hodge and prym tau functions jenkins strebel differentials and combinatorial model of mathcal m_ g n the goal of the paper is to apply the theory of integrable systems to construct explicit sections of line bundles over the combinatorial model of the moduli space of pointed riemann surfaces based on strebel differentials these line bundles are tensor products of the determinants of the hodge or prym vector bundles with the standard tautological line bundles mathcal l j and the sections are constructed in terms of tau functions the combinatorial model is interpreted as the real slice of a complex analytic moduli space of quadratic differentials where the phase of each tau function provides a section of a circle bundle the phase of the ratio of the prym and hodge tau functions gives a section of the kappa 1 circle bundle by evaluating the increment of the phase around co dimension 2 sub complexes we identify the poincare dual cycles to the chern classes of the corresponding line bundles they are expressed explicitly as combination of witten s cycle w 5 and kontsevich s boundary this provides combinatorial analogues of mumford s relations on mathcal m g n and penner s relations in the hyperbolic combinatorial model the free homotopy classes of loops around w 5 are interpreted as pentagon moves while those of loops around kontsevich s boundary as combinatorial dehn twists throughout the paper we exploit the classical description of the combinatorial model in terms of strebel differentials parametrized in terms of period or homological coordinates we show that they provide darboux coordinates for the symplectic structure introduced by kontsevich we also express the latter as the intersection pairing in the odd homology of the canonical double cover

hochschild cohomology and atiyah classes abstract in this paper we prove that on a smooth algebraic variety the hkr morphism twisted by the square root of the todd genus gives an isomorphism between the sheaf of poly vector fields and the sheaf of poly differential operators both considered as derived gerstenhaber algebras in particular we obtain an isomorphism between hochschild cohomology and the cohomology of poly vector fields which is compatible with the lie bracket and the cupproduct the latter compatibility is an unpublished result by kontsevich our proof is set in the framework of lie algebroids and so applies without modification in much more general settings as well

on the existence of strong nash equilibria this paper investigates the existence of strong nash equilibria sne in continuous and convex games we show that the concavity and an additional condition on payoff functions together with the compactness of strategy space permit the existence of strong nash equilibria these conditions are satisfied in many economic games and are quite simple to check we also characterize the existence of sne by providing a necessary and sufficient condition moreover we suggest a procedure that can be used to efficiently compute strong nash equilibrium the result is illustrated with an application to an economy with multilateral environmental externalities and to the simple oligopoly static model

moments of random matrices and hypergeometric orthogonal polynomials we establish a new connection between moments of n times n random matrices xn and hypergeometric orthogonal polynomials specifically we consider moments mathbb e rm tr x n s as a function of the complex variable s in mathbb c whose analytic structure we describe completely we discover several remarkable features including a reflection symmetry or functional equation zeros on a critical line in the complex plane and orthogonality relations an application of the theory resolves part of an integrality conjecture of cunden et al j math phys 57 111901 2016 on the time delay matrix of chaotic cavities in each of the classical ensembles of random matrix theory gaussian laguerre jacobi we characterise the moments in terms of the askey scheme of hypergeometric orthogonal polynomials we also calculate the leading order n rightarrow infty asymptotics of the moments and discuss their symmetries and zeroes we discuss aspects of these phenomena beyond the random matrix setting including the mellin transform of products and wronskians of pairs of classical orthogonal polynomials when the random matrix model has orthogonal or symplectic symmetry we obtain a new duality formula relating their moments to hypergeometric orthogonal polynomials

hochschild cohomology of abelian categories and ringed spaces this paper continues the development of the deformation theory of abelian categories introduced in a previous paper by the authors we show first that the deformation theory of abelian categories is controlled by an obstruction theory in terms of a suitable notion of hochschild cohomology for abelian categories we then show that this hochschild cohomology coincides with the one defined by gerstenhaber schack and swan in the case of module categories over diagrams and schemes and also with the hochschild cohomology for exact categories introduced recently by keller in addition we show in complete generality that hochschild cohomology satisfies a mayer vietoris property and that for constantly ringed spaces it coincides with the cohomology of the structure sheaf

a note on abreu matsushima mechanisms in a stimulating recent paper abreu and matsushima 1992 hereafter a m show how a class of social choice functions can be virtually implemented in iteratively undominated strategies this work has several important features the mechanisms used are finite and not too difficult to understand and so less objectionable in this regard than many in the literature the class of social choice functions implemented is large and the solution concept nash equilibrium determined uniquely by iterative elimination of strongly dominated strategies is relatively uncontroversial the point of this note is to argue that the mechanisms used by a m unfortunately tend to generate games in which the iterative removal of strongly dominated strategies sometimes is indeed or ought to be controversial 2 we proceed by first examining an example of a related but simpler implementation problem in which the argument is easily exposed then indicating how the argument applies generally in the a m setup consider the following much discussed two player coordination game

riemann hilbert problem associated to frobenius manifold structures on hurwitz spaces irregular singularity solutions to the riemann hilbert problems with irregular singularities naturally associated to semisimple frobenius manifold structures on hurwitz spaces moduli spaces of meromorphic functions on a riemann surface are constructed the solutions are given in terms of meromorphic bidifferentials defined on the underlying riemann surface the relationship between different classes of frobenius manifolds structures on hurwitz spaces real doubles deformations is described on the level of the corresponding riemann hilbert problems

solution of matrix riemann hilbert problems with quasi permutation monodromy matrices in this paper we solve an arbitrary matrix riemann hilbert inverse monodromy problem with irreducible quasi permutation monodromy representation outside of a divisor in the space of monodromy data this divisor is characterized in terms of the theta divisor on the jacobi manifold of an auxiliary compact riemann surface realized as an appropriate branched covering of open image in new windowp1 the solution is given in terms of a generalization of szego kernel on the riemann surface in particular our construction provides a new class of solutions of the schlesinger system the isomonodromy tau function of these solutions is computed up to a nowhere vanishing factor independent of the elements of monodromy matrices results of this work generalize the results of papers 13 and 14 where the 2 2 case was solved

mellin transforms of a generalization of legendre polynomials abstract oberle m k s l scott g t gilbert r l hatcher and d f addis mellin transforms of a generalization of legendre polynomials journal of computational and applied mathematics 45 1993 367 369 we show that the zeros and poles of the mellin transforms of polynomials on 1 1 orthogonal with respect to the weight brvbar x 2r with r gt 1 2 are real and simple

the mukai pairing ii the hochschild kostant rosenberg isomorphism abstract we continue the study of the hochschild structure of a smooth space that we began in our previous paper examining implications of the hochschild kostant rosenberg theorem the main contributions of the present paper are we introduce a generalization of the usual notions of mukai vector and mukai pairing on differential forms that applies to arbitrary manifolds we give a proof of the fact that the natural chern character map k 0 x hh 0 x becomes after the hkr isomorphism the usual one k 0 x h i x x i and we present a conjecture that relates the hochschild and harmonic structures of a smooth space similar in spirit to the tsygan formality conjecture

strong asymptotics for cauchy biorthogonal polynomials with application to the cauchy two matrix model we apply the nonlinear steepest descent method to a class of 3 3 riemann hilbert problems introduced in connection with the cauchy two matrix random model the general case of two equilibrium measures supported on an arbitrary number of intervals is considered in this case we solve the riemann hilbert problem for the outer parametrix in terms of sections of a spinorial line bundle on a three sheeted riemann surface of arbitrary genus and establish strong asymptotic results for the cauchy biorthogonal polynomials

virtual implementation in nash equilibrium the two person case needs to be considered separately we provide a complete characterization of virtually implementable two person social choice functions while not all two person social choice functions are virtually implementable our necessary and sufficient condition is simple this contrasts with the rather complex necessary and sufficient conditions for exact implementation we show how our results can be extended to implementation in strict nash equilibrium and coalition proof nash equilibrium to social choice correspondences which map from cardinal preference profiles to lotteries and to environments with a continuum of pure alternatives

baker akhiezer spinor kernel and tau functions on moduli spaces of meromorphic differentials in this paper we study the baker akhiezer spinor kernel on moduli spaces of meromorphic differentials on riemann surfaces we introduce the baker akhiezer tau function which is related to both the bergman tau function which was studied before in the context of hurwitz spaces and spaces of holomorphic abelian and quadratic differentials and the kp tau function on such spaces

on the algebro geometric integration of the schlesinger equations a new approach to the construction of isomonodromy deformations of 2 2 fuchsian systems is presented the method is based on a combination of the algebro geometric scheme and riemann hilbert approach of the theory of integrable systems for a given number 2g 1 g 1 of finite regular singularities the method produces a 2g parameter submanifold of the fuchsian monodromy data for which the relevant riemann hilbert problem can be solved in closed form via the baker akhiezer function technique this in turn leads to a 2g parameter family of solutions of the corresponding schlesinger equations explicitly described in terms of riemann theta functions of genus g in the case g 1 the solution found coincides with the general elliptic solution of the particular case of the painleve vi equation first obtained by n j hitchin h1

virtual bayesian implementation allowing for incomplete information this paper characterizes the social choice functions that can be approximated by the equilibrium outcomes of a mechanism incentive compatibility is necessary and almost sufficient for virtual bayesian implementability in conjunction with a second condition bayesian incentive consistency incentive compatibility is also sufficient this new condition is weak under standard topological and informational assumptions it is satisfied by every social choice function the type sets of the agents are taken to be arbitrary possibly infinite measurable spaces an example shows that there are virtually in fact exactly bayesian implementable social choice functions that are not virtually implementable in iteratively undominated strategies

formality for lie algebroids using dolgushev s generalization of fedosov s method for deformation quantization we give a positive answer to a question of p xu can one prove a formality theorem for lie algebroids as a direct application of this result we obtain that any triangular lie bialgebroid is quantizable

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

accessory parameters for punctured spheres this paper contains some qualitative results about the accessory parameters for punctured spheres with signature we show that the fuchsian uniformizing connection and hence also the accessory parameters for the surface depends real analytically on moduli we also show that the important invariants of a uniformization of a punctured sphere such as the accessory parameters fuchsian groups poincare metrics and covering maps vary continuously under degenerations such as coalescing of punctures the branched holomorphic universal covering group of a riemann surface s of finite analytic type g n with 2g 2 n 0 can be described as the monodromy group of the schwarzian differential equation where o is a meromorphic connection on s with singularities of very simple type and only at the punctures of s the connection o depends on 3g 3 n unknown constants called accessory parameters despite numerous attempts see hejhal he2 for some historical remarks these parameters remain elusive they can be determined only in the cases where the covering group is elementary or the surface has no moduli see 2 2 nevertheless the study of the parameters has led to some surprising results hejhal he2 has used variations of accessory parameters to study poincare series and in the work of zograf and takhtadzhyan zt1 zt2 zt3 these accessory parameters appear in calculations of the potential for the weil petersson metric on teichmuller space in this paper we study the simplest case that is n punctured spheres g 0 in this case the connection o is a rational function we show that o and hence also the accessory parameters depends real analytically on moduli and that various objects attached to a uniformization such as the uniformizing received by the editors november 30 1987 1980mathematicssubject classification 1985 revision primary 30f10 30f35 32g15 14h15 research partially supported by national science foundation grants 8401280 and 8120790 1989 american mathematical society 0002 9947 89 1 00 25 per page

the berry keating hamiltonian and the local riemann hypothesis the local riemann hypothesis states that the zeros of the mellin transform of a harmonic oscillator eigenfunction on a real or p adic configuration space have a real part 1 2 for the real case we show that the imaginary parts of these zeros are the eigenvalues of the berry keating hamiltonian projected onto the subspace of oscillator eigenfunctions of a lower level this gives a spectral proof of the local riemann hypothesis for the reals in the spirit of the hilbert polya conjecture the p adic case is also discussed

zeros of symmetric quasi definite orthogonal polynomials abstract it is well known that all zeros of real polynomials orthogonal with respect to a positive definite moment functional are real in the first section of this paper we show that the zeros of certain symmetric quasi definite orthogonal polynomials are real or purely imaginary in the second section we apply our finding to mellin transforms of laguerre polynomials generalizing a result of bump and ng 1

virtual implementation in iteratively undominated strategies complete information we investigate the implementation of social choice functions in complete information environments we consider social choice functions scf s which map from a finite set of preference profiles to lotteries over alternatives and require virtual implementation in iteratively undominated strategies an scf x is virtually implementable in iteratively undominated strategies if for all e 0 there exists an scf y which is e close to x that is for all preference profiles x and y map to lotteries which are within e of one another and which is exactly implementable in iteratively undominated strategies under very weak domain restrictions we show that if there are three or more players any scf is virtually implementable in iteratively undominated strategies a noteworthy feature of our constructions is that we only employ finite mechanisms as a corollary we obtain virtual implementation in pure and mixed strategy nash equilibrium using well behaved in particular finite mechanisms the literature on implementation in nash equilibrium and its refinements is compromised by its reliance on game forms with unnatural features for example integer games or modulo constructions with mixed strategies arbitrarily excluded in contrast our results allow for mixed strategies and do not rely on mechanisms with obviously suspicious features

genericity and robustness of full surplus extraction we study whether priors that admit full surplus extraction fse are generic an issue that becomes a gauge to evaluate the validity of the current mechanism design paradigm we consider the space of priors on the universal type space and thereby relax the assumption of a fixed finite number of types made by cremer and mclean 1988 we show that fse priors are topologically generic contrary to the result of heifetz and neeman 2006 that fse is generically impossible both geometrically and measure theoretically instead of using the bdp approach or convex combinations of priors adopted in heifetz and neeman 2006 we prove our genericity results by showing a robustness property of cremer mclean mechanisms

singular value decomposition of a finite hilbert transform defined on several intervals and the interior problem of tomography the riemann hilbert problem approach we study the asymptotics of singular values and singular functions of a finite hilbert transform fht which is defined on several intervals transforms of this kind arise in the study of the interior problem of tomography we suggest a novel approach based on the technique of the matrix riemann hilbert problem rhp and the steepest descent method of deift zhou we obtain a family of matrix rhps depending on the spectral parameter and show that the singular values of the fht coincide with the values of for which the rhp is not solvable expressing the leading order solution as 0 of the rhp in terms of the riemann theta functions we prove that the asymptotics of the singular values can be obtained by studying the intersections of the locus of zeroes of a certain theta function with a straight line this line can be calculated explicitly and it depends on the geometry of the intervals that define the fht the leading order asymptotics of the singular functions and singular values are explicitly expressed in terms of the riemann theta functions and of the period matrix of the corresponding normalized differentials respectively we also obtain the error estimates for our asymptotic results 2016 wiley periodicals inc

asymptotics for the partition function in two cut random matrix models we obtain large n asymptotics for the random matrix partition function z n v int mathbb r n prod i j x i x j 2 prod j 1 ne nv x j dx j in the case where v is a polynomial such that the random matrix eigenvalues accumulate on two disjoint intervals the two cut case we compute leading and sub leading terms in the asymptotic expansion for log zn v up to terms that are small as n to infty our approach is based on the explicit computation of the first terms in the asymptotic expansion for a quartic symmetric potential v afterwards we use deformation theory of the partition function and of the associated equilibrium measure to generalize our results to general two cut potentials v the asymptotic expansion of log zn v as n to infty contains terms that depend analytically on the potential v and that have already appeared in the literature in addition our method allows us to compute the v independent terms of the asymptotic expansion of log zn v which to the best of our knowledge had not appeared before in the literature we use rigorous orthogonal polynomial and riemann hilbert techniques which had to this point only been successful to compute asymptotics for the partition function in the one cut case

partition function for multi cut matrix models we consider the partition function z n of a random matrix model with polynomial potential v t 1 t 2 2 t 2d 2d it is known that the second logarithmic derivative of z n with respect to the times t k can be expressed in terms of the recurrence coefficients of the related orthogonal polynomials an explicit formula for the recurrence coefficients of the orthogonal polynomials in the limit n oo for multi cut regular v has been derived in 10 through the riemann hilbert approach the expression for z n in the limit n has been derived in 7 through a mean field approach we show that the above asymptotic formulae satisfy the same relations that hold for finite n

a degeneration of two phase solutions of the focusing nonlinear schrodinger equation via riemann hilbert problems two phase solutions of focusing nls equation are classically constructed out of an appropriate riemann surface of genus two and expressed in terms of the corresponding theta function we show here that in a certain limiting regime such solutions reduce to some elementary ones called solitons on unstable condensate this degeneration turns out to be conveniently studied by means of basic tools from the theory of riemann hilbert problems in particular no acquaintance with riemann surfaces and theta function is required for such analysis

computing the strong nash equilibrium for markov chains games present a novel method for finding the strong nash equilibriumintroduce tikhonov s regularization method to guarantee convergence to a single equilibriumprove the existence of a unique strong nash equilibriumpropose the euler method with penalty function for converging to the strong nash equilibriumpresent the convergence conditions of the step size parameter and tikhonov s regularizator in this paper we present a novel method for finding the strong nash equilibrium the approach consists on determining a scalar and the corresponding strategies d fixing specific bounds min and max that belong to the pareto front bounds correspond to restrictions imposed by the player over the pareto front that establish a specific decision area where the strategies can be selected we first exemplify the pareto front of the game in terms of a nonlinear programming problem adding a set of linear constraints for the markov chain game based on the c variable method for solving the strong nash equilibrium problem we propose to employ the euler method and a penalty function with regularization the tikhonov s regularization method is used to guarantee the convergence to a single strong equilibrium point then we established a nonlinear programming method to solve the successive single objective constrained problems that arise from taking the regularized functional of the game to achieve the goal we implement the gradient method to solve the first order optimality conditions starting from an utopia point pareto optimal point given an initial of the individual objectives the method solves an optimization problem adding linear constraints required to find the optimal strong strategy d we show that in the regularized problem the functional of the game decrease and finally converges proving the existence and uniqueness of strong nash equilibrium pareto optimal nash equilibrium in addition we present the convergence conditions and compute the estimated rate of convergence of variables and corresponding to the step size parameter of the gradient method and the tikhonov s regularization respectively moreover we provide all the details needed to implement the method in an efficient and numerically stable way the usefulness of the method is successfully demonstrated by a numerical example

on global deformation quantization in the algebraic case abstract we give a proof of yekutieli s global algebraic deformation quantization result which does not rely on the choice of local sections of the bundle of affine coordinate systems instead we use an argument inspired by algebraic de rham cohomology

on the generic im possibility of full surplus extraction in mechanism design a number of studies most notably cremer and mclean 1985 1988 have shown that in harsanyi type spaces of a fixed finite size it is generically possible to design mechanisms that extract all the surplus from players and as a consequence implement any outcome as if the players private information were commonly known in contrast we show that within the set of common priors on the universal type space the subset of priors that permit the extraction of the players full surplus is shy shyness is an otion of smallness for convex subsets of infinite dimensional topological vector spaces in our case the set of common priors which generalizes the usual notion of zero lebesgue measure in finite dimensional spaces

theta functions modular invariance and strings we use quillen s theorem and algebraic geometry to investigate the modular transformation properties of some quantities of interest in string theory in particular we show that the spin structure dependence of the chiral dirac determinant on a riemann surface is given by riemann s theta function we use this result to investigate the modular invariance of multiloop heterotic string amplitudes

relative formality theorem and quantisation of coisotropic submanifolds we prove a relative version of kontsevich s formality theorem this theorem involves a manifold m and a submanifold c and reduces to kontsevich s theorem if c m it states that the dgla of multivector fields on an infinitesimal neighbourhood of c is l quasiisomorphic to the dgla of multidifferential operators acting on sections of the exterior algebra of the conormal bundle applications to the deformation quantisation of coisotropic submanifolds are given the proof uses a duality transformation to reduce the theorem to a version of kontsevich s theorem for supermanifolds which we also discuss in physical language the result states that there is a duality between the poisson sigma model on a manifold with a d brane and the poisson sigma model on a supermanifold without branes or more properly with a brane which extends over the whole supermanifold

tau functions hodge classes and discriminant loci on moduli spaces of hitchin s spectral covers we define two tau functions and on moduli spaces of spectral covers of gl n hitchin systems analyzing the properties of we express the rational divisor class of the universal hitchin s discriminant in terms of standard generators of the rational picard group of the moduli spaces of spectral covers with variable base the function is used to compute the divisor of canonical 1 forms with multiple zeros

covariant and equivariant formality theorems abstract we give a proof of kontsevich s formality theorem for a general manifold using fedosov resolutions of algebras of polydifferential operators and polyvector fields the main advantage of our construction of the formality quasi isomorphism is that it is based on the use of covariant tensors unlike kontsevich s original proof which is based on jets of polydifferential operators and polyvector fields using our construction we prove that if a group g acts smoothly on a manifold m and m admits a g invariant affine connection then there exists a g equivariant quasi isomorphism of formality this result implies that if a manifold m is equipped with a smooth action of a finite or compact group g or equipped with a free action of a lie group g then m admits a g equivariant formality quasi isomorphism in particular this gives a solution of the deformation quantization problem for an arbitrary poisson orbifold

from local to global deformation quantization of poisson manifolds we give an explicit construction of a deformation quantization of the algebra of functions on a poisson manifold based on m kontsevich s local formula the deformed algebra of functions is realized as the algebra of horizontal sections of a vector bundle with flat connection

holonomy groupoids of singular foliations we give a new construction of lie groupoids which is particularly well adapted to the generalization of holonomy groupoids to singular foliations given a family of local lie groupoids on open sets of a smooth manifold m satisfying some hypothesis we construct a lie groupoid which contains the whole family this construction involves a new way of considering local morita equivalences not only as equivalence relations but also as generalized isomorphisms in particular we prove that almost injective lie algebroids are integrable

remarks on derived equivalences of ricci flat manifolds after a finite etale cover any ricci flat kahler manifold decomposes into a product of complex tori irreducible holomorphic symplectic manifolds and calabi yau manifolds we present results indicating that this decomposition is an invariant of the derived category the main idea to distinguish the derived category of an irreducible holomorphic symplectic manifold from that of a calabi yau manifold is that point sheaves do not deform in certain non commutative deformations of the former whereas they do for the latter on the way we prove a conjecture of c ld raru on the module structure of the hochschild kostant rosenberg isomorphism for manifolds with trivial canonical bundle as a direct consequence of recent work by calaque van den bergh and ramadoss

todd class via homotopy perturbation theory abstract we compute the quantized cycle class of a closed embedding of complex manifolds defined by grivaux using homotopy perturbation theory in the case of a diagonal embedding our approach provides a novel perspective of the usual todd class of a complex manifold

commuting difference operators spinor bundles and the asymptotics of orthogonal polynomials with respect to varying complex weights abstract the paper has three parts in the first part we apply the theory of commuting pairs of pseudo difference operators to the formal asymptotics of orthogonal polynomials using purely geometrical arguments we show heuristically that the asymptotics for large degrees of orthogonal polynomial with respect to varying weights is intimately related to certain spinor bundles on a hyperelliptic algebraic curve reproducing formulae appearing in the works of deift et al on the subject in the second part we show that given an arbitrary nodal hyperelliptic curve satisfying certain conditions of admissibility we can reconstruct a sequence of polynomials orthogonal with respect to semiclassical complex varying weights supported on several curves in the complex plane the strong asymptotics of these polynomials will be shown to be given by the spinors introduced in the first part using a riemann hilbert analysis in the third part we use strebel theory of quadratic differentials and the procedure of welding to reconstruct arbitrary admissible hyperelliptic curves as a result we can obtain orthogonal polynomials whose zeroes may become dense on a collection of jordan arcs forming an arbitrary forest of trivalent loop free trees

a local riemann hypothesis i daniel bump1 kwok kwong choi2 par kurlberg3 jeffrey vaaler4 1 department of mathematics stanford university stanford ca 94305 2125 usa e mail bump math stanford edu 2 department of mathematics the university of hong kong pokfulam road hong kong hong kong e mail choi maths hku hk 3 department of mathematics university of georgia athens ga 30602 usa e mail kurlberg math uga edu 4 university of texas at austin department of mathematics rlm 8 100 austin tx 78712 1082 usa e mail vaaler math utexas edu

formality theorems for hochschild chains in the lie algebroid setting in this paper we prove lie algebroid versions of tsygan s formality conjecture for hochschild chains both in the smooth and holomorphic settings in the holomorphic setting our result implies a version of tsygan s formality conjecture for hochschild chains of the structure sheaf of any complex manifold and in the smooth setting this result allows us to describe quantum traces for an arbitrary poisson lie algebroid the proofs are based on the use of kontsevich s quasi isomorphism for hochschild cochains of r y 1 y d shoikhet s quasi isomorphism for hochschild chains of r y 1 y d and fedosov s resolutions of the natural analogues of hochschild co chain complexes associated with a lie algebroid

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

an infinite family of zeta functions indexed by hermite polynomials abstract riemann s hypothesis has challenged many mathematicians and yet still remains an open problem this work deals with the construction of an infinite family of zeta functions denoted 2 q 0 s for q 0 1 2 which 1 satisfy riemann s functional equation and 2 generalize many properties of the riemann zeta function s in section 3 we show that 2 q 0 s 2 q s p q s where p q s is a polynomial in the variable s of degree q it is true that for small q the zeroes of p q s have real part equal to one half are simple and are distinct we conjecture that for all q 1 2 the zeroes of p q s have real part equal to one half are simple and are distinct and we hope to prove this conjecture in a paper which is to follow it is important to note that before one can establish the properties of 2 q 0 s one must first generalize the jacobi inversion formula this generalization can be found in section 2

deformation quantization in algebraic geometry abstract we study deformation quantizations of the structure sheaf o x of a smooth algebraic variety x in characteristic 0 our main result is that when x is d affine any formal poisson structure on x determines a deformation quantization of o x canonically up to gauge equivalence this is an algebro geometric analogue of kontsevich s celebrated result

isomonodromic deformations with an irregular singularity and hyperelliptic curves in this paper we extend the result of kitaev and korotkin 1998 int math res notices 17 877 905 to the case where a monodromy preserving deformation has an irregular singularity for the monodromy preserving deformation we obtain the function whose deformation parameters are the positions of regular singularities and the parameter t of an irregular singularity furthermore the function is expressed by the hyperelliptic function moving the argument and the period where t and the positions of regular singularities move z and respectively

the stokes groupoids we construct and describe a family of groupoids over complex curves which serve as the universal domains of definition for solutions to linear ordinary differential equations with singularities as a consequence we obtain a direct functorial method for resumming formal solutions to such equations

symplectic manifolds and isomonodromic deformations abstract we study moduli spaces of meromorphic connections with arbitrary order poles over riemann surfaces together with the corresponding spaces of monodromy data involving stokes matrices natural symplectic structures are found and described both explicitly and from an infinite dimensional viewpoint generalising the atiyah bott approach this enables us to give an intrinsic symplectic description of the isomonodromic deformation equations of jimbo miwa and ueno thereby putting the existing results for the six painleve equations and schlesinger s equations into a uniform framework

twisted deformation quantization of algebraic varieties abstract let x be a smooth algebraic variety over a field k containing the real numbers we introduce the notion of twisted associative resp poisson deformation of the structure sheaf o x these are stack like versions of usual deformations we prove that there is a twisted quantization operation from twisted poisson deformations to twisted associative deformations which is canonical and bijective on gauge equivalence classes this result extends work of kontsevich and our own earlier work on deformation quantization of algebraic varieties