tyz expansion for the kepler manifold the main goal of the paper is to address the issue of the existence of kempf s distortion function and the tian yau zelditch tyz asymptotic expansion for the kepler manifold an important example of non compact manifold motivated by the recent results for compact manifolds we construct kempf s distortion function and derive a precise tyz asymptotic expansion for the kepler manifold we get an exact formula finite asymptotic expansion of n 1 terms and exponentially small error terms uniformly with respect to the discrete quantization parameter m rightarrow infty hbar m 1 rightarrow 0 standing for planck s constant and x rightarrow infty x in mathbb c n moreover the coefficients are calculated explicitly and they turned out to be homogeneous functions with respect to the polar radius in the kepler manifold we show that our estimates are sharp by analyzing the nonharmonic behaviour of tm for m rightarrow infty the arguments of the proofs combine geometrical methods quantization tools and functional analytic techniques for investigating asymptotic expansions in the framework of analytic gevrey spaces

on rigidity of grauert tubes over riemannian manifolds of constant curvature it is well known that a real analytic manifold x admits a complexification xc a complex manifold that contains x as the fixed point set of an antiholomorphic involution this can be seen as follows the transition functions defining the manifold x are real analytic local diffeomorphisms of rn the taylor expansions of these transition functions can be considered as local biholomorphisms of cn hence they serve as transition functions of a complex manifold the germ of the complexification xc is unique every sufficiently small tubular neighborhood of x in the tangent bundle tx admits a real analytic diffeomorphism into xc that fixes x therefore a sufficiently small tubular neighborhood ofx in tx has a complex structure and can be considered as a complexification of x in general the complex structure on the tubular neighborhood is not unique since there are manyways to embed it intoxc there have been a lot of interest in finding canonical complex structures on tubular neighborhoods ofx in tx with additional datum of a real analytic riemannian metric g on x a canonical complex structure can be specified for sufficiently small tubular neighborhoods of x in tx see gs ls s1 there is a unique complex structure on such that the map f i i c is holomorphic wherever it is defined

the segal bargmann coherent state transform for compact lie groups abstract let k be an arbitrary compact connected lie group we describe on k an analog of the segal bargmann coherent state transform and we prove theorem 1 that this generalized coherent state transform maps l 2 k isometrically onto the space of holomorphic functions in l 2 g where g is the complexification of k and where is an appropriate heat kernel measure on g the generalized coherent state transform is defined in terms of the heat kernel on the compact group k and its analytic continuation to the complex group g we also define a k averaged version of the coherent state transform and we prove a similar result for it theorem 2 in the appendix we describe the coherent states which motivate the definitions of these transforms in section 9 we obtain inversion formulas for both the coherent state transform and the k averaged coherent state transform theorems 3 4 as a consequence we obtain formulas for certain bergman kernels on the complex group g theorems 5 6 in section 10 we derive a general paley wiener theorem for k and we exhibit a family of convolution transforms each of which is an isometric isomorphism of l 2 k onto a certain l 2 space of holomorphic functions on g these transforms include the k averaged coherent state transform as a special case we also discuss the analogous results for r 1 the results of section 10 are theorems 7 10 finally in section 11 we discuss the case of quotient spaces we prove theorem 11 that both the coherent state transform and the k averaged coherent state transform pass in a natural way from k to k h where h is a closed connected subgroup of k

complex symplectomorphisms and pseudo kahler islands in the quantization of toric manifolds let p be a delzant polytope we show that the quantization of the corresponding toric manifold x p in toric kahler polarizations and in the toric real polarization are related by analytic continuation of hamiltonian flows evaluated at time t sqrt 1 s we relate the quantization of x p in two different toric kahler polarizations by taking the time sqrt 1 s hamiltonian flow of strongly convex functions on the moment polytope p by taking s to infinity we obtain the quantization of x p in the singular real toric polarization recall that x p has an open dense subset which is biholomorphic to mathbb c n the quantization of x p in a toric kahler polarization can also be described by applying the complexified hamiltonian flow of the abreu guillemin symplectic potential g at time t sqrt 1 to an appropriate finite dimensional subspace of quantum states in the quantization of t mathbb t n in the vertical polarization by taking other imaginary times t k sqrt 1 k in mathbb r we describe toric kahler metrics with cone singularities along the toric divisors in x p for convex hamiltonian functions and sufficiently negative imaginary part of the complex time we obtain degenerate kahler structures which are negative definite in some regions of x p we show that the pointwise and l 2 norms of quantum states are asymptotically vanishing on negative definite regions

the complex monge ampere equation zoll metrics and algebraization let m be a real analytic riemannian manifold an adapted complex structure on tm is a complex structure on a neighborhood of the zero section such that the leaves of the riemann foliation are complex submanifolds this structure is called entire if it may be extended to the whole of tm we prove here that the only real analytic zoll metric on the n sphere with an entire adapted complex structure on tm is the round sphere using similar ideas we answer a special case of an algebraization question raised by the first author characterizing some stein manifolds as affine algebraic in terms of plurisubharmonic exhaustion functions satisfying the homogeneous complex monge ampere equation the result presented here is an extension to higher dimensions of an observation attributed to w stoll for the case of riemann surfaces

adapted complex structures and the geodesic flow in this paper we give a new construction of the adapted complex structure on a neighborhood of the zero section in the tangent bundle of a compact real analytic riemannian manifold motivated by the complexifier approach of t thiemann as well as certain formulas of v guillemin and m stenzel we obtain the polarization associated to the adapted complex structure by applying the imaginary time geodesic flow to the vertical polarization meanwhile at the level of functions we show that every holomorphic function is obtained from a function that is constant along the fibers by composition with the imaginary time geodesic flow we give several equivalent interpretations of this composition including a convergent power series in the vector field generating the geodesic flow

direct images fields of hilbert spaces and geometric quantization geometric quantization often produces not one hilbert space to represent the quantum states of a classical system but a whole family h s of hilbert spaces and the question arises if the spaces h s are canonically isomorphic axelrod et al j diff geo 33 787 902 1991 and hitchin commun math phys 131 347 380 1990 suggest viewing h s as fibers of a hilbert bundle h introduce a connection on h and use parallel transport to identify different fibers here we explore to what extent this can be done first we introduce the notion of smooth and analytic fields of hilbert spaces and prove that if an analytic field over a simply connected base is flat then it corresponds to a hermitian hilbert bundle with a flat connection and path independent parallel transport second we address a general direct image problem in complex geometry pushing forward a hermitian holomorphic vector bundle e to y along a non proper map y to s we give criteria for the direct image to be a smooth field of hilbert spaces third we consider quantizing an analytic riemannian manifold m by endowing tm with the family of adapted kahler structures from lempert and sz ke bull lond math soc 44 367 374 2012 this leads to a direct image problem when m is homogeneous we prove the direct image is an analytic field of hilbert spaces for certain such m but not all the direct image is even flat which means that in those cases quantization is unique

orbits of the geodesic flow and chains on the boundary of a grauert tube a c omega riemannian manifold x g determines an integrable complex structure on a tubular neighborhood t epsilon x of the zero section in t x and a cr structure on the boundary m epsilon there are two natural families of curves on m epsilon the orbits of the geodesic flow and a cr invariant family called chains it is natural to ask whether they are related we show that if orbits of the geodesic flow are chains on m epsilon for all epsilon sufficiently small then x g is einstein as a partial converse we show that if x g is harmonic then orbits of the geodesic flow are chains to prove this we study the fefferman metric associated with m epsilon

invariant domains in the complexification of a noncompact riemannian symmetric space let g k be a noncompact riemannian symmetric space and let gc kc be its complexification then g acts on gc kc by left translations we study the invariant cr structure of the closed g orbits of maximal dimension and determine which ones can lie in the boundary of an invariant stein domain in this way we obtain information on the g invariant stein domains in gc kc

singular monge ampere foliations this paper generalizes results of lempert and szoke on the structure of the singular set of a solution of the homogeneous monge ampere equation on a stein manifold their a priori assumption that the singular set has maximum dimension is shown to be a consequence of regularity of the solution in addition their requirement that the square of the solution be c 3 everywhere is replaced by a smoothness condition on the blowup of the singular set under these conditions the singular set is shown to inherit a finsler metric which in the real analytic case uniquely determines the solution of the monge ampere equation these results are proved using techniques from contact geometry

coherent states for the quantum mechanics on a compact manifold we review our recent observations concerning quantum mechanics on such compact manifolds as a circle torus and sphere

canonical complex structures associated to connections and complexifications of lie groups the notion of adapted complex structure is extended from riemannian manifolds to general koszul connections the case of the canonical connection of a lie group and the levi civita connection of a pseudo riemannian manifold is studied

an inversion formula for the segal bargmann transform on a symmetric space of non compact type abstract we prove analogs of the heat kernel transform inversion formulae of golse leichtnam and the author e leichtnam f golse m stenzel intrinsic microlocal analysis and inversion formulae for the heat equation on compact real analytic riemannian manifolds ann sci ecole norm sup 4 29 6 1996 669 736 mr1422988 97h 58153 theorems 0 3 0 4 in the setting of a riemannian symmetric space of helgason s non compact type

on some classes of mkdv periodic solutions we obtain exact periodic solutions of the positive and negative modified kortweg de vries mkdv equations we examine the dynamical stability of these solitary wave lattices through direct numerical simulations while the positive mkdv breather lattice solutions are found to be unstable the two soliton lattice solution of the same equation is found to be stable similarly a negative mkdv lattice solution is found to be stable we also touch upon the implications of these results for the kdv equation

breather lattice and its stabilization for the modified korteweg de vries equation we obtain an exact solution for the breather lattice solution of the modified korteweg de vries equation numerical simulation of the breather lattice demonstrates its instability due to the breather breather interaction however such multibreather structures can be stabilized through the concurrent application of ac driving and viscous damping terms

phase space bounds for quantum mechanics on a compact lie group let k be a compact connected lie group and k bbb c its complexification i consider the hilbert space cal h l 2 left k bbb c nu t right of holomorphic functions introduced in h1 where the parameter t is to be interpreted as planck s constant in light of l s the complex group k bbb c may be identified canonically with the cotangent bundle of k using this identification i associate to each f in cal h l 2 left k bbb c nu t right a phase space probability density the main result of this paper is theorem 1 which provides an upper bound on this density which holds uniformly over all f and all points in phase space specifically the phase space probability density is at most a t left 2 pi t right n where n dim k and a t is a constant which tends to one exponentially fast as t tends to zero at least for small t this bound cannot be significantly improved

a systematical way to find breather lattice solutions to the positive mkdv equation in this paper dependent and independent variable transformations are introduced to solve the positive mkdv equation systematically by using knowledge of elliptic equation and jacobian elliptic functions it is shown that different kinds of solutions can be obtained to the positive mkdv equation including many kinds of breather lattice solutions

maximal complexifications of certain riemannian homogeneous manifolds let m g k be a homogeneous riemannian manifold with dim c g c dim r g where g c denotes the universal complexification of g under certain extensibility assumptions on the geodesic flow of m we give a characterization of the maximal domain of definition in tm for the adapted complex structure and show that it is unique for instance this can be done for generalized heisenberg groups and naturally reductive homogeneous riemannian spaces as an application it is shown that the case of generalized heisenberg groups yields examples of maximal domains of definition for the adapted complex structure which are neither holomorphically separable nor holomorphically convex

the complex time segal bargmann transform abstract we introduce a new form of the segal bargmann transform for a connected lie group k of compact type we show that the heat kernel t x t 0 x k has a space time analytic continuation to a holomorphic function c z re 0 z k c where k c is the complexification of k the new transform is defined by the integral b f z k c z k 1 f k d k z k c if s 0 and d s s the disk of radius s centered at s this integral defines a holomorphic function on k c for each f l 2 k s we construct a heat kernel density s on k c such that for all s as above b s b l 2 k s is an isometric isomorphism from l 2 k s onto the space of holomorphic functions in l 2 k c s when t s the transform b t t coincides with the one introduced by the second author for compact groups and extended by the first author to groups of compact type when t 0 2 s the transform b s t coincides with the one introduced by the first two authors

maximal complexifications of certain homogeneous riemannian manifolds let m g k be a homogeneous riemannian manifold with dim c g c dim r g where g c denotes the universal complexification of g under certain extensibility assumptions on the geodesic flow of m we give a characterization of the maximal domain of definition in tm for the adapted complex structure and show that it is unique for instance this can be done for generalized heisenberg groups and naturally reductive homogeneous riemannian spaces as an application it is shown that the case of generalized heisenberg groups yields examples of maximal domains of definition for the adapted complex structure which are neither holomorphically separable nor holomorphically convex

subelliptic mathrm spin _ mathbb c dirac operators iii the atiyah weinstein conjecture in this paper we extend the results obtained in 9 10 to manifolds with spin c structures defined near the boundary by an almost complex structure we show that on such a manifold with a strictly pseudoconvex boundary there are modified neumann boundary conditions defined by projection operators r eo which give subelliptic fredholm problems for the spin c dirac operator eo we introduce a generalization of fredholm pairs to the tame category in this context we show that the index of the graph closure of eo r eo equals

complex time evolution in geometric quantization and generalized coherent state transforms abstract for the cotangent bundle t k of a compact lie group k we study the complex time evolution of the vertical tangent bundle and the associated geometric quantization hilbert space l 2 k under an infinite dimensional family of hamiltonian flows for each such flow we construct a generalized coherent state transform cst which is a unitary isomorphism between l 2 k and a certain weighted l 2 space of holomorphic functions for a particular set of choices we show that this isomorphism is naturally decomposed as a product of a heisenberg type evolution for complex time within l 2 k followed by a polarization changing geometric quantization evolution for complex time in this case our construction yields the usual generalized segal bargmann transform of hall we show that the infinite dimensional family of hamiltonian flows can also be understood in terms of thiemann s complexifier method which generalizes the construction of adapted complex structures

the asymptotic expansion of a cr invariant and grauert tubes cr manifolds the abstract models of real hypersurfaces in complex manifolds are 2n 1 dimensional manifolds m with a codimension one subbundle h of the tangent bundle which carries a complex structure the cr refers to cauchy riemann because for m c c n l the subbundle h consists of induced cauchy riemann operators there is a wealth of geometry and analysis associated with these structures especially when the cr manifolds are strictly pseudoconvex for example two strictly pseudoconvex domains are biholomorphically equivalent if and only if their boundaries are cr equivalent a fundamental problem in cr geometry is to find computable invariants associated with the cr structures the global cr invariant we will consider in this paper is the chem simons type invariant discovered by bums and epstein b e 1 it is a real valued global cr invariant of a compact 3dimensional strictly pseudo convex cr manifold whose holomorphic tangent bundle is trivial cheng and lee independently found this invariant and extend the definition of b e invariant to a relative invariant on an arbitrary compact 3 dimensional cr manifold cf c l we will evaluate this g asymptotically on the boundary of small grauert tubes before posing the question in a more precise form we will first say a few workds about grauert tubes let x be a real analytic manifold then every coordinate patch u c r n can be thickened to obtain an open set r c c n since the coordinate changes of x are real analytic maps by taking power series expansions and by shrinking c u to get convergence they can be extended holomorphically to such enlarged domains and thus they can be used as holomorphic transition

the segal bargmann transform on a symmetric space of compact type abstract we study the segal bargmann transform on a symmetric space x of compact type mapping l 2 x into holomorphic functions on the complexification x c we invert this transform by integrating against a dual heat kernel measure in the fibers of a natural fibration of x c over x we prove that the segal bargmann transform is an isometry from l 2 x onto the space of holomorphic functions on x c which are square integrable with respect to a natural measure these results extend those of b hall in the compact group case

iterated function systems and control languages valuations morphisms from e to 0 1 are a generalization of bernoulli morphisms introduced by eilenberg automata languages and machines academic press new york 1974 here we show how to generalize the notion of entropy of a language in order to obtain new formulas to determine the hausdorff dimension of fractal sets also in euclidean spaces especially defined via regular languages by doing this we can sharpen and generalize earlier results in two ways first we treat the case where the underlying basic iterated function system contains noncontractive mappings and second we obtain results valid for nonregular languages as well

diophantine tori and weyl laws for non selfadjoint operators in dimension two we study the distribution of eigenvalues for non selfadjoint perturbations of selfadjoint semiclassical analytic pseudodifferential operators in dimension two assuming that the classical flow of the unperturbed part is completely integrable an asymptotic formula of weyl type for the number of eigenvalues in a spectral band bounded from above and from below by levels corresponding to diophantine invariant lagrangian tori is established the weyl law is given in terms of the long time averages of the leading non selfadjoint perturbation along the classical flow of the unperturbed part

complex monge ampere equations and totally real submanifolds abstract we study the dirichlet problem for complex monge ampere equations in hermitian manifolds with general non pseudoconvex boundary our main result theorem 1 1 extends the classical theorem of caffarelli kohn nirenberg and spruck in c n we also consider the equation on compact manifolds without boundary attempting to generalize yau s theorems in the kahler case as applications of the main result we study some connections between the homogeneous complex monge ampere hcma equation and totally real submanifolds and a special dirichlet problem for the hcma equation related to donaldson s conjecture on geodesics in the space of kahler metrics

isometry theorem for the segal bargmann transform on a noncompact symmetric space of the complex type we consider the segal bargmann transform on a noncompact symmetric space of the complex type we establish isometry and surjectivity theorems for the transform in a form as parallel as possible to the results in the dual compact case the isometry theorem involves integration over a tube of radius r in the complexification followed by analytic continuation with respect to r a cancellation of singularities allows the relevant integral to have a nonsingular extension to large r even though the function being integrated has singularities

on the kakutani ito segal gross and segal bargmann hall isomorphisms recently gross has shown that the kakutani ito segal isomorphism theorem has an extension from the setting of gaussian measure on a vector space to heat kernel measure pt on a simply connected lie group g of compact type the isomorphism relates l2 pt to a certain completion of the universal enveloping algebra of g lie g gross proves this result using the kakutani ito segal theorem and an infinite dimensional calculus associated to g valued brownian motion hijab has greatly simplified and clarified gross proof hiiab s proof avoids most but not all of the infinite dimensional analysis in the original proof in this paper we will build on hijab s proof to give a completely finite dimensional non probabilistic proof of gross isomorphism theorem the proof given here relies heavily on hall s beautiful extension of the segal bargmann transformation to the setting of compact lie groups this theorem relating l2 pt to a certain l2 space of holomorphic functions l2 gc h on the complexified lie group gc will also be generalized to lie groups of compact type in the process it is shown how to characterize in terms of summability conditions on all the derivatives at the identity in gc those holomorphic functions which are in l2 gc

cartan decomposition of the moment map we investigate a class of actions of real lie groups on complex spaces using moment map techniques we establish the existence of a quotient and a version of luna s slice theorem as well as a version of the hilbert mumford criterion a global slice theorem is proved for proper actions we give new proofs of results of mostow on decompositions of groups and homogeneous spaces

geometric quantization and the generalized segal bargmann transform for lie groups of compact type let k be a connected lie group of compact type and let t k be its cotangent bundle this paper considers geometric quantization of t k first using the vertical polarization and then using a natural kahler polarization obtained by identifying t k with the complexified group k the first main result is that the hilbert space obtained by using the kahler polarization is naturally identifiable with the generalized segal bargmann space introduced by the author from a different point of view namely that of heat kernels the second main result is that the pairing map of geometric quantization coincides with the generalized segal bargmann transform introduced by the author this means that the pairing map in this case is a constant multiple of a unitary map for both results it is essential that the half form correction be included when using the kahler polarization

infinite word languages and continuous mappings abstract languages with infinite words have often been studied as subsets of a topological space the topology used for this purpose was always the natural topology of sequences this paper suggests that another topology may be more useful it is obtained by taking lim w as the derived set of w the resulting topological space is shown to be completely regular strongly zero dimensional and not normal the languages appearing in the theory of automata are shown to be closely related to functionally closed sets continuous inverse images of the set 0 as a consequence a number of known results can be expressed and proved in terms of continuous mappings

tangentially positive isometric actions and conjugate points let m g be a complete riemannian manifold with no conjugate points and f m g b g b a principal g bundle where g is a lie group acting by isometries and b the smooth quotient with g b the riemannian submersion metric we obtain a characterization of conjugate point free quotients b g b in terms of symplectic reduction and a canonical pseudo riemannian metric on the tangent bundle tm from which we then derive necessary conditions involving g and m for the quotient metric to be conjugate point free particularly for m a reducible riemannian manifold let g tm with the lie algebra of g be the moment map of the tangential g action on tm and let gp be the canonical pseudo riemannian metric on tm defined by the symplectic form d and the map f tm m x m f z exp z exp z first we prove a theorem stating that if gp is not positive definite on the action vector fields for the tangential action along g 1 0 then b g b acquires conjugate points we proved the converse result in 2005 then we characterize self parallel vector fields on m in terms of the positivity of the gp length of their tangential lifts along certain canonical subsets of tm we use this to derive some necessary conditions on g and m for actions to be tangentially positive on relevant subsets of tm which we then apply to isometric actions on complete conjugate point free reducible riemannian manifolds when one of the irreducible factors satisfies certain curvature conditions

shift invariant topologies for the cantor space x the space of one sided infinite words plays a crucial rle in several parts of theoretical computer science usually it is convenient to regard this space as a metric space the cantor space it turned out that for several purposes topologies other than the one of the cantor space are useful e g for studying fragments of first order logic over infinite words or for a topological characterisation of random infinite words it is shown that these topologies refine the topology of the cantor space moreover from common features of these topologies we extract properties which characterise a large class of topologies it turns out that for this general class of topologies the corresponding closure and interior operators respect the shift operations and also to some extent the definability of sets of infinite words by finite automata

sampling of real multivariate polynomials and pluripotential theory we consider the problem of stable sampling of multivariate real polynomials of large degree in a general framework where the polynomials are defined on an affine real algebraic variety m equipped with a weighted measure in particular this framework contains the well known setting of trigonometric polynomials when m is a torus equipped with its invariant measure where the limit of large degree corresponds to a high frequency limit as well as the classical setting of one variable orthogonal algebraic polynomials when m is the real line equipped with a suitable measure where the sampling nodes can be seen as generalizations of the zeros of the corresponding orthogonal polynomials it is shown that a necessary condition for sampling in the general setting is that the asymptotic density of the sampling points is greater than the density of the corresponding weighted equilibrium measure of m as defined in pluripotential theory this result thus generalizes the well known landau type results for sampling on the torus where the corresponding critical density corresponds to the nyqvist rate as well as the classical result saying that the zeros of orthogonal polynomials become equidistributed with respect to the logarithmic equilibrium measure as the degree tends to infinity

positivity of direct image bundles and convexity on the space of kahler metrics we develop some results from 4 on the positivity of direct image bundles in the particular case of a trivial fibration over a one dimensional base we also apply the results to study variations of kahler metrics

complex zeros of real ergodic eigenfunctions we determine the limit distribution as of complex zeros for holomorphic continuations to grauert tubes of real eigenfunctions of the laplacian on a real analytic compact riemannian manifold m g with ergodic geodesic flow if phi j k is an ergodic sequence of eigenfunctions we prove the weak limit formula frac 1 lambda j z phi j k mathbb c to frac i pi partial bar partial xi g where z phi j k mathbb c is the current of integration over the complex zeros and where overline partial is with respect to the adapted complex structure of lempert szoke and guillemin stenzel

coherent states on spheres we describe a family of coherent states and an associated resolution of the identity for a quantum particle whose classical configuration space is the d dimensional sphere sd the coherent states are labeled by points in the associated phase space t sd these coherent states are not of perelomov type but rather are constructed as the eigenvectors of suitably defined annihilation operators we describe as well the segal bargmann representation for the system the associated unitary segal bargmann transform and a natural inversion formula although many of these results are in principle special cases of the results of hall and stenzel we give here a substantially different description based on ideas of thiemann and of kowalski and rembielinski all of these results can be generalized to a system whose configuration space is an arbitrary compact symmetric space we focus on the sphere case in order to carry out the calculations in a self contained and explicit way

berezin toeplitz quantization on lie groups let k be a connected compact semisimple lie group and kc its complexification the generalized segal bargmann space for kc is a space of square integrable holomorphic functions on kc with respect to a k invariant heat kernel measure this space is connected to the schrodinger hilbert space l 2 k by a unitary map the generalized segal bargmann transform this paper considers certain natural operators on l 2 k namely mul tiplication operators and differential operators conjugated by the general ized segal bargmann transform the main results show that the resulting operators on the generalized segal bargmann space can be represented as toeplitz operators the symbols of these toeplitz operators are expressed in terms of a certain subelliptic heat kernel on kc i also examine some of the results from an infinite dimensional point of view based on the work of l gross and p malliavin

on the bks pairing for kahler quantizations of the cotangent bundle of a lie group a natural one parameter family of kahler quantizations of the cotangent bundle t k of a compact lie group k taking into account the half form correction was studied in fmmn in the present pa per it is shown that the associated blattner kostant sternberg bks pairing map is unitary and coincides with the parallel transport of the quantum connection introduced in our previous work from the point of view of adpw the bks pairing map is a composition of uni tary coherent state transforms of k introduced in ha1 continuity of the hermitian structure on the quantum bundle in the limit when one of the kahler polarizations degenerates to the vertical real polar ization leads to the unitarity of the corresponding bks pairing map this is in agreement with the unitarity up to scaling with respect to a rescaled inner product of this pairing map established by hall

the segal bargmann transform for noncompact symmetric spaces of the complex type we consider the generalized segal bargmann transform defined in terms of the heat operator for a noncompact symmetric space of the complex type for radial functions we show that the segal bargmann transform is a unitary map onto a certain l 2 space of meromorphic functions for general functions we give an inversion formula for the segal bargmann transform involving integration against an unwrapped version of the heat kernel for the dual compact symmetric space both results involve delicate cancellations of singularities

kolmogorov complexity and hausdorff dimension abstract in this paper various relationships between the kolmogorov complexity of infinite strings and measures of information content are given the general approach taken here is to bound the complexity of a maximally complex string in a given set of strings by the hausdorff dimension or the entropy of that set it turns out that hausdorff dimension yields lower bounds to the kolmogorov complexity whereas under certain recursiveness constraints on the structure of the respective sets their entropy yields upper bounds more detailed investigations result in a generalization of p martin lof s theorems on the kolmogorov complexity of random strings to maximally complex strings in regularly structured sets of infinite strings

harmonic analysis with respect to heat kernel measure this paper surveys developments over the last decade in harmonic analysis on lie groups relative to a heat kernel measure these include analogs of the hermite expansion the segal bargmann transform and the taylor ex pansion some of the results can be understood from the standpoint of geo metric quantization others are intimately related to stochastic analysis

holomorphic geometric models for representations of c algebras abstract representations of banach lie groups are realized on hilbert spaces formed by sections of holomorphic homogeneous vector bundles these sections are obtained by means of a new notion of reproducing kernel which is suitable for dealing with involutive diffeomorphisms defined on the base spaces of the bundles the theory involves considering complexifications of homogeneous spaces acted on by groups of unitaries and applies in particular to representations of c algebras endowed with conditional expectations in this way we present holomorphic geometric models for the stinespring dilations of completely positive maps the general results are further illustrated by a discussion of several specific topics including similarity orbits of representations of amenable banach algebras similarity orbits of conditional expectations geometric models of representations of cuntz algebras the relationship to endomorphisms of b h and non commutative stochastic analysis