orbifolds and exact solutions of strongly coupled matrix models we find an exact solution to strongly coupled matrix models with a single trace monomial potential our solution yields closed form expressions for the partition function as well as averages of schur functions the results are fully factorized into a product of terms linear in the rank of the matrix and the parameters of the model we extend our formulas to include both logarithmic and finite difference deformations thereby generalizing the celebrated selberg and kadell integrals we conjecture a formula for correlators of two schur functions in these models and explain how our results follow from a general orbifold like procedure that can be applied to any one matrix model with a single trace potential

some recursive formulas for selberg type integrals a set of recursive relations satisfied by selberg type integrals involving monomial symmetric polynomials are derived generalizing previous results in aomoto 1987 siam j math anal 18 545 49 and iguri 2009 lett math phys 89 141 58 these formulas provide a well defined algorithm for computing selberg schur integrals whenever the kostka numbers relating schur functions and the corresponding monomial polynomials are explicitly known we illustrate the usefulness of our results discussing some interesting examples

asymptotic expansions of witten reshetikhin turaev invariants for some simple 3 manifolds the link l this paper investigates the invariant z m 0 when g sk for a simple family of rational homology 3 spheres obtained by integer surgery around 2 n type torus knots in particular we find a closed formula for a formal power series z m eq h in h q 1 from which zf m 0 may be derived for all sufficiently large primes r we show that this formal power series may be viewed as the asymptotic expansion around 4 1 of a multivalued holomorphic function of q with 1 contained on the boundary of its domain of definition for these particular manifolds most of which are not z homology spheres this extends work of ohtsuki and murakami in which the existence of power series with rational coefficients related to z m 0 was demonstrated for rational homology spheres the coefficients in the formal power series z m are expected to be identical to those obtained from a perturbative expansion of the witten chem simons path integral formula for z m 0 0 199 5 american institute of physics

noncolliding brownian motion with drift and time dependent stieltjes wigert determinantal point process using the determinantal formula of biane bougerol and o connell we give multitime joint probability densities to the noncolliding brownian motion with drift where the number of particles is finite we study a special case such that the initial positions of particles are equidistant with a period a and the values of drift coefficients are well ordered with a scale we show that at each time t 0 the single time probability density of particle system is exactly transformed to the biorthogonal stieltjes wigert matrix model in the chern simons theory introduced by dolivet and tierz here one parameter extensions extensions of the stieltjes wigert polynomials which are themselves q extensions of the hermite polynomials play an essential role the two parameters a and of the process combined with time t are mapped to the parameters q and of the biorthogonal polynomials by the transformation of normalization factor of our probability density the partition function of the chern simons matrix model is readily obtained we study the determinantal structure of the matrix model and prove that at each time t 0 the present noncolliding brownian motion with drift is a determinantal point process in the sense that any correlation function is given by a determinant governed by a single integral kernel called the correlation kernel using the obtained correlation kernel we study time evolution of the noncolliding brownian motion with drift

integrable almost symplectic hamiltonian systems we extend the notion of liouville integrability which is peculiar to hamiltonian systems on symplectic manifolds to hamiltonian systems on almost symplectic manifolds namely manifolds equipped with a nondegenerate but not closed 2 form the key ingredient is to require that the hamiltonian vector fields of the integrals of motion in involution or equivalently the generators of the invariant tori are symmetries of the almost symplectic form we show that under this hypothesis essentially all of the structure of the symplectic case from quasiperiodicity of motions to an analog of the action angle coordinates and of the isotropic coisotropic dual pair structure characteristic of the fibration by the invariant tori carries over to the almost symplectic case

numerical generation of periodic traveling wave solutions of some nonlinear dispersive wave systems in this paper a numerical procedure to generate approximations to periodic traveling wave profiles of some nonlinear dispersive wave systems is introduced the method is based on a suitable modification of a fixed point algorithm of petviashvili type and solves several drawbacks of some previous algorithms presented in the literature by way of illustration the method is applied to generate numerically periodic traveling waves of two problems of interest the fractional kdv type equations and the extended boussinesq system we propose a method to compute periodic traveling waves the method can be applied to some nonlinear dispersive systems it is checked in fractional kdv equations and boussinesq systems

a conceptual approach to the problem of action angle variables in this paper we develop a general conceptual approach to the problem of existence of action angle variables for dynamical systems which establishes and uses the fundamental conservation property of associated torus actions anything which is preserved by the system is also preserved by the associated torus actions this approach allows us to obtain among other things a the shortest and most easy to understand conceptual proof of the classical arnold liouville mineur theorem b basically all known results in the literature about the existence of action angle variables in various contexts can be recovered in a unifying way with simple proofs using our approach c new results on action angle variables in many different contexts including systems on contact manifolds systems on presymplectic and dirac manifolds action angle variables near singularities stochastic systems and so on even when there are no natural action variables our approach still leads to useful normal forms for dynamical systems which are not necessarily integrable

a large k asymptotics of witten s invariant of seifert manifolds we calculate a largek asymptotic expansion of the exact surgery formula for witten ssu 2 invariant of some seifert manifolds the contributions of all flat connections are identified an agreement with the 1 loop formula is checked a contribution of the irreducible connections appears to contain only a finite number of terms in the asymptotic series a 2 loop correction to the contribution of the trivial connection is found to be proportional to casson s invariant

from weak to strong coupling in abjm theory the partition function of mathcal n 6 supersymmetric chern simons matter theory known as abjm theory on mathbb s 3 as well as certain wilson loop observables are captured by a zero dimensional super matrix model this super matrix model is closely related to a matrix model describing topological chern simons theory on a lens space we explore further these recent observations and extract more exact results in abjm theory from the matrix model in particular we calculate the planar free energy which matches at strong coupling the classical iia supergravity action on rm ads 4 times mathbb c mathbb p 3 and gives the correct n3 2 scaling for the number of degrees of freedom of the m2 brane theory furthermore we find contributions coming from world sheet instanton corrections in mathbb c mathbb p 3 we also calculate non planar corrections both to the free energy and to the wilson loop expectation values this matrix model appears also in the study of topological strings on a toric calabi yau manifold and an intriguing connection arises between the space of couplings of the planar abjm theory and the moduli space of this calabi yau in particular it suggests that in addition to the usual perturbative and strong coupling ads expansions a third natural expansion locus is the line where one of the two t hooft couplings vanishes and the other is finite this is the conifold locus of the calabi yau and leads to an expansion around topological chern simons theory we present some explicit results for the partition function and wilson loop observables around this locus

theory of tensor invariants of integrable hamiltonian systems ii theorem on symmetries and its applications the theorem on symmetries is proved that states that a liouville integrable hamiltonian system is non degene rate in kolmogorov s sense and has compact invariant submanifolds if and only if the corresponding lie algebra of symmetries is abelian the theorem on symmetries has applications to the characterization problem to the integrable hierarchies problem to the necessary conditions for the strong dynamical compatibility problem and to the problem on master symmetries the invariant necessary conditions for the non degenerate c integrability in kolmogorov s sense of a given dynamical system v are derived it is proved that the c integrable hamiltonian system is non degenerate in the iso energetic sense if and only if the corresponding lie algebra of the iso energetic conformal symmetries is abelian an extended concept of integrability of hamiltonian systems on the symplectic manifolds m n n 2k is introduced the concept of integrability describes the hamiltonian systems that have quasi periodic dynamics on tori or on toroidal cylinders of an arbitrary dimension this concept includes as a particular case all hamiltonian systems that are integrable in liouville s classical sense for which the a b c cohomologies are introduced for dynamical systems on smooth manifolds

energy dissipating flows for solving nonlinear eigenpair problems abstract this work is concerned with computing nonlinear eigenpairs which model solitary waves and various other physical phenomena we aim at solving nonlinear eigenvalue problems of the general form t u q u in our setting t is a variational derivative of a convex functional such as the laplacian operator with respect to the dirichlet energy q is an arbitrary bounded nonlinear operator and is an unknown real eigenvalue we introduce a flow that numerically generates an eigenpair solution by its steady state analysis for the general case is performed showing a monotone decrease in the convex functional throughout the flow when t is the laplacian operator a complete discretized version is presented and anlalyzed we implement our algorithm on korteweg and de vries kdv and nonlinear schrodinger nls equations in one and two dimensions the proposed approach is very general and can be applied to a large variety of models moreover it is highly robust to noise and to perturbations in the initial conditions compared to classical petiashvili based methods

superstrings and topological strings at large n we embed the large n chern simons topological string duality in ordinary superstrings this corresponds to a large n duality between generalized gauge systems with n 1 supersymmetry in four dimensions and superstrings propagating on noncompact calabi yau manifolds with certain fluxes turned on we also show that in a particular limit of the n 1 gauge theory system certain superpotential terms in the n 1 system including deformations if spacetime is noncommutative are captured to all orders in 1 n by the amplitudes of noncritical bosonic strings propagating on a circle with self dual radius we also consider d brane anti d brane system wrapped over vanishing cycles of compact calabi yau manifolds and argue that at large n they induce a shift in the background to a topologically distinct calabi yau which we identify as the ground state system of the brane anti brane system

residue formulas for the large k asymptotics of witten s invariants of seifert manifolds the case of su 2 we derive the largek asymptotics of the surgery formula forsu 2 witten s invariants of general seifert manifolds the contributions of connected components of the moduli space of flat connections are identified the contributions of irreducible connections are presented in the residue form this allows us to express them in terms of intersection numbers on their moduli spaces

extended integrability and bi hamiltonian systems the current notion of integrability of hamiltonian systems was fixed by liouville in a famous 1855 paper it describes systems in a 2k dimensional phase space whose trajectories are dense on tori q or wind on toroidal cylinders m q m within liouville s construction the dimension q cannot exceed k and is the main invariant of the system in this paper we generalize liouville integrability so that trajectories can be dense on tori k of arbitrary dimensions q 1 2k 1 2k and an additional invariant v 2 q k v 2 q 2 can be recovered the main theorem classifies all k k 1 2 canonical forms of hamiltonian systems that are integrable in a newly defined broad sense an integrable physical problem having engineering origin is presented the notion of extended compatibility of two poisson structures is introduced the corresponding bi hamiltonian systems are shown to be integrable in the broad sense

theory of tensor invariants of integrable hamiltonian systems i incompatible poisson structures this paper develops a new theory of tensor invariants of a completely integrable non degenerate hamiltonian system on a smooth manifoldmn the central objects in this theory are supplementary invariant poisson structurespc which are incompatable with the original poisson structurep1 for this hamiltonian system a complete classification of invariant poisson structures is derived in a neighbourhood of an invariant toroidal domain this classification resolves the well known inverse problem that was brought into prominence by magri s 1978 paper deveoted to the theory of compatible poisson structures applications connected with the kam theory with the kepler problem with the basic integrable problem of celestial mechanics and with the harmonic oscillator are pointed out a cohomology is defined for dynamical systems on smooth manifolds the physically motivated concepts of dynamical compatibility and strong dynamical compatibility of pairs of poisson structures are introduced to study the diversity of pairs of poisson structures incompatible in magri s sense it is proved that if a dynamical systemv preserves two strongly dynamically compatible poisson structuresp1 andp2 in a general position then this system is completely integrable such a systemv generates a hierarchy of integrable dynamical systems which in general are not hamiltonian neither with respect top1 nor with respect top2 necessary conditions for dynamical compatibility and for strong dynamical compatibility are derived which connect these global properties with new local invariants of an arbitrary pair of incompatible poisson structures

a boussinesq system for two way propagation of interfacial waves the theory of internal waves between two layers of immiscible fluids is important both for its applications in oceanography and engineering and as a source of interesting mathematical model equations that exhibit nonlinearity and dispersion a boussinesq system for two way propagation of interfacial waves in a rigid lid configuration is derived in most cases the nonlinearity is quadratic however when the square of the depth ratio is close to the density ratio the coefficients of the quadratic nonlinearities become small and cubic nonlinearities must be considered the propagation as well as the collision of solitary waves and or fronts is studied numerically

computer calculation of witten s 3 manifold invariant witten s 2 1 dimensional chern simons theory is exactly solvable we compute the partition function a topological invariant of 3 manifolds on generalized seifert spaces thus we test the path integral using the theory of 3 manifolds in particular we compare the exact solution with the asymptotic formula predicted by perturbation theory we conclude that this path integral works as advertised and gives an effective topological invariant

chern simons theory matrix integrals and perturbative three manifold invariants the universal perturbative invariants of rational homology spheres can be extracted from the chern simons partition function by combining perturbative and nonperturbative results we spell out the general procedure to compute these invariants and we work out in detail the case of seifert spaces by extending some previous results of lawrence and rozansky the chern simons partition function with arbitrary simply laced group for these spaces is written in terms of matrix integrals the analysis of the perturbative expansion amounts to the evaluation of averages in a gaussian ensemble of random matrices as a result explicit expressions for the universal perturbative invariants of seifert homology spheres up to order five are presented

chern simons witten invariants of lens spaces and torus bundles and the semiclassical approximation we derive explicit formulas for the chern simons witten invariants of lens spaces and torus bundles overs1 for arbitrary values of the levelk most of our results are for the groupg su 2 though some are for more general compact groups we explicitly exhibit agreement of the limiting values of these formulas ask with the semiclassical approximation predicted by the chern simons path integral

remodeling the b model we propose a complete new formalism to compute unambiguously b model open and closed amplitudes in local calabi yau geometries including the mirrors of toric manifolds the formalism is based on the recursive solution of matrix models recently proposed by eynard and orantin the resulting amplitudes are non perturbative in both the closed and the open moduli the formalism can then be used to study stringy phase transitions in the open closed moduli space at large radius this formalism may be seen as a mirror formalism to the topological vertex but it is also valid in other phases in the moduli space we develop the formalism in general and provide an extensive number of checks including a test at the orbifold point of ap fibrations where the amplitudes compute the t hooft expansion of vevs of wilson loops in chern simons theory on lens spaces we also use our formalism to predict the disk amplitude for the orbifold mathbb c 3 mathbb z 3

petviashvili type methods for traveling wave computations i analysis of convergence in this paper a family of fixed point algorithms for the numerical resolution of some systems of nonlinear equations is designed and analyzed the family introduced here generalizes the petviashvili method and can be applied to the numerical generation of traveling waves in some nonlinear dispersive systems conditions for the local convergence are derived and numerical comparisons between different elements of the family are carried out

two dimensional yang mills theory painleve equations and the six vertex model we show that the chiral partition function of two dimensional yang mills theory on the sphere can be mapped to the partition function of the homogeneous six vertex model with domain wall boundary conditions in the ferroelectric phase a discrete matrix model description in both cases is given by the meixner ensemble leading to a representation in terms of a stochastic growth model we show that the partition function is a particular case of the z measure on the set of young diagrams yielding a unitary matrix model for chiral yang mills theory on s2 and the identification of the partition function as a tau function of the painlev v equation we describe the role played by generalized non chiral yang mills theory on s2 in relating the meixner matrix model to the toda chain hierarchy encompassing the integrability of the six vertex model we also argue that the thermodynamic behaviour of the six vertex model in the disordered and antiferroelectric phases are captured by particular q deformations of two dimensional yang mills theory on the sphere

characteristic polynomials of random matrices number theorists have studied extensively the connections between the distribution of zeros of the riemann function and of some generalizations with the statistics of the eigenvalues of large random matrices it is interesting to compare the average moments of these functions in an interval to their counterpart in random matrices which are the expectation values of the characteristic polynomials of the matrix it turns out that these expectation values are quite interesting for instance the moments of order 2k scale for unitary invariant ensembles as the density of eigenvalues raised to the power k 2 the prefactor turns out to be a universal number i e it is independent of the specific probability distribution an equivalent behaviour and prefactor had been found as a conjecture within number theory the moments of the characteristic determinants of random matrices are computed here as limits at coinciding points of multi point correlators of determinants these correlators are in fact universal in dyson s scaling limit in which the difference between the points goes to zero the size of the matrix goes to infinity and their product remains finite

witten reshetikhin turaev invariants of seifert manifolds for seifert homology spheres we derive a holomorphic function of k whose value at integer k is the sl2 witten reshetikhin turaev invariant zk at q exp 2 i k this function is expressed as a sum of terms which can be naturally corresponded to the contributions of flat connections in the stationary phase expansion of the witten chern simons path integral the trivial connection contribution is found to have an asymptotic expansion in powers of k 1 which for k an odd prime power converges k adically to the exact total value of the invariant zk at that root of unity evaluations at rational k are also discussed using similar techniques an expression for the coloured jones polynomial of a torus knot is obtained providing a trivial connection contribution which is an analytic function of the colour this demonstrates that the stationary phase expansion of the chern simons witten theory is exact for seifert manifolds and for torus knots in s3 the possibility of generalising such results is also discussed

analytic integrable systems analytic normalization and embedding flows abstract in this paper we mainly study the existence of analytic normalization and the normal form of finite dimensional complete analytic integrable dynamical systems more details we will prove that any complete analytic integrable diffeomorphism f x b x f x in c n 0 with b having eigenvalues not modulus 1 and f x o x 2 is locally analytically conjugate to its normal form meanwhile we also prove that any complete analytic integrable differential system x a x f x in c n 0 with a having nonzero eigenvalues and f x o x 2 is locally analytically conjugate to its normal form furthermore we will prove that any complete analytic integrable diffeomorphism defined on an analytic manifold can be embedded in a complete analytic integrable flow we note that parts of our results are the improvement of moser s one in j moser the analytic invariants of an area preserving mapping near a hyperbolic fixed point comm pure appl math 9 1956 673 692 and of poincare s one in h poincare sur l integration des equations differentielles du premier order et du premier degre ii rend circ mat palermo 11 1897 193 239 these results also improve the ones in xiang zhang analytic normalization of analytic integrable systems and the embedding flows j differential equations 244 2008 1080 1092 in the sense that the linear part of the systems can be nonhyperbolic and the one in n t zung convergence versus integrability in poincare dulac normal form math res lett 9 2002 217 228 in the way that our paper presents the concrete expression of the normal form in a restricted case

one dimensional quaternion homogeneous polynomial differential equations in this paper we study the dynamics of real four dimensional polynomial differential equations of the forms q aqkq m or q qkaqm where q a are quaternions q is the conjugate of q and k and m are non negative integers in many parts of the work the quaternion structure is used along the proofs

greene s residue criterion for the breakup of invariant tori of volume preserving maps abstract invariant tori play a fundamental role in the dynamics of symplectic and volume preserving maps codimension one tori are particularly important as they form barriers to transport such tori foliate the phase space of integrable volume preserving maps with one action and d angles for the area preserving case greene s residue criterion is often used to predict the destruction of tori from the properties of nearby periodic orbits even though kam theory applies to the three dimensional case the robustness of tori in such systems is still poorly understood we study a three dimensional reversible volume preserving analogue of chirikov s standard map with one action and two angles we investigate the preservation and destruction of tori under perturbation by computing the residue of nearby periodic orbits we find tori with diophantine rotation vectors in the spiral mean cubic algebraic field the residue is used to generate the critical function of the map and find a candidate for the most robust torus

newton s method with deflation for isolated singularities of polynomial systems we present a modification of newton s method to restore quadratic convergence for isolated singular solutions of polynomial systems our method is symbolic numeric we produce a new polynomial system which has the original multiple solution as a regular root using standard bases a tool for the symbolic computation of multiplicities we show that the number of deflation stages is bounded by the multiplicity of the isolated root our implementation performs well on a large class of applications

on the virasoro constraints for torus knots we construct a virasoro algebra of differential operators for the matrix model for torus knots these operators generate various relations between wilson loops then we discuss the operators constructed and corresponding relations in the stability limit we also give a series of examples

torus homflypt as the hall littlewood polynomials we show that the homflypt polynomials for the torus knots t m n in all fundamental representations are equal to the hall littlewood polynomials in a representation which depends on m and with a quantum parameter which depends on n this makes the long anticipated interpretation of wilson averages in 3d chern simons theory as characters precise at least for the torus knots and calls for further studies in this direction this fact is deeply related to the hall littlewood macdonald duality of character expansion of superpolynomials found in mironov et al 2012 j geom phys 62 148 55 in fact the relation continues to hold for extended polynomials but the symmetry between m and n is broken then m is the number of strands in the braid besides the homflypt case with q t the torus superpolynomials are reduced to the single hall littlewood characters in the two other distinguished cases q 0 and t 0

schur polynomials and biorthogonal random matrix ensembles the study of the average of schur polynomials over a stieltjes wigert ensemble has been carried out by dolivet and tierz j math phys 48 023507 2007 e print arxiv hep th 0609167 where it was shown that it is equal to quantum dimensions using the same approach we extend the result to the biorthogonal case we also study using the littlewood richardson rule some particular cases of the quantum dimension result finally we show that the notion of giambelli compatibility of schur averages introduced by borodin et al adv appl math 37 209 2006 e print arxiv math ph 0505021 also holds in the biorthogonal setting

the uses of the refined matrix model recursion we study matrix models in the ensemble by building on the refined recursion relation proposed by chekhov and eynard we present explicit results for the first deformed corrections in the one cut and the two cut cases as well as two applications to supersymmetric gauge theories the calculation of superpotentials in n 1 gauge theories and the calculation of vevs of surface operators in superconformal n 2 theories and their liouville duals finally we study the deformation of the chern simons matrix model our results indicate that this model does not provide an appropriate description of the deformed topological string on the resolved conifold and therefore that the deformation might provide a different generalization of topological string theory in toric calabi yau backgrounds

oscillatory matrix model in chern simons theory and jacobi theta determinantal point process the partition function of the chern simons theory on the three sphere with the unitary group u n provides a one matrix model the corresponding n particle system can be mapped to the determinantal point process whose correlation kernel is expressed by using the stieltjes wigert orthogonal polynomials the matrix model and the point process are regarded as q extensions of the random matrix model in the gaussian unitary ensemble and its eigenvalue point process respectively we prove the convergence of the n particle system to an infinite dimensional determinantal point process in n in which the correlation kernel is expressed by jacobi s theta functions we show that the matrix model obtained by this limit realizes the oscillatory matrix model in chern simons theory discussed by de haro and tierz

the liouville arnold nekhoroshev theorem for non compact invariant manifolds under certain conditions generalized action angle coordinates can be introduced near non compact invariant manifolds of completely and partially integrable hamiltonian systems

matrix integration and combinatorics of modular groups integration over gaussian matrix ensembles was used to obtain the virtual euler characteristics of mapping class groups we present some simplifications in the combinatorial part of the calculation

bi hamiltonian partially integrable systems given a first order dynamical system possessing a commutative algebra of dynamical symmetries we show that under certain conditions there exists a poisson structure on an open neighborhood of its regular not necessarily compact invariant manifold which makes this dynamical system into a partially integrable hamiltonian system this poisson structure is by no means unique bi hamiltonian partially integrable systems are described in some detail as an outcome we state the conditions of quasiperiodic stability the kam theorem for partially integrable hamiltonian systems

existence of periodic travelling wave solutions for a regularized benjamin ono system abstract in this paper we discuss the existence of periodic travelling wave solutions of a regularized benjamin ono system by using the topological degree theory of positive operators on banach spaces furthermore we use a high accuracy pseudospectral solver based on a fourier decomposition to construct numerical approximations of these stationary solutions the numerical simulations are in perfect agreement with the theoretical results

a numerical scheme for periodic travelling wave simulations in some nonlinear dispersive wave models a numerical method for simulating periodic travelling wave solutions of some nonlinear dispersive wave equations is proposed the construction of the scheme is based on an efficient computation of the elements that characterize these solutions the initial profile and the velocity of the wave

nonlocal models for nonlinear dispersive waves abstract considered herein are model equations for the unidirectional propagation of small amplitude nonlinear dispersive long waves such as those governed by the classical korteweg de vries equation of special interest are physical situations in which the linear dispersion relation is not appropriately approximated by a polynomial so that the operator modelling dispersive effect is nonlocal particular cases in view here are the benjamin ono equation and the intermediate long wave equation which arise in internal wave theory and smith s equation which governs certain types of continental shelf waves the initial value problem for these equations is shown to be globally well posed in the classical sense including continuous dependence upon the initial data and in certain cases upon the modelling of nonlinear and dispersive effects whilst the results are stated for the specific equations listed above the techniques utilized are seen to have a considerable range of generality as regards application to nonlinear dispersive evolution equations particularly worthy of note is our theorem implying that solutions of the intermediate long wave equation converge strongly to solutions of the korteweg de vries equation or to solutions of the benjamin ono equation in appropriate asymptotic limits

chern simons matrix models and stieltjes wigert polynomials employing the random matrix formulation of chern simons theory on seifert manifolds we show how the stieltjes wigert orthogonal polynomials are useful in exact computations in chern simons matrix models we construct a biorthogonal extension of the stieltjes wigert polynomials not available in the literature necessary to study chern simons matrix models when the geometry is a lens space we also study the relationship between stieltjes wigert and rogers szego polynomials and the corresponding equivalence with a unitary matrix model finally we give a detailed proof of a result that relates quantum dimensions with averages of schur polynomials in the stieltjes wigert ensemble

a contribution of the trivial connection to the jones polynomial and witten s invariant of 3d manifolds ii we extend the results of our previous paper 1 from knots to links by using a formula for the jones polynomial of a link derived recently by n reshetikhin we establish a relation between the parameters of this formula and the multivariable alexander polynomial this relation is illustrated by an example of a torus link we check that our expression for the alexander polynomial satisfies some of its basic properties finally we derive a link surgery formula for the loop corrections to the trivial connection contribution to witten s invariant of rational homology spheres

chern simons perturbation theory ii we study the perturbation theory for three dimensional chern simons quantum field theory on a general compact three manifold without boundary we show that after a simple change of variables the action obtained by brs gauge fixing in the lorentz gauge has a superspace formulation the basic properties of the propagator and the feynman rules are written in a precise manner in the language of differential forms using the explicit description of the propagator singularities we prove that the theory is finite finally the anomalous metric dependence of the 2 loop partition function on the riemannian metric which was introduced to define the gauge fixing can be cancelled by a local counterterm as in the 1 loop case in fact the counterterm is equal to the chern simons action of the metric connection normalized precisely as one would expect based on the framing dependence of witten s exact solution invited talk at the xxth conference on differential geometric methods in physics new york june 1991

noncommutative integrability on noncompact invariant manifolds the mishchenko fomenko theorem on noncommutative completely integrable hamiltonian systems on a symplectic manifold is extended to the case of noncompact invariant submanifolds

wall crossing as seen by matrix models the number of bps bound states of d branes on a calabi yau manifold depends on two sets of data the bps charges and the stability conditions for d0 and d2 branes bound to a single d6 brane wrapping a calabi yau 3 fold x both are naturally related to the kahler moduli space mathcal m x we construct unitary one matrix models which count such bps states for a class of toric calabi yau manifolds at infinite t hooft coupling the matrix model for the bps counting on x turns out to give the topological string partition function for another calabi yau manifold y whose kahler moduli space mathcal m y contains two copies of mathcal m x one related to the bps charges and another to the stability conditions the two sets of data are unified in mathcal m y the matrix models have a number of other interesting features they compute spectral curves and mirror maps relevant to the remodeling conjecture for finite t hooft coupling they give rise to yet more general geometry widetilde y containing y

the selberg jack symmetric functions abstract k aomoto has recently given a simple proof of an extension of a selberg s integral we prove the following generalization of aomoto s theorem for eachk 0 there exists a family sk t of homogeneous symmetric functions such that if the integrand of selberg s integral is multiplied bysk t1 tn then the integral has a certain closed form sincesk t is a renormalized jack symmetric function with 1 k we call sk t the selberg jack symmetric functions r p stanley and i g macdonald have extended many of the basic properties of the i j schur functions to the jack symmetric functions stanley s extensions of the pieri formula in a form obtained by macdonald s extension of the duality property and the combinatorial representation arise naturally in our proof we give proofs of these and other results of stanley and macdonald we give two constant term identities which are equivalent to the integration formula these results extend w g morris theorem foranand the macdonald morris constant term identity forbcn by exploring selberg s proof of his integral we give a recurrence relation outline a method for calculating the selberg jack symmetric functions and show that a constant term orthogonality relation recently proved by macdonald arises naturally we give an expansion which allows us to evaluate the normalization of this orthogonality thus giving an alternative proof of the integration formula

on the rogers szego polynomials an orthogonality relation on the full real line for the rogers szego polynomials is discussed it is argued that fourier transformation with the standard exponential kernel exp ixy relates the rogers szego and stieltjes wigert functions

hamiltonian systems with a discrete symmetry abstract if m m is an antisymplectic involution of a symplectic manifold m then the fixed set of is a lagrangian submanifold l m moreover there exist cotangent bundle coordinates in a neighborhood of l in m such that in these coordinates maps a covector into its negative thus classical examples which have a discrete symmetry such as the restricted three body problems are locally like a reversible system

nonintegrability of dynamical systems with homo and heteroclinic orbits abstract we consider general n dimensional systems of differential equations having an n 2 dimensional locally invariant manifold on which there exist equilibria connected by heteroclinic orbits for n 3 the system may be non hamiltonian and have no saddle centers and the equilibria are allowed to be the same and connected by a homoclinic orbit under additional assumptions we prove that the monodromy group for the normal variational equation which is represented by components of the variational equation normal to the locally invariant manifold and defined on a riemann surface is diagonalizable or infinitely cyclic if the system is real meromorphically integrable in the meaning of bogoyavlenski we apply our theory to a three dimensional volume preserving system describing the streamline of a steady incompressible flow with two parameters and show that it is real meromorphically nonintegrable for almost all values of the two parameters

e8 spectral curves i provide an explicit construction of spectral curves for the affine mathrm e 8 relativistic toda chain their closed form expression is obtained by determining the full set of character relations in the representation ring of mathrm e 8 for the exterior algebra of the adjoint representation this is in turn employed to provide an explicit construction of both integrals of motion and the action angle map for the resulting integrable system i consider two main areas of applications of these constructions on the one hand i consider the resulting family of spectral curves in the context of the correspondences between toda systems 5d seiberg witten theory gromov witten theory of orbifolds of the resolved conifold and chern simons theory to establish a version of the b model gopakumar vafa correspondence for the mathrm sl n l e murakami ohtsuki invariant of the poincar e integral homology sphere to all orders in 1 n on the other i consider a degenerate version of the spectral curves and prove a 1 dimensional landau ginzburg mirror theorem for the frobenius manifold structure on the space of orbits of the extended affine weyl group of type mathrm e 8 introduced by dubrovin zhang equivalently the orbifold quantum cohomology of the type mathrm e 8 polynomial mathbb c p 1 orbifold this leads to closed form expressions for the flat co ordinates of the saito metric the prepotential and a higher genus mirror theorem based on the chekhov eynard orantin recursion i will also show how the constructions of the paper lead to a generalisation of a conjecture of norbury scott to ade mathbb p 1 orbifolds and a mirror of the dubrovin zhang construction for all weyl groups and choices of marked roots

form invariant poisson brackets of hydrodynamic type with several spatial variables form invariant poisson brackets of hydrodynamic type with several spatial variables x1 xp are introduced for any commuting vector fields u1 um on a manifold mn the incompressibility equation for certain parameters of the poisson brackets is derived poisson brackets connected with an arbitrary dynamical system on mn are constructed a gauge transform for the poisson brackets with one spatial variable x is defined and its action onto the tensor invariants is studied a general identity between the coefficients bkij u and gij u is found