dirac geometry quasi poisson actions and d g valued moment maps we study dirac structures associated with manin pairs d g and give a dirac geometric approach to hamiltonian spaces with d g valued moment maps originally introduced by alekseev and kosmann schwarzbach 3 in terms of quasi poisson structures we explain how these two distinct frameworks are related to each other proving that they lead to isomorphic categories of hamiltonian spaces we stress the connection between the viewpoint of dirac geometry and equivariant differential forms the paper discusses various examples including q hamiltonian spaces and poisson lie group actions explaining how presymplectic groupoids are related to the notion of double in each context

groupoid equivariant prequantization in their 2005 paper c laurent gengoux and p xu define prequantization for pre hamiltonian actions of quasi presymplectic lie groupoids in terms of s1 central extensions of lie groupoids the definition requires that the quasi presymplectic structure be exact i e the closed 3 form on the unit space of the lie groupoid must be exact in the present paper we define prequantization for pre hamiltonian actions of not necessarily exact quasi presymplectic lie groupoids in terms of dixmier douady bundles the definition is a natural adaptation of e meinrenken s treatment of prequantization for quasi hamiltonian lie group actions with group valued moment map the definition given in this paper is shown to be compatible with the definition of laurent gengoux and xu when the underlying quasi preysmplectic structure is exact properties related to morita invariance and symplectic reduction are established

quantization of hamiltonian loop group spaces we prove a fredholm property for spin c dirac operators mathsf d on non compact manifolds satisfying a certain condition with respect to the action of a semi direct product group k ltimes gamma with k compact and gamma discrete we apply this result to an example coming from the theory of hamiltonian loop group spaces in this context we prove that a certain index pairing mathcal x cap mathsf d yields an element of the formal completion r infty t of the representation ring of a maximal torus t subset h the resulting element has an additional antisymmetry property under the action of the affine weyl group indicating mathcal x cap mathsf d corresponds to an element of the ring of projective positive energy representations of the loop group

quasi bialgebras and dynamical r matrices abstract we study the relationship between general dynamical poisson groupoids and lie quasi bialgebras for a class of lie quasi bialgebras g naturally compatible with a reductive decomposition we extend the description of the moduli space of classical dynamical r matrices of etingof and schiffmann we construct in each gauge orbit an explicit analytic representative l can we translate the notion of duality for dynamical poisson groupoids into a duality for lie quasi bialgebras it is shown that duality maps the dynamical poisson groupoid for l can and g to the dynamical poisson groupoid for l can and the dual quasi bialgebra g

singular value decomposition analysis of source illumination in seismic interferometry by multidimensional deconvolution we have developed a method to analytically evaluate the relationship between the source receiver configuration and the retrievedwavefieldinseismicinterferometryperformedbymultidimensional deconvolution mdd the mdd method retrieves thewavefieldwiththedesiredsource receiverconfigurationfrom the observed wavefield without source information we used a singular value decomposition svd approach to solve the inverse problem of mdd by introducing svd into mdd we obtained quantities that revealed the characteristics of the mdd inverse problem and interpreted the effect of the initial sourcereceiverconfigurationforasurveydesign wenumericallysimulated the wavefield with a 2d model and investigated the rank of the incident field matrix of the mdd inverse problem with a source array of identical length a sparse and a dense source distribution resulted in an incident field matrix of the same rank and retrievedthesamewavefield therefore theoptimumsourcedistribution can be determined by analyzing the rank of the incident field matrix of the inverse problem in addition the introduction ofscatterersintothemodelimprovedthesourceilluminationand effectively increased the rank enabling mdd to retrieve a better wavefield we found that the ambiguity of thewavefield inferred from the model resolution matrix was a good measure of the amount of illumination of each receiver by the sources we used the field data recorded at the two boreholes from the surface sourcestosupportourresultsofthenumericalmodeling weevaluated the rank of incident field matrix with the dense and sparse source distribution we discovered that these two distributions resulted in an incident field matrix of almost the same rank and retrieved almost the samewavefield as the numerical modeling thisiscrucialinformationfordesigningseismicexperiments using the mdd based approach

a geometric framework for mixed quantum states based on a kahler structure in this paper we introduce a geometric framework for mixed quantum states based on a kahler structure the geometric framework includes a symplectic form an almost complex structure and a riemannian metric that characterize the space of mixed quantum states we argue that the almost complex structure is integrable we also in detail discuss a visualizing application of this geometric framework by deriving a geometric uncertainty relation for mixed quantum states the framework is computationally effective and it provides us with a better understanding of general quantum mechanical systems

retrieving the green s function in an open system by cross correlation a comparison of approaches l we compare two approaches for deriving the fact that the green s function in an arbitrary inhomogeneous open system can be obtained by cross correlating recordings of the wave field at two positions one approach is based on physical arguments exploiting the principle of time reversal invariance of the acoustic wave equation the other approach is based on rayleigh s reciprocity theorem using a unified notation we show that the result of the time reversal approach can be obtained as an approximation of the result of the reciprocity approach

on poisson actions of compact lie groups on symplectic manifolds let g p be a compact simple poisson lie group equipped with a poisson structure p and m omega be a symplectic manifold assume that m carries a poisson action of g p and there is an equivariant moment map in the sense of lu and weinstein which maps

norm square localization and the quantization of hamiltonian loop group spaces abstract in an earlier article we introduced a new definition for the quantization of a hamiltonian loop group space m involving the equivariant l 2 index of a dirac type operator d on a non compact finite dimensional submanifold y of m in this article we study a deformation of this operator similar to the work of tian zhang and ma zhang we obtain a formula for the index with infinitely many non trivial contributions indexed by the components of the critical set of the norm square of the moment map this is the main part of a new proof of the q r 0 theorem for hamiltonian loop group spaces

geometric quantization and multiplicities of group representations the heisenberg uncertainty principle says that it is impossible to determine simultaneously the position and momentum of a quantum mechanical particle this can be rephrased as follows the smallest subsets of classical phase space in which the presence of a quantum mechanical particle can be detected are its lagrangian submanifolds for this reason it makes sense to regard the lagrangian submanifolds of phase space as being its true points see weinstein 17 now let g be a compact lie group and g x x x a hamiltonian action of g on x see w for definitions it is well known that the fixed points of this action form a symplectic submanifold of x see for instance guillemin and sternberg 5 however what can one say about the fixed points of g we will show that they are also the points of a symplectic manifold xc this manifold is the marsden weinstein reduction of x with respect to the zero orbit in g and will be described in sect 2 it was introduced in a completely different context fl om ours by marsden and weinstein 12 problems in classical mechanics can often be reduced to the study of hamiltonian systems on symplectic manifolds and problems in quantum mechanics to the study of linear operators on hilbert space this fact has inspired a number of efforts to quantize symplectic geometry by devising schemes for associating hilbert space to symplectic manifolds the no go theorems of groenwald and van hove impose some embarrassing limitations on all such schemes however it seems to be a useful idea heuristically to think of every symplectic manifold x a ic 1 as being symbiotically associated with a hilbert space xq a t in such a way that the classical observables on the first space correspond to quantum observables on the second space the heuristics further suggests that if g is a group of symmetries of x la i j it should also be a group of symmetries of xq a tum in this heuristic spirit we will state the main conjecture of this paper

the quantization conjecture revisited a strong version of the quantization conjecture of guillemin and sternberg is proved for a reductive group action on a smooth compact polarized variety x l the cohomologies of l over the git quotient x g equal the invariant part of the cohomologies over x this generalizes the theorem of gs on global sections and strengthens its subsequent extensions jk li to riemannroch numbers remarkable by products are the invariance of cohomology of vector bundles over x g under a small change in the defining polarization or under shift desingularization as well as a new proof of boutot s theorem also studied are equivariant holomorphic forms and the equivariant hodgeto de rham spectral sequences for x and its strata whose collapse is shown one application is a new proof of the borel weil bott theorem of ti for the moduli stack of c bundles over a curve and of analogous statements for the moduli stacks and spaces of bundles with parabolic structures collapse of the hodge to de rham sequences for these stacks is also shown

equivalence of the virtual source method and wave field deconvolution in seismic interferometry seismic interferometry and the virtual source method are related approaches for extracting the green s function that accounts for wave propagation between receivers by making suitable combinations of the waves recorded at these two receivers these waves can either be excited by active controlled sources or by natural incoherent sources we compare this technique with the deconvolution of the wave field recorded at different receivers we show that the deconvolved wave field is a solution of the same wave equation as that for the physical system but that the deconvolved wave forms may satisfy different boundary conditions than those of the original system we apply this deconvolution approach to the wave motion recorded at various levels in a building after an earthquake and show how to extract the building response for different boundary conditions extracting the response of the system with different boundary conditions can be used to enhance or suppress the normal mode response in seismic exploration this principle can be used for the suppression of surface related multiples

an analysis of 3d particle path integration algorithms several techniques for the numerical integration of particle paths in steady and unsteady vector velocity fields are analyzed most of the analysis applies to unsteady vector fields however some results apply to steady vector field integration multistep multistage and some hybrid schemes are considered it is shown that due to initialization errors many unsteady particle path integration schemes are limited to third order accuracy in time multistage schemes require at least three times more internal data storage than multistep schemes of equal order however for timesteps within the stability bounds multistage schemes are generally more accurate a linearized analysis shows that the stability of these integration algorithms are determined by the eigenvalues of the local velocity tensor thus the accuracy and stability of the methods are interpreted with concepts typically used in critical point theory this paper shows how integration schemes can lead to erroneous classification of critical points when the timestep is finite and fixed for steady velocity fields we demonstrate that timesteps outside of the relative stability region can lead to similar integration errors from this analysis guidelines for accurate timestep sizing are suggested for both steady and unsteady flows in particular using simulation data for the unsteady flow around a tapered cylinder we show that accurate particle path integration requires timesteps which are at most on the order of the physical timescale of the flow

on the rigidity theorems of witten in this paper we prove the rigidity theorems predicted by witten in 1986 about the index of certain elliptic operators on manifolds with an s1 action w witten s insight was the culmination of an interesting interchange of ideas between him and hopkins landweber ochanine and stong for the detailed history we refer the reader to la the present account is essentially a reinterpretation of the second author s taubes t original proof of the theorem the senior author s contribution was solely to notice that the rather densely written arguments of the original manuscript could be formulated in terms of the well known fixed point formulae of equivariant index theory and equivariant cohomology in this context the final proof then appears in direct lineage of the atiyah hirzebruch theorem concerning the vanishing of the a genus of spin manifolds admitting a circle action a h and of the even older idea of lusztig concerning circle actions in the complex case there remains however the beautiful physics inspired novelty of connecting these techniques with elliptic function theory

nonlinear particle tracking for high order elements abstract the problem of calculating particle trajectories on unstructured meshes using a high order polynomial approximation of the velocity field is addressed the calculation of the particle trajectory is based on a runge kutta integration in time a convenient way of implementing high order approximations is to employ an auxiliary mapping that transforms a finite element into a topologically equivalent parent element within a normalized parametric space this presents two possible choices of space in which to perform the time integration of the particle position the physical space or the parametric space we present algorithms for implementing both particle tracking strategies using high order elements and discuss their merits the main drawback of both methods is their reliance on nonlinear procedures to calculate the particle trajectory a novel alternative hybrid approach that advances a particle in both the physical and the parametric space without requiring nonlinear iterations is proposed the error introduced by the alternative linearized procedures and their effect in the rate of convergence of the time integration is discussed finally the performance of the different algorithms is compared using a set of analytical and computational linear and high order velocity fields

a representation for green s function retrieval by multidimensional deconvolution green s function retrieval by crosscorrelation may suffer from irregularities in the source distribution asymmetric illumination intrinsic losses etc multidimensional deconvolution mdd may overcome these limitations a unified representation for green s function retrieval by mdd is proposed from this representation it follows that the traditional crosscorrelation method gives a green s function of which the source is smeared in space and time this smearing is quantified by a space time point spread function psf which can be retrieved from measurements at an array of receivers mdd removes this psf and thus deblurs and deghosts the source of the green s function obtained by correlation

surjectivity for hamiltonian loop group spaces let g be a compact lie group and let lg denote the corresponding loop group let x be a weakly symplectic banach manifold consider a hamiltonian action of lg on x and assume that the moment map mu rm x to l g is proper we consider the function left mu right 2 x to mathbb r and use a version of morse theory to showthat the inclusion map j mu 1 left 0 right to x induces a surjection j h g left x right to h g left mu 1 left 0 right right in analogywithkirwan s surjectivity theorem in the finite dimensional case we also prove a version of this surjectivity theorem for quasi hamiltonian g spaces

the yang mills heat flow on the moduli space of framed bundles on a surface we study the analog of the yang mills heat flow on the moduli space of framed bundles on a cut surface existence and convergence of the heat flow give a stratification of morse type invariant under the action of the loop group we use the stratification to prove versions of kahler quantization commutes with reduction and kirwan surjectivity

quantization of symplectic dynamical r matrices and the quantum composition formula in this paper we quantize symplectic dynamical r matrices over a possibly nonabelian base the proof is based on the fact that the existence of a star product with a nice property called strong invariance is sufficient for the existence of a quantization we also classify such quantizations and prove a quantum analogue of the classical composition formula for coboundary dynamical r matrices

green s function estimation using secondary sources in a shallow water environment this work provides a new way to measure the green s function between two points in an acoustic channel without emitting a pulse by any of the two points the green s function between a and b is obtained from a set of secondary sources in the guide by averaging either the correlation or the convolution of the signals received in a and b a theoretical approach based on mode propagation in a monochromatic regime is presented results are then extended to the time domain estimation of the green s function is performed numerically in a range independent and a range dependent environment application to discreet acoustic communications is discussed

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

quantum line bundles on s2 and method of orbits for suq 2 the quantum sphere is covered by two quantum complex planes quantum line bundle on s2 is introduced by specifying a transition function the structure of the quantum complex planes enables one to construct representations of suq 2 acting in spaces of holomorphic sections in the quantum case dressing transformation replaces coadjoint action

brst quantization of quasisymplectic manifolds and beyond we consider a class of factorizable poisson brackets which includes almost all reasonable poisson structures a particular case of the factorizable brackets are those associated with symplectic lie algebroids the brst theory is applied to describe the geometry underlying these brackets as well as to develop a deformation quantization procedure in this particular case this can be viewed as an extension of the fedosov deformation quantization to a wide class of irregular poisson structures in a more general case the factorizable poisson brackets are shown to be closely connected with the notion of n algebroid a simple description is suggested for the geometry underlying the factorizable poisson brackets based on construction of an odd poisson algebra bundle equipped with an abelian connection it is shown that the zero curvature condition for this connection generates all the structure relations for the n algebroid as well as a generalization of the yang baxter equation for the symplectic structure

a simple and exact acoustic wavefield modeling code for data processing imaging and interferometry applications improvements in industrial seismic seismological acoustic or interferometric theory and applications often result in quite subtle changes in sound quality seismic images or information which are nevertheless crucial for improved interpretation or experience when evaluating new theories and algorithms using synthetic data an important aspect of related researchisthereforethat numerical errors due towavefield modeling are reduced to a minimum we present a new matlab code based on the foldy method that models theoretically exact direct and scattered parts of a wavefield its main advantage lies in the fact that while all multiple scattering interactions aretakenintoaccount unlikefinite difference or finite element methods numerical dispersion errors are avoided the method is therefore ideal for testing new theory in industrial seismics seismology acoustics and in wavefield interferometry in particular because the latter is particularly sensitive to the dynamics of scattering interactions we present the theory behind the foldy acoustic modeling method and provide examples of its implementation we also benchmark the code against a good finite difference code because our foldy code was written and optimized to test new theory in seismic interferometry examples of its application to seismicinterferometryarealsopresented showing its validity and importance when exact modeling results are needed

the multiplicative eigenvalue problem and deformed quantum cohomology abstract we construct deformations of the small quantum cohomology rings of homogeneous spaces g p and obtain an irredundant set of inequalities determining the multiplicative eigen polytope for the compact form k of g

k area hofer metric and geometry of conjugacy classes in lie groups given a closed symplectic manifold m we introduce a certain quantity associated to a tuple of conjugacy classes in the universal cover of the group ham m by means of the hofer metric on ham m we use pseudo holomorphic curves involved in the definition of the multiplicative structure on the floer cohomology of a symplectic manifold m to estimate this quantity in terms of actions of some periodic orbits of related hamiltonian flows as a corollary we get a new way to obtain agnihotri belkale woodward inequalities for eigenvalues of products of unitary matrices as another corollary we get a new proof of the geodesic property with respect to the hofer metric of hamiltonian flows generated by certain autonomous hamiltonians our main technical tool is k area defined for hamiltonian fibrations over a surface with boundary in the spirit of l polterovich s work on hamiltonian fibrations over s2

classification of real three dimensional poisson lie groups all real three dimensional poisson lie pl groups are explicitly constructed and fully classified under group automorphisms by making use of their one to one correspondence with the complete classification of real three dimensional lie bialgebras given in gomez 2000 j math phys 41 4939 many of these 3d pl groups are non coboundary structures whose poisson brackets are given here for the first time casimir functions for all three dimensional pl groups are given and some features of several pl structures are commented

the hyperbolic moduli space of flat connections and the isomorphism of symplectic multiplicity spaces the hyperbolic moduli space of flat connections and the isomorphism of symplectic multiplicity spaces

decomposable representations and lagrangian submanifolds of moduli spaces associated to surface groups the importance of explicit examples of lagrangian submanifolds of moduli spaces is revealed by papers such as dostoglou and salamon ann of math 2 139 3 581 640 1994 and salamon proceedings of the international congress of mathematicians vol 1 2 zurich 1994 pp 526 536 birkhauser basel 1995 given a 3 manifold m with boundary m dostoglou and salamon use such examples to obtain a proof of the atiyah floer conjecture relating the symplectic floer homology of the representation space hom 1 m u u associated to an explicit pair of lagrangian submanifolds of this representation space and the instanton homology of the 3 manifold m in the present paper we construct a lagrangian submanifold of the space of representations mathcal m g l hom mathcal c pi g l u u of the fundamental group g lof a punctured riemann surface g l into an arbitrary compact connected lie group u this lagrangian submanifold is obtained as the fixed point set of an anti symplectic involution hat beta defined on mathcal m g l we show that the involution hat beta is induced by a form reversing involution defined on the quasi hamiltonian space u times u g times mathcal c 1 times cdots times mathcal c l the fact that hat beta has a non empty fixed point set is a consequence of the real convexity theorem for group valued momentum maps proved in schaffhauser a real convexity theorem for quasi hamiltonian actions submitted 25 p 2007 http arxiv org abs math 0705 0858 the notion of decomposable representation provides a geometric interpretation of the lagrangian submanifold thus obtained

green s function for migration continuous recording geometry analytical formulas are derived to give the point scatterer responses for the zero offset and nonzero offset migration operators these formulas are derived under the far field approximation for a continuous distribution of sources and receivers and are denoted as migration green s functions by specifying a homogeneous velocity model with irregular layers these green s functions can be used to quickly compute the synthetic migrated sections without having to compute the modeled data thus these formulas provide a fundamental understanding of migration artifacts and their relationship to the image point location and source receiver configuration this understanding lays the foundation for the next step elimination of these artifacts by migration deconvolution

an improved known vicinity algorithm based on geometry test for particle localization in arbitrary grid the known vicinity algorithm based on the geometry test for the particle localization problem in the hybrid eulerian lagrangian model was extended and enhanced aiming at the connected grids with convex polygon polyhedral cells such extensions were achieved by proposing novel improvements specifically a new side function to determine the relative position of the particle and the cell was introduced to build a more formal test process in addition a binary search method was developed to accelerate the particle in cell test and trajectory face intersection test for grids consisting of arbitrary polygon polyhedral cells further the particle location problem without the known vicinity position was established and solved by special boundary treatment through considering the internal external boundary and larger particle displacement in one single lagrangian step the improved algorithm was applied to the particle location problem with both two dimensional and three dimensional eulerian grids additionally the proposed algorithm was compared with the previous ones to exhibit its higher efficiency and broader application sample cases focusing the water impingement computation for aircraft icing were solved by adopting this algorithm assisted by the lagrangian particle dynamics model and the computational results were verified by the experiments

lie group valued moment maps we develop a theory of quasi hamiltonian g spaces for which the moment map takes values in the group g itself rather than in the dual of the lie algebra the theory includes counterparts of hamiltonian reductions the guillemin sternberg symplectic cro

mesh generation in curvilinear domains using high order elements the ability to construct suitable computational meshes is currently a significant limiting factor in the development of compact high order algorithms in very complex geometries compact high order algorithms such as spectral element or p type finite element techniques offer the potential of high accuracy if the solution is smooth and a well behaved mapping exists between the local sub domains and a standard region in this paper we address the problem of coupling high order algorithms with unstructured mesh generation and cad representation techniques we present three complementary strategies to alleviate the problem of generation of elemental regions with singular elemental mappings the first strategy investigates the influence of an anisometric parametric surface representation on the quality of high order unstructured meshes we show that a straightforward splitting of a standard linear mesh for use in a high order method can lead to the generation of distorted possibly invalid meshes even for simple computational domains such as cubes the second strategy uses hybrid meshes applying prismatic elements near curvilinear boundaries in the third and final strategy we investigate the use of curvature driven surface discretization to incorporate higher order information about the surface into the mesh generation using the computational reconstruction of the arterial bypass graft as an example computational domain we demonstrate how the combination of all three strategies leads to a valid high order discretization of the computational domain copyright 2001 john wiley sons ltd

double bracket equations and geodesic flows on symmetric spaces in this paper we consider the geometry of hamiltonian flows on the cotangent bundle of coadjoint orbits of compact lie groups and on symmetric spaces a key idea here is the use of the normal metric to define the kinetic energy this leads to hamiltonian flows of the double bracket type we analyze the integrability of geodesic flows according to the method of thimm we obtain via the double bracket formalism a quite explicit form of the relevant commuting flows and a correspondingly transparent proof of involutivity we demonstrate for example integrability of the geodesic flow on the real and complex grassmannians we also consider right invariant systems and the generalized rigid body equations in this setting

quantized multiplicative quiver varieties abstract beginning with the data of a quiver q and its dimension vector d we construct an algebra d q d q mat d q which is a flat q deformation of the algebra of differential operators on the affine space mat d q the algebra d q is equivariant for an action by a product of quantum general linear groups acting by conjugation at each vertex we construct a quantum moment map for this action and subsequently define the hamiltonian reduction a d q of d q with moment parameter we show that a d q is a flat formal deformation of lusztig s quiver varieties and their multiplicative counterparts for all dimension vectors satisfying a flatness condition of crawley boevey indeed the product on a d q yields a fedosov quantization the of symplectic structure on multiplicative quiver varieties as an application we give a description of the category of representations of the spherical double affine hecke algebra of type a n 1 and its generalization constructed by etingof oblomkov and rains in terms of a quotient of the category of equivariant d q modules by a serre subcategory of aspherical modules

left and right centers in quasi poisson geometry of moduli spaces abstract we introduce left central and right central functions and left and right leaves in quasi poisson geometry generalizing central or casimir functions and symplectic leaves from poisson geometry they lead to a new type of quasi poisson reduction which is both simpler and more general than known quasi hamiltonian reductions we study these notions in detail for moduli spaces of flat connections on surfaces where the quasi poisson structure is given by an intersection pairing on homology

toric structures on the moduli space of flat connections on a riemann surface volumes and the moment map abstract in earlier papers we constructed a hamiltonian torus action on an open dense set in the moduli space of flat su 2 connections on a compact riemann surface where the dimension of the torus is half the dimension of the moduli space this torus action shows that this set can be viewed symplectically as a noncompact toric variety the number of integral points of the moment map for the torus action turns out to be identical to the verlinde dimension d g k as an application we furnish a new proof of the relation between the large k limit of d g k and the volume of the moduli space from our point of view this relation follows from the equality between the symplectic volume of a toric variety and the euclidean volume of the image of the moment map similar considerations are shown to give rise to the volumes of moduli spaces of parabolic bundles on a riemann surface knowledge of these volumes has been shown to allow a proof of the verlinde formula for the dimension of the space of holomorphic sections of line bundles on this space

trace functionals on noncommutative deformations of moduli spaces of flat connections abstract we describe an efficient construction of a canonical noncommutative deformation of the algebraic functions on the moduli spaces of flat connections on a riemann surface the resulting algebra is a variant of the quantum moduli algebra introduced by alekseev grosse and schomerus and buffenoir and roche we construct a natural trace functional on this algebra and show that it is related to the canonical trace in the formal index theory of fedosov and nest and tsygan via verlinde s formula

quantized algebras of functions on homogeneous spaces with poisson stabilizers let g be a simply connected semisimple compact lie group with standard poisson structure k a closed poisson lie subgroup 0 q 1 we study a quantization c gq kq of the algebra of continuous functions on g k using results of soibelman and dijkhuizen stokman we classify the irreducible representations of c gq kq and obtain a composition series for c gq kq we describe closures of the symplectic leaves of g k refining the well known description in the case of flag manifolds in terms of the bruhat order we then show that the same rules describe the topology on the spectrum of c gq kq next we show that the family of c algebras c gq kq 0 q 1 has a canonical structure of a continuous field of c algebras and provides a strict deformation quantization of the poisson algebra mathbb c g k finally extending a result of nagy we show that c gq kq is canonically kk equivalent to c g k

determination of effective brain connectivity from functional connectivity using propagator based interferometry and neural field theory with application to the corticothalamic system it is shown how to compute both direct and total effective connection matrices decms and tecms which embody the strengths of neural connections between regions from correlation based functional cms using propagator based interferometry a method that stems from geophysics and acoustics coupled with the recent identification of decms and tecms with bare and dressed propagators respectively the approach incorporates excitatory and inhibitory connections multiple structures and populations and measurement effects the propagator is found for a generalized scalar wave equation derived from neural field theory and expressed in terms of neural activity correlations and covariances and wave damping rates it is then related to correlation matrices that are commonly used to express functional and effective connectivities in the brain the results are illustrated in analytically tractable test cases

the witten reshetikhin turaev invariant for links in finite order mapping tori i abstract we state asymptotic expansion and growth rate conjectures for the witten reshetikhin turaev invariants of arbitrary framed links in 3 manifolds and we prove these conjectures for the natural links in mapping tori of finite order automorphisms of marked surfaces our approach is based upon geometric quantisation of the moduli space of parabolic bundles on the surface which we show coincides with the construction of the witten reshetikhin turaev invariants using conformal field theory as was recently completed by andersen and ueno

dirac geometry of the holonomy fibration in this paper we solve the problem of giving a gauge theoretic description of the natural dirac structure on a lie group which plays a prominent role in the theory of d branes for the wess zumino witten model as well as the theory of quasi hamiltonian spaces we describe the structure as an infinite dimensional reduction of the space of connections over the circle our insight is that the formal poisson structure on the space of connections is not an actual poisson structure but is itself a dirac structure due to the fact that it is defined by an unbounded operator we also develop general tools for reducing courant algebroids and morphisms between them allowing us to give a precise correspondence between hamiltonian loop group spaces and quasi hamiltonian spaces

asymptotic properties of the quantum representations of the mapping class group we establish two results on the large level limit of projective quantum representations of surface mapping class groups obtained by quantizing moduli spaces of flat su n bundle with hitchin s connection first we prove that these projective representations lift to asymptotic representations second we show that under an infinitesimal rigidity assumption the characters of these representations have an asymptotic expansion the leading order term of this expansion agrees with the formula derived heuristically by e witten in quantum field theory and the jones polynomial

the geometry of loop groups l espace g des boucles basees sur un groupe de lie compact admet une metrique de kahler on exprime sa courbure en termes d operateurs de toeplitz et on definit des classes de chern par analogie avec la theorie de chern weil aux dimensions finies il y a une structure de fredhohn naturelle sur g induite a partir de la famille des operateurs de toeplitz on obtient des formules explicites pour su n

deformation quantization of heisenberg manifolds form a smooth manifold equipped with a poisson bracket we formulate ac algebra framework for deformation quantization including the possibility of invariance under a lie group of diffeomorphisms preserving the poisson bracket we then show that the much studied non commutative tori give examples of such deformation quantizations invariant under the usual action of ordinary tori going beyond this the main results of the paper provide a construction of invariant deformation quantizations for those poisson brackets on heisenberg manifolds which are invariant under the action of the heisenberg lie group and for various generalizations suggested by this class of examples interesting examples are obtained of simplec algebras on which the heisenberg group acts ergodically

ginzburg weinstein via gelfand zeitlin let u n be the unitary group and u n the dual of its lie al gebra equipped with the kirillov poisson structure in their 1983 paper guillemin sternberg introduced a densely defined hamil tonian action of a torus of dimension n 1 n 2 on u n with mo ment map given by the gelfand zeitlin coordinates a few years later flaschka ratiu described a similar multiplicative gelfand zeitlin system for the poisson lie group u n by the ginzburg weinstein theorem u n is isomorphic to u n as a poisson manifold flaschka ratiu conjectured that one can choose the ginzburg weinstein diffeomorphism in such a way that it intertwines the linear and nonlinear gelfand zeitlin systems our main result gives a proof of this conjecture and produces a canonical ginzburg weinstein diffeomorphism

a family of poisson structures on hermitian symmetric spaces we investigate the compatibility of symplectic kirillov kostant souriau structure and poisson lie structure on coadjoint orbits of semisimple lie group we prove that they are compatible for an orbit compact lie group iff the orbit is hermitian symmetric space we prove also the compatibility statement for non compact hermitian symmetric space as an example we describe a structure of symplectic leaves oncp n for this family these leaves may be considered as a perturbation of schubert cells possible applications to infinite dimensional examples are discussed