencer operators motivated by our attempt to recast cartan s work on lie pseudogroups in a more global and modern language we are brought back to the question of understanding the linearization of multiplicative forms on groupoids and the corresponding integrability problem from this point of view the novelty of this paper is that we study forms with coefficients however the main contribution of this paper is conceptual the discovery of the relationship between multiplicative forms and cartan s work is explored here to provide a completely new approach to integrability theorems for multiplicative forms the multiplicative point of view shows that modulo lie s functor the cartan pfaffian system itself a multiplicative form with coefficients is the same thing as the classical spencer operator

extended hamiltonian systems in multisymplectic field theories we consider hamiltonian systems in first order multisymplectic field theories we review the properties of hamiltonian systems in the so called restricted multimomentum bundle including the variational principle which leads to the hamiltonian field equations in an analogous way to how these systems are defined in the so called extended symplectic formulation of nonautonomous mechanics we introduce hamiltonian systems in the extended multimomentum bundle the geometric properties of these systems are studied the hamiltonian equations are analyzed using integrable multivector fields the corresponding variational principle is also stated and the relation between the extended and the restricted hamiltonian systems is established all these properties are also adapted to certain kinds of submanifolds of the multimomentum bundles in order to cover the case of almost regular field theories

symplectic groupoids and poisson manifolds on definit la notion de groupoide symplectique on donne la variete de poisson de ce groupoide symplectique on presente une construction

boutet de monvel operators on lie manifolds with boundary abstract we introduce and study a general pseudodifferential calculus for boundary value problems on a class of non compact manifolds with boundary so called lie manifolds with boundary this is accomplished by constructing a suitable generalization of the boutet de monvel calculus for boundary value problems the data consists of a compact manifold with corners m that is endowed with a lie structure of vector fields 2 v a so called lie manifold the manifold m is split into two equal parts x and x which intersect in an embedded hypersurface y x our goal is to describe a transmission boutet de monvel calculus for boundary value problems compatible with the structure of lie manifolds starting with the example of b vector fields we show that there are two groupoids integrating the lie structures on m and on y respectively these two groupoids form a bibundle or a groupoid correspondence and under some mild assumptions these groupoids are morita equivalent with the help of the bibundle structure and canonically defined manifolds with corners which are blow ups in particular cases we define a class of boutet de monvel type operators we then define the representation homomorphism for these operators and show closedness under composition with the help of a representation theorem finally we consider appropriate fredholm conditions and construct the parametrices for elliptic operators in the calculus

lie algebroid structures on a class of affine bundles we introduce the notion of a lie algebroid structure on an affine bundle whose base manifold is fibered over r it is argued that this is the framework which one needs for coming to a time dependent generalization of the theory of lagrangian systems on lie algebroids an extensive discussion is given of a way one can think of forms acting on sections of the affine bundle it is further shown that the affine lie algebroid structure gives rise to a coboundary operator on such forms the concept of admissible curves and dynamical systems whose integral curves are admissible brings an associated affine bundle into the picture on which one can define in a natural way a prolongation of the original affine lie algebroid structure

random walk in random scenery and self intersection local times in dimensions d 5 let s k k 0 be a symmetric random walk on mathbb z d and eta x x in mathbb z d an independent random field of centered i i d random variables with tail decay p eta x t approx exp t alpha we consider a random walk in random scenery that is x n eta s 0 dots eta s n we present asymptotics for the probability over both randomness that x n n for 1 2 and 1 to obtain such asymptotics we establish large deviations estimates for the self intersection local times process sum x l n 2 x where l n x is the number of visits of site x up to time n