action of the loop group on the self dual yang mills equation recently kac moody symmetry has played an important role in mathematical physics dolan and chau ge and wu discovered an infinitesimal action of the kac moody lie algebra on the space of solutions of sdym we have discovered an action of the loop group on the space of generalized solutions of sdym which exponentiates the kac moody action the group acts by adding a special type of source onto the solution the action is a geometric construction using the twistor picture

from template analysis to generating partitions ii characterization of the symbolic encodings we give numerical evidence of the validity of a previously described algorithm for constructing symbolic encodings of chaotic attractors from a template analysis we verify that the different solutions that can be found are dynamically equivalent and that our approach yields results that are consistent with those obtained from methods based on homoclinic tangencies this is further confirmed by verifying directly that the computed partitions are generating to a high degree of accuracy and that they can be used to estimate precisely the metric entropy it is also shown that the correct number of symbols needed to describe the dynamics is naturally provided and that a compact parameterization of a partition can easily be determined which makes our algorithm suitable for applications such as real time encoding

kahlerian twistor spaces the work of r penrose has shown the deep relationship between the conformal structure of minkowski space and the complex geometry of lines in projective three space this transformation applies to a more general class of conformal structures and in particular there is a riemannian version described in 5 which is itself of interest to mathematical physicists basically to each oriented riemannian 4manifold x one associates canonically a 6 manifold z the twistor space of x together with an almost complex structure if this structure is integrable then z becomes a complex 3 manifold with a distinguished family of projective lines on it whose geometry describes the conformal geometry of the manifold x it is natural to ask what sort of complex manifold one obtains by this process in this paper we ask the more specific question which compact 4c manifolds have ka hlerian twistor spaces 1 the answer rather surprisingly is that there are just two examples the 4 sphere s and the complex projective plane p2 c each with their standard homogeneous conformal structures the twistor space of s 4 is the 3 dimensional projective space p3 c and that of p 2 q is the manifold f2 c of flags in c 3 the essential point of the proof of this result is the remark that if z is a twistor space with a kahler metric then the canonical bundle k is negative using vanishing theorems and the riemann roch theorem we then narrow down the possibilities for z to four cases projective space the flag manifold the intersection of two quadrics in p5 and the double covering of p3 branched over a non singular quartic surface the last two candidates cannot be realized however because the value of their euler characteristic is incompatible with their being twistor spaces throughout the proof we make essential use of a real structure on z which reduces considerably the occurrence of singularities at various stages aside from this our methods are the standard ones of algebraic geometry the four examples to which the proof leads us are all fano threefolds if we drop the requirement of compactness on x then it is quite possible for an open set of a fano threefold to be a twistor space in particular we show that the family of conies lying on the intersection of two

a finiteness theorem for harmonic maps into hilbert grassmannians in this article we demonstrate that every harmonic map from a closed riemannian manifold into a hilbert grassmannian has image contained within a finite dimensional grassmannian

backlund transformations and loop group actions we construct a local action of the group of rational maps from 2 to gl n on local solutions of flows of the zs akns sl n hierarchy we show that the actions of simple elements linear fractional transformations give local backlund transformations and we derive a permutability formula from different factorizations of a quadratic element we prove that the action of simple elements on the vacuum may give either global smooth solutions or solutions with singularities however the action of the subgroup of the rational maps that satisfy the u n reality condition on the space of global rapidly decaying solutions of the flows in the u n hierarchy is global and the action of a simple element gives a global backlund transformation the actions of certain elements in the rational loop group on the vacuum give rise to explicit time periodic multisolitons multibreathers we show that this theory generalizes the classical backlund theory of the sine gordon equation the group structures of backlund transformations for various hierarchies are determined by their reality conditions we identify the reality conditions the group structures for the sl n u k n k kdv kupershmidt wilson and gel fand dikii hierarchies the actions of linear fractional transformations that satisfy a reality condition modulo the center of the group of rational maps give backlund and darboux transformations for the hierarchy defined by the reality condition since the factorization cannot always be carried out under this reality condition the action is again local and backlund transformations only generate local solutions for these hierarchies unless singular solutions are allowed 2000 john wiley sons inc

calabi yau generalized complete intersections and aspects of cohomology of sheaves we consider generalized complete intersection manifolds in the product space of projective spaces and work out useful aspects pertaining to the cohomology of sheaves over them first we present and prove a vanishing theorem on the cohomology groups of sheaves for subvarieties of the ambient product space of projective spaces we then prove a birational equivalence between configuration matrices of complete intersection calabi yau manifolds we also present a formula of the genus of curves in generalized complete intersection manifolds some of these curves arise as the fixed point locus of certain symmetry group action on the generalized complete intersection calabi yau manifolds we also make a blowing up along curves by which one can generate new calabi yau manifolds moreover an approach on spectral sequences is used to compute hodge numbers of generalized complete intersection calabi yau manifolds and the genus of curves therein

extremal bundles on calabi yau threefolds we study constructions of stable holomorphic vector bundles on calabi yau threefolds especially those with exact anomaly cancellation which we call extremal by going through the known databases we find that such examples are rare in general and can be ruled out for the spectral cover construction for all elliptic threefolds we then introduce a general hartshorne serre construction and use it to find extremal bundles of general ranks and study their stability as well as computing their chern numbers based on both existing and our new constructions we revisit the dry conjecture for the existence of stable sheaves on calabi threefolds and provide theoretical and numerical evidence for its correctness our construction can be easily generalized to bundles with no extremal conditions imposed

comparison of several generating partitions of the henon map the existence and uniqueness of generating partitions in non hyperbolic dynamical systems is usually studied in a simple exemplary system namely the henon map at standard parameter values a 1 4 and b 0 3 we compare its standard partition with three other binary partitions which are quite different from the standard partition but also appear to be generating one of these partitions passes twice through the same orbit of homoclinic tangencies providing a counterexample to a recent conjecture by jaeger and kantz j phys a math gen 30 l567 introducing some simple rules to manipulate symbolic sequences we show how to translate symbolic sequences produced by one partition into sequences produced by the other partitions this proves that all these partitions are as good approximations to generating partitions as the standard partition we also construct an infinite number of binary partitions which are all quite similar to the standard partition derive their translation rules and prove the same equivalence it is not known for sure whether any of these partitions is indeed generating but if one of them is generating then they all are

soliton surfaces associated with sigma models differential and algebraic aspects in this paper we consider both differential and algebraic properties of surfaces associated with sigma models it is shown that surfaces defined by the generalized weierstrass formula for immersion for solutions of the sigma model with finite action defined in the riemann sphere are themselves solutions of the euler lagrange equations for sigma models on the other hand we show that the euler lagrange equations for surfaces immersed in the lie algebra with conformal coordinates that are extremals of the area functional subject to a fixed polynomial identity are exactly the euler lagrange equations for sigma models in addition to these differential constraints the algebraic constraints in the form of eigenvalues of the immersion functions are systematically treated the spectrum of the immersion functions for different dimensions of the model as well as its symmetry properties and its transformation under the action of the ladder operators are discussed another approach to the dynamics is given i e description in terms of the unitary matrix which diagonalizes both the immersion functions and the projectors constituting the model

compact self dual manifolds with positive scalar curvature on etudie l existence d une metrique auto duale a courbure scalaire positive sur une variete compacte simplement connexe qui n est pas conformement equivalente a ces metriques standard sur s 4 ou cp 2

a loop group method for affine harmonic maps into lie groups we generalize the uhlenbeck segal theory for harmonic maps into compact semi simple lie groups to general lie groups equipped with torsion free bi invariant connection

harmonic maps into lie groups classical solutions of the chiral model on considere deux aspects de la structure algebrique de 7b m l espace des applications harmoniques d un domaine a 2 dimensions simplement connexe dans un groupe de lie reel g r la forme reelle d un groupe complexe g on construit une representation du groupe boucle 7b a s 1 g r sur 7b m correspondant a l algebre de lie de kacmoody des deformations infinitesimales on decrit l action sur 7b m et l action d un sous groupe sur l espace des applications harmoniques dans des gaussmanniens dans une seconde partie on se restreint a une theorie qui s applique seulement quand est un domaine riemannien a 2 dimensions simplement connexe et le groupe est g r u n

stability of special instanton bundles on ℙ 2n 1 we prove that the special instanton bundles of rank 2n on p2n 1 c with a symplectic structure studied by spindler and trautmann are stable in the sense of mumford takemoto this implies that the generic special instanton bundle is stable moreover all instanton bundles on p5 are stable we get also the stability of other related vector bundles

structure of generating partitions for two dimensional maps we discuss the non uniqueness of the generating partitions of two dimensional maps and we give new explanations for ambiguities that are to be expected in the construction of generating partitions if a parameter is varied both points are discussed with the help of the connecting lines of homoclinic tangencies in the entire phase space i e also outside the attractor that can be calculated very easily with a new algorithm introduced recently

su n and so n invariant chiral fields one and two dimensional subspaces applying the harmonic map ansatz we find new classes of su n and so n chiral fields we write the corresponding one dimensional subgroups of su n and so n chiral fields in terms of one and two harmonic maps and the corresponding two dimensional subgroups in terms of su 2 and so 3 chiral fields in other words we reduce the su n and so n chiral equations to harmonic maps in one and two dimensional spaces

homoclinic tangencies and non normal jacobians effects of noise in nonhyperbolic chaotic systems abstract a general study of the effects of noise interacting with the deterministic dynamics of nonhyperbolic chaotic 2 d maps and 3 flows is presented noise in these systems can have dramatic effects on the invariant measure of the system in regions of homoclinic tangencies perturbations are transported away from the neighborhood of the attractor leading to deformations which can be one to two orders of magnitude larger than the noise level a qualitative understanding is attained by a study of the universal properties of the unstable and stable foliation of the phase space in the vicinity of homoclinic tangencies through the investigation of nontrivial structures of the tangent space at homoclinic tangencies related to the non normal jacobians we obtain a quantitative description of these noise induced deformations of the attractor local expansion rates which are much larger than the maximal lyapunov exponent can be used as a measure of the system s structural instability under perturbations we exemplify our general results on the henon ikeda and duffing system a new and effective algorithm to calculate the homoclinic tangencies in the entire phase space based on the results of this paper is presented

on the topology of the henon map topological invariants of the henon map are investigated by means of the pruning front first a long sequence of primary homoclinic tangencies is computed confirming the monotonicity of the front an algorithm to extract forbidden sequences is then introduced and discussed forbidden sequences of increasing lengths are used to construct a hierarchy of regular grammars represented by directed graphs which approximate the exact grammar arbitrarily well the topological entropy is estimated as the largest eigenvalue of their adjacency matrix it exhibits an exponential convergence towards the asymptotic value with an exponent in agreement with a previous conjecture based on the growth rate of the number of forbidden words

minimal topological chaos coexisting with a finite set of homoclinic and periodic orbits abstract in this note we explain how to find the minimal topological chaos relative to finite set of homoclinic and periodic orbits the main tool is the pruning method which is used for finding a hyperbolic map obtained uncrossing pieces of the invariant manifolds whose basic set contains all orbits forced by the finite set under consideration then we will show applications related to transport phenomena and to the problem of determining the orbits structure coexisting with a finite number of periodic orbits arising from the bouncing ball model

from template analysis to generating partitions i periodic orbits knots and symbolic encodings abstract we present a detailed algorithm to construct symbolic encodings for chaotic attractors of three dimensional flows it is based on a topological analysis of unstable periodic orbits embedded in the attractor and follows the approach proposed by lefranc et al phys rev lett 73 1994 1364 for each orbit the symbolic names that are consistent with its knot theoretic invariants and with the topological structure of the attractor are first obtained using template analysis this information and the locations of the periodic orbits in the section plane are then used to construct a generating partition by means of triangulations we provide numerical evidence of the validity of this method by applying it successfully to sets of more than 1500 periodic orbits extracted from numerical simulations and obtain partitions whose border is localized with a precision of 0 01 a distinctive advantage of this approach is that the solution is progressively refined using higher period orbits which makes it robust to noise and suitable for analyzing experimental time series furthermore the resulting encodings are by construction consistent in the corresponding limits with those rigorously known for both one dimensional and hyperbolic maps

twistors kahler manifolds and bimeromorphic geometry i by considering deformations of the moishezon twistor spaces of cp2 cp2 constructed in 20 we showthat theblowupof c2 at n points in general position admits an asymptotically flat scalar flat kahler metric in each kahler class at least provided that the given points are nearly collinear department of mathematics suny at stony brook stony brook new york 11794 e mail address claude math sunysb edu this content downloaded from 157 55 39 153 on mon 19 sep 2016 04 58 00 utc all use subject to http about jstor org terms

a construction of non kahler calabi yau manifolds and new solutions to the strominger system abstract we propose a new construction of compact non kahler calabi yau manifolds with balanced metrics and study the strominger system on them in particular we obtain explicit solutions to the strominger system with degeneracies on g t 4 where g is an immersed minimal surface of genus g 3 in flat t 3

heterotic string compactification and new vector bundles we propose a construction of kahler and non kahler calabi yau manifolds by branched double covers of twistor spaces in this construction we use the twistor spaces of four manifolds with self dual conformal structures with the examples of connected sum of n mathbb p 2 s we also construct k3 fibered calabi yau manifolds from the branched double covers of the blow ups of the twistor spaces these manifolds can be used in heterotic string compactifications to four dimensions we also construct stable and polystable vector bundles some classes of these vector bundles can give rise to supersymmetric grand unified models with three generations of quarks and leptons in four dimensions

equivariant connected sums of compact self dual manifolds due to a recent theorem of taubes self dual conformal structures on compact four dimensional manifolds exist in abundance 43 therefore finding examples in general is no longer an issue in this subject to deepen our understanding of self dual conformal geometry we take the path of trying to understand the possible symmetry groups of this geometry i e the groups of orientation preserving conformal transformations the first systematic approach which constructs a large collection of compact self dual manifolds with a non trivial group of symmetries is lebrun s hyperbolic ansatz 22 the condition of this ansatz is that the symmetry group contains an s l u 1 which acts on the manifold semi freely meaning that the isotropy group at any point of the manifold is either trivial or the entire group s i with this condition on the group action and some favourable topological conditions on the underlying manifold the geometrical equations of self duality on the given four dimensional manifold are reduced to the s 1 monopole equations on hyperbolic 3 space with this setup lebrun explicitly constructed a large number of compact self dual manifolds with semi free s l symmetry including the fubinistudy metric on the complex projective plane ci 2 and a one parameter family of self dual metrics on c c first found in 38 we shall call the self dual metrics on nci 2 n 1 constructed by lebrun s hyperbolic ansatz the lebrun metrics after this very successful ansatz it is natural to try to find new examples which are beyond the scope of that method

on the geometric structure of non hyperbolic attractors the authors discuss the f alpha spectrum of non hyperbolic attractors of the henon type they elucidate the origin of the phase transition found in a previous paper and give a lower bound to the spectrum in the non hyperbolic phase where kaplan yorke type formulae no longer hold their results disagree with other recent attempts numerical simulations for the henon map agree with analytical estimates

vector bundles and f theory to understand in detail duality between heterotic string and f theory compactifications it is important to understand the construction of holomorphic g bundles over elliptic calabi yau manifolds for various groups g in this paper we develop techniques to describe these bundles and make several detailed comparisons between the heterotic string and f theory

the construction of harmonic maps into complex grassmannians on developpe une technique d analyse des applications harmoniques d une surface de riemann dans un grassmannien complexe en utilisant des diagrammes qui etablissent plusieurs resultats

universal encoding for unimodal maps the standard encoding procedure to describe the chaotic orbits of unimodal maps is accurately investigated we show that the grammatical rules of the underlying language can be easily classified in a compact form by means of a universal parameter two procedures to construct finite graphs which approximate non markovian cases are discussed showing also the intimate relation with the corresponding construction of approximate markov partitions the convergence of the partial estimates of the topological entropy is discussed proving that the error decreases exponentially with the length of the sequences considered the rate is shown to coincide with the topological entropyh itself finally a superconvergent method to estimateh is introduced

scaling laws for invariant measures on hyperbolic and nonhyperbolic atractors the analysis of dynamical systems in terms of spectra of singularities is extended to higher dimensions and to nonhyperbolic systems prominent roles in our approach are played by the generalized partial dimensions of the invariant measure and by the distribution of effective liapunov exponents for hyperbolic attractors the latter determines the metric entropies and provides one constraint on the partial dimensions for nonhyperbolic attractors there are important modifications we discuss them for the examples of the logistic and henon map we show in particular that the generalized dimensions have singularities with noncontinuous derivative similar to first order phase transitions in statistical mechanics

the cauchy problem of the ward equation with mixed scattering data we solve the cauchy problem of the ward equation with both continuous and discrete scattering data

estimating the information content of symbol sequences and efficient codes several variants of an algorithm for estimating shannon entropies of symbol sequences are presented they are all related to the lempel ziv algorithm 1976 1977 and to recent algorithms for estimating hausdorff dimensions the average storage and running times increase as n and nlog n respectively with the sequence length n these algorithms proceed basically by constructing efficient codes they seem to be the optimal algorithms for sequences with strong long range correlations e g natural languages an application to written english illustrates their use

explicit construction of harmonic two spheres into the complex grassmannian we present an explicit description of all harmonic maps of finite uniton number from a riemann surface into a complex grassmannian namely starting from a constant map q and a collection of meromorphic functions and their derivatives we show how to algebraically construct all harmonic maps from the two sphere into a given grassmannian g p mathbb c n in this setting the uniton number depends on q and p and we obtain a sharp estimate for it

anti self dual yang mills connections over complex algebraic surfaces and stable vector bundles on presente une correspondance entre la geometrie algebrique et la geometrie differentielle des fibres vectoriels soit une surface algebrique projective x qui a un plongement donne x cp n et soit une metrique de kahler sur x dont la classe de cohomologie associee est duale a la classe de section d hyperplan h un fibre sur x est stable par rapport au plongement projectif si et seulement si il admet une connexion irreductible d hermite einstein par rapport a la metrique cette connexion est alors unique

the symmetries of solitons in this article we will retrace one of the great mathematical adventures of this century the discovery of the soliton and the gradual explanation of its remarkable properties in terms of hidden symmetries we will take an historical approach starting with a famous numerical experiment carried out by fermi pasta and ulam on one of the first electronic computers and with zabusky and kruskal s insightful explanation of the surprising results of that experiment and of a follow up experiment of their own in terms of a new concept they called solitons solitons however raised even more questions than they answered in particular the evolution equations that govern solitons were found to be hamiltonian and have infinitely many conserved quantities pointing to the existence of many non obvious symmetries we will cover next the elegant approach to solitons in terms of the inverse scattering transform and lax pairs and finally explain how those ideas led step by step to the discovery that loop groups acting by dressing transformations give a conceptually satisfying explanation of the secret soliton symmetries

homoclinic tangencies generating partitions and curvature of invariant manifolds a method to compute the curvature of the unstable manifold is introduced and applied to henon map and duffing attractor showing that it allows the authors to locate the homoclinic tangencies and in turn to construct a generating partition the probability distribution of curvature values is investigated showing a power law decay finally the shape of the multifractal spectrum of effective liapunov exponents in nonhyperbolic systems is discussed

invariant recurrence relations for mathbb c p n 1 models in this paper we present invariant recurrence relations for the completely integrable euclidean sigma model in two dimensions defined on the riemann sphere s2 when its action functional is finite we determine the links between successive projection operators wavefunctions of the linear spectral problem and immersion functions of surfaces in the algebra together with the outlines of the proofs our formulation preserves the conformal and scaling invariance of these quantities certain geometrical aspects of these relations are described we also discuss the singularities of the meromorphic solutions of the model and show that they do not affect the invariant quantities we illustrate the construction procedure through the examples of the and models