stable bundles and integrable systems on considere la geometrie symplectique des fibres cotangents aux espaces de modules de fibres vectoriels stables sur une surface de riemann on montre que ce sont des systemes dynamiques hamiltoniens algebriquement completement integrables

the g generalized hitchin systems and prym varieties in this article we consider the generalized hitchin systems introduced by bottacin and markman for an arbitrary reductive complex group g they have the structure of lagrangian fibrations pr u by generalized prym varieties over sets u parametrizing families of weyl invariant curves the g generalized hitchin systems satisfy a rank two condition and one can find invariant varieties x which distinguish these systems it is then shown that there is a correspondence between these integrable systems and the varieties x k d h which are equipped with an appropriate two form with values in the cartan subalgebra h

a comparison of polynomial time completeness notions ladner lynch and selman 1975 showed the differences among the power of several types of polynomial time reductions for dext c 0dtime 2cn but questions concerning complete degrees w r t these reductions have remained open here we show in dext almost all the possible differences between the completeness notions w r t any pair of the following reductions pt ptt pp n tt pc pd pbtt pk tt pm differences between several types of truth table completenesses we introduce the notion f easy subset for each pr reduction we first prove the property that all pr complete sets for dext have f easy subsets where the class f depends on each reduction type r then the construction of a ps complete set which is not pr complete becomes a simple diagonalization argument these constructions also work for all deterministic superpolynomial complexity classes which have a pm complete set in order to construct a set which is pt complete for dext but not ptt complete for dext we use strings which are not polynomially producible from small sized inputs these strings are said to have large generalized kolmogorov complexity cf hartmanis 1983

separation of np completeness notions we use hypotheses of structural complexity theory to separate various np completeness notions in particular we introduce an hypothesis from which we describe a set in np that is mbox leq rm p rm t complete but not mbox leq rm p tt complete we provide fairly thorough analyses of the hypotheses that we introduce

hilbert schemes separated variables and d branes we explain sklyanin s separation of variables in geometrical terms and construct it for hitchin and mukai integrable systems we construct hilbert schemes of points on t for c c or elliptic curve and on c 2 and show that their complex deformations are integrable systems of calogero sutherland moser type we present the hyperkahler quotient constructions for hilbert schemes of points on cotangent bundles to the higher genus curves utilizing the results of hurtubise kronheimer and nakajima finally we discuss the connections to physics of d branes and string duality

parabolic higgs bundles and representations of the fundamental group of a punctured surface into a real group abstract we study parabolic g higgs bundles over a compact riemann surface with fixed punctures when g is a real reductive lie group and establish a correspondence between these objects and representations of the fundamental group of the punctured surface in g with arbitrary holonomy around the punctures this generalizes simpson s results for gl n c to arbitrary complex and real reductive lie groups three interesting features are the relation between the parabolic degree and the tits geometry of the boundary at infinity of the symmetric space the treatment of the case when the logarithm of the monodromy is on the boundary of a weyl alcove and the correspondence of the orbits encoding the singularity via the kostant sekiguchi correspondence we also describe some special features of the moduli spaces when g is a split real form or a group of hermitian type

symplectic structures of moduli space of higgs bundles over a curve and hilbert scheme of points on the canonical bundle the moduli space of triples of the form e s are considered where e is a higgs bundle on a fixed riemann surface x and s is a nonzero holomorphic section of e such a moduli space admits a natural map to the moduli space of higgs bundles simply by forgetting s if y l is the spectral data for the higgs bundle e then s defines a section of the line bundle l over y the divisor of this section gives a point of a hilbert scheme parametrizing 0 dimensional subschemes of the total space of the canonical bundle k x since y is a curve on k x the main result says that the pullback of the symplectic form on the moduli space of higgs bundles to the moduli space of triples coincides with the pullback of the natural symplectic form on the hilbert scheme using the map that sends any triple e s to the divisor of the corresponding section of the line bundle on the spectral curve

the limit of the hermitian yang mills flow on reflexive sheaves abstract in this paper we study the asymptotic behavior of the hermitian yang mills flow on a reflexive sheaf we prove that the limiting reflexive sheaf is isomorphic to the double dual of the graded sheaf associated to the harder narasimhan seshadri filtration this answers a question by bando and siu

on cyclic higgs bundles in this paper we derive a maximum principle for a type of elliptic systems and apply it to analyze the hitchin equation for cyclic higgs bundles we show several domination results on the pullback metric of the possibly branched minimal immersion f associated to cyclic higgs bundles also we obtain a lower and upper bound of the extrinsic curvature of the image of f as an application we give a complete picture for maximal sp 4 mathbb r representations in the 2g 3 gothen components and the hitchin components

physical constraints on hypercomputation many attempts to transcend the fundamental limitations to computability implied by the halting problem for turing machines depend on the use of forms of hypercomputation that draw on notions of infinite or continuous as opposed to bounded or discrete computation thus such schemes may include the deployment of actualised rather than potential infinities of physical resources or of physical representations of real numbers to arbitrary precision here we argue that such bases for hypercomputation are not materially realisable and so cannot constitute new forms of effective calculability

the nature of the extended analog computer abstract this paper defines the relationship between rubel s extended analog computer eac model and indiana university s implementation of it a fundamental distinction that is necessary to understand the design implementation operation and applications of the eac beginning with the detailed structure of the current eac its method of computing is illustrated with diagrams and three applications next the digraph a labeled directed graph is introduced to show how unconventional and conventional computers relate nature mathematics and computer architecture the digraph defines the paradigms of analogy and algorithm illustrates how applications for the eac are analogies developed by choosing the semantics for a machine configuration and suggests how partial differential equations might be compiled to eac configurations a difficult problem compared to compiling a digital computer program and one that is still not solved the digraph also distinguishes between the eac s explicit functions whose operation is defined in the structure of its components and implicit functions whose operation is inherent in the properties of matter and energy brief case studies show how industry and academia are adopting the eac the paper closes with topics and open questions for future research

computability of analog networks we define a general concept of a network of analog modules connected by channels processing data from a metric space a and operating with respect to a global continuous clock t the inputs and outputs of the network are continuous streams u t a and the input output behaviour of the network with system parameters from a is modelled by a function f a rxc t a p c t a q p q 0 r 0 where c t a is the set of all continuous streams equipped with the compact open topology we give an equational specification of the network and a semantics which involves solving a fixed point equation over c t a using a contraction principle based on the fact that c t a can be approximated locally by metric spaces we show that if the module functions are continuous then so is the network function f we analyse in detail two case studies involving mechanical systems finally we introduce a custom made concrete computation theory over c t a and show that if the module functions are concretely computable then so is f

concrete models of computation for topological algebras a concrete model of computation for a topological algebra is based on a representation of the algebra made from functions on the natural numbers the functions computable in a concrete model are computable in the representation in the classical sense of the chruch turing thesis moreover the functions turn out to be continuous in the topology of the algebra in this paper we consider different concrete models for computing in topological algebras and prove their mutual equivalence in certain commonly occurring circumstances for topological algebras the concrete models we use are effective representation by algebraic domains stoltenberg hansen and tucker effective representation by continuous domains edelat effective representation by type two recursion on baire space weihrauch and for metric and normed algebras we use effective metric spaces moschovakis and computability structures pour el and richards the result are evidence that computability theory for topological algebras is a stable theory independent of the specific models of computation just as classical computability theory for discrete algebras is stable

recursion theory on the reals and continuous time computation abstract we define a class of recursive functions on the reals analogous to the classical recursive functions on the natural numbers corresponding to a conceptual analog computer that operates in continuous time this class turns out to be surprisingly large and includes many functions which are uncomputable in the traditional sense we stratify this class of functions into a hierarchy according to the number of uses of the zero finding operator at the lowest level are continuous functions that are differentially algebraic and computable by shannon s general purpose analog computer at higher levels are increasingly discontinuous and complex functions we relate this hierarchy to the arithmetical and analytical hierarchies of classical recursion theory

complete problems and strong polynomial reducibilities a set a is m reducible to a set b if and only if there is a polynomial time computable function f such that for all x x in a leftrightarrow f x in b a set c is m complete for a class s if c in s and all sets in s are m reducible to c one reducibility and one completeness can be defined by requiring f to be one one two sets a and b are p isomorphic if the function f can be taken one to one onto and polynomially invertible in this paper it is shown that all the m complete sets are one one complete for operatorname dtime 2 mathcal o n operatorname ntime 2 mathcal o n and the class of recursively enumerable sets further all the sets complete for operatorname ntime 2 mathcal o n under 1 l or two way dfa reductions are p isomorphic all the m complete sets for operatorname dtime 2 mathcal o n are p isomorphic if and only if all the m complete sets for dtime 2 text poly are p isomorphic

path queries on functions abstract let f 1 n 1 n be a function and l 1 n 1 indicate a label assigned to each element of the domain we design several compact data structures that answer various kinds of summary queries on the labels of paths in f for example we can find either the minimum label in f k i for a given i and any k 0 in a given range k 1 k 2 or the minimum label in f k i for a given i and k 0 using n lg n n lg o n lg n bits and time o n the inverse ackermann function within similar space we can count in time o lg n lg lg n the number of labels within a range and report each element with such labels in o lg n lg lg n additional time several other tradeoffs and possible queries are considered such as selection top r queries and majorities finally we consider queries that allow us navigate on the graph of the function such as the nearest common successor of two elements or the nearest successor or predecessor of an element within a range of labels

a kobayashi hitchin correspondence for i holomorphic bundles abstract in this paper we introduce the notions of hermitian einstein metric and stability for i holomorphic vector bundles on bi hermitian manifolds moreover we establish a kobayashi hitchin correspondence for i holomorphic vector bundles on bi hermitian manifolds examples of such vector bundles include generalized holomorphic bundles on generalized kahler manifolds we also show that stability of a vector bundle in this sense can depend on the parameter

comparing reductions to np complete sets under the assumption that np does not have p measure 0 we investigate reductions to np complete sets and prove the following 1 adaptive reductions are more powerful than nonadaptive reductions there is a problem that is turing complete for np but not truth table complete 2 strong nondeterministic reductions are more powerful than deterministic reductions there is a problem that is snp complete for np but not turing complete 3 every problem that is many one complete for np is complete under length increasing reductions that are computed by polynomial size circuits the first item solves one of lutz and mayordomo s twelve problems in resource bounded measure 1999 we also show that every many one complete problem for ne is complete under one to one length increasing reductions that are computed by polynomial size circuits

the self duality equations on a riemann surface in this paper we shall study a special class of solutions of the self dual yang mills equations the original self duality equations which arose in mathematical physics were defined on euclidean 4 space the physically relevant solutions were the ones with finite action the so called instantons the same equations may be dimensionally reduced to euclidean 3 space by imposing invariance under translation in one direction these equations also have physical relevance the solutions which have finite action in three dimensions are the magnetic monopoles if we take the reduction process one step further and consider solutions which are invariant under two translations we obtain a set of equations in the plane here however there is no clear physical meaning and indeed attempts to find finite action solutions have failed nevertheless these are the equations we shall consider despite the lack of interesting solutions in r2 the equations have the important property conformal invariance which allows them to be defined on manifolds modelled on r2 by conformal maps namely riemann surfaces we shall consider here solutions of the self duality equations defined on a compact riemann surface there are in fact solutions as we shall show and the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure which will be the focus of our study it brings together in a harmonious way the subjects of riemannian geometry topology algebraic geometry and symplectic geometry illuminating all these facets of the same object accounts for the length of this paper the self duality equations are equations from gauge theory geometrically they are defined in terms of connections on principal bundles while the group of the principal bundle may be chosen arbitrarily for the equations to make sense we restrict attention here to the simplest case of su 2 or so 3 there are two reasons for this the first and most obvious is that it simplifies calculations and avoids the use of inductive processes which are inherent in the consideration of a general lie group of higher rank the second reason is that solutions for su 2 have an intimate relationship with the internal structure of the riemann surface as a consequence of results we shall prove about solutions to the self duality equations we learn something about the moduli space of complex structures on the surface itself namely teichmuller space

the symplectic nature of fundamental groups of surfaces soit 7b g la categorie des groupes de lie g avec une forme bilineaire symetrique non singuliere b sur l algebre de lie soit le groupe fondamental d une surface orientee close de genre 1 et soit 7b s la categorie des out espaces symplectiques alors la correspondance g b hom g g b definit un foncteur 7b g 7b s ou b est la structure symplectique sur hom g g definie par b

conformal limits and the bialynicki birula stratification of the space of λ connections abstract the bialynicki birula decomposition of the space of connections restricts to the morse stratification on the moduli space of higgs bundles and to the partial oper stratification on the de rham moduli space of holomorphic connections for both the morse and partial oper stratifications every stratum is a holomorphic lagrangian fibration over a component of the space of complex variations of hodge structure in this paper we generalize known results for the hitchin section and the space of opers to arbitrary strata these include the following a biholomorphic identification of the fibers of the two strata over a stable variation of hodge structure via the conformal limit of gaiotto a proof that the fibers of the morse and partial oper stratifications are transverse at the base point and an explicit parametrization of the fibers as half dimensional affine spaces

continuity of operators on continuous and discrete time streams abstract we consider the semantics of networks processing streams of data from a complete metric space we consider two types of data streams those based on continuous time used in networks of physical components and analog devices and those based on discrete time used in concurrent algorithms the networks are both governed by global clocks and together model a huge range of systems previously we have investigated these two types of networks separately here we combine their study in a unified theory of stream transformers given as fixed points of equations we begin to develop this theory by using the standard mathematical techniques of topology to prove certain computationally desirable properties of these semantic functions notably continuity which is significant for models of a physical system according to hadamard s principle

automata over continuous time the principal objective of this paper is to lift basic concepts of the classical automata theory from discrete to continuous real time it is argued that the set of finite memory retrospective functions is the set of functions realized by finite state devices we show that the finite memory retrospective functions are speed independent i e they are invariant under stretchings of the time axis therefore such functions cannot deal with metrical aspects of the reals we classify and analyze phenomena which appear at continuous time and are invisible at discrete time

gauge theories on four dimensional riemannian manifolds this paper develops the riemannian geometry of classical gauge theories yang mills fields coupled with scalar and spinor fields on compact four dimensional manifolds some important properties of these fields are derived from elliptic theory regularity an energy gap theorem the manifold structure of the configuration space and a bound for the supremum of the field in terms of the energy it is then shown that finite energy solutions of the coupled field equations cannot have isolated singularities this extends a theorem of k uhlenbeck

compressibility and resource bounded measure we give a new definition of resource bounded measure based on compressibility of infinite binary strings we prove that the new definition is equivalent to the one commonly used this new characterization offers us a different way to look at resource bounded measure shedding more light on the meaning of measure zero results and providing one more tool to prove such results the main contribution of the paper is the new definition and the proofs leading to the equivalence result we then show how this new characterization can be used to prove that the class of linear autoreducible sets has p measure 0 we also prove that the class of sets that are truth table reducible to a p selective set has p measure 0 and that the class of sets that turing reduce to a subpolynomial dense set has p measure 0 this strengthens various results

on γ reducibility versus polynomial time many one reducibility abstract we prove that each element of a class of functions denoted by npc t p whose graphs can be accepted in nondeterministic polynomial time can be evaluated in deterministic polynomial time if and only if reducibility is equivalent to polynomial time many one reducibility we also modify the proof technique used to obtain part of this result to obtain the stronger result that if every reduction can be replaced by a polynomial time turing reduction then every function in npc t p can be evaluated in deterministic polynomial time

ends of the moduli space of higgs bundles we associate to each stable higgs pair a 0 phi 0 on a compact riemann surface x a singular limiting configuration a infinity phi infinity assuming that det phi has only simple zeroes we then prove a desingularization theorem by constructing a family of solutions a t t phi t to hitchin s equations which converge to this limiting configuration as t infinity this provides a new proof via gluing methods for elements in the ends of the higgs bundle moduli space and identifies a dense open subset of the boundary of the compactification of this moduli space

inverting onto functions we look at the hypothesis that all honest onto polynomial time computable functions have a polynomial time computable inverse we show this hypothesis equivalent to several other complexity conjectures including in polynomial time one can find accepting paths of nondeterministic polynomial time turing machines that accept every total multivalued nondeterministic function has a polynomial time computable refinement in polynomial time one can compute satisfying assignments for any polynomial time computable set of satisfiable formulae in polynomial time one can convert the accepting computations of any nondeterministic turing machine that accepts sat to satisfying assignments we compare these hypotheses with several other important complexity statements we also examine the complexity of these statements where we only require a single bit instead of the entire inverse

multi hamiltonian structures for r matrix systems for the rational elliptic and trigonometric r matrices we exhibit the links between three levels of poisson spaces a some finite dimensional spaces of matrix valued meromorphic functions on the complex line b spaces of spectral curves and sheaves supported on them and c symmetric products of a surface we have at each level a linear space of compatible poisson structures and the maps relating the levels are poisson this leads in a natural way to the nijenhuis coordinates for these spaces at level b there are hamiltonian systems on these spaces which are integrable for each poisson structure in the family and which are such that the lagrangian leaves are the intersections of the symplectic leaves over the poisson structures in the family specific examples include many of the well known integrable systems

bihermitian surfaces with odd first betti number compact bihermitian surfaces are considered that is compact oriented conformal four manifolds admitting two distinct compatible complex structures it is shown that if the first betti number is odd then with respect to either complex structure such a manifold belongs to class vii of the enriques kodaira classification moreover it must be either a special hopf or an inoue surface in the strongly bihermitian case or obtained by blowing up a minimal class vii surface with curves in the non strongly bihermitian case

surface group representations and u p q higgs bundles using the l2 norm of the higgs field as a morse function we study the moduli spaces of u p q higgs bundles over a riemann surface we require that the genus of the surface be at least two but place no constraints on p q a key step is the identification of the function s local minima as moduli spaces of holomorphic triples in a companion paper 7 we prove that these moduli spaces of triples are nonempty and irreducible because of the relation between flat bundles and fundamental group representations we can interpret our conclusions as results about the number of connected components in the moduli space of semisimple pu p q representations the topological invariants of the flat bundles are used to label subspaces these invariants are bounded by a milnor wood type inequality for each allowed value of the invariants satisfying a certain coprimality condition we prove that the corresponding subspace is nonempty and connected if the coprimality condition does not hold our results apply to the closure of the moduli space of irreducible representations 1 members of vbac vector bundles on algebraic curves which is partially supported by eager ec fp5 contract no hprn ct 2000 00099 and by edge ec fp5 contract no hprn ct 2000 00101 2 partially supported by the national science foundation under grant dms0072073 3 partially supported by the ministerio de ciencia y tecnolo ia spain under grant bfm2000 0024 4 partially supported by the fundacao para a ciencia e a tecnologia portugal through the centro de matematica da universidade do porto and through grant no sfrh bpd 1606 2000 5 partially supported by the portugal spain bilateral programme acciones integradas grant nos hp2000 0015 and ai 01 24 6 partially supported by a british epsrc grant october december 2001 received 11 27 2002

unifying computers and dynamical systems using the theory of synchronous concurrent algorithms a synchronous concurrent algorithm sca is a parallel deterministic algorithm based on a network of modules and channels computing and communicating data in parallel and synchronised by a global clock with discrete time many types of algorithms computer architectures and mathematical models of physical and biological systems are examples of scas for example conventional digital hardware is made from components that are scas and many computational models possess the essential features of scas including systolic arrays neural networks cellular automata and coupled map lattices in this paper we formalise the general concept of an sca equipped with a global clock in order to analyse precisely i specifications of their spatio temporal behaviour and ii the senses in which the algorithms are correct we start the mathematical study of sca computation specification and correctness using methods based on computation on many sorted topological algebras and equational logic we show that specifications can be given equationally and hence that the correctness of scas can be reduced to the validity of equations in certain computable algebras since the idea of an sca is general our methods and results apply to each of the particular classes of algorithms and dynamical systems above

separating coordinates for the generalized hitchin systems and the classical r matrices we exhibit natural darboux coordinates for the generalized hitchin systems studied by bottacin and markman these systems are defined on spaces of stable pairs consisting of a vector bundle and a form valued meromorphic endomorphism of the bundle in special cases genus zero genus one the bundles are rigid and one has the rational trigonometric and elliptic gaudin systems explicit formulae are given in these cases

hyperkahler metrics and supersymmetry we describe two constructions of hyperkahler manifolds one based on a legendre transform and one on a sympletic quotient these constructions arose in the context of supersymmetric nonlinear models but can be described entirely geometrically in this general setting we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience

monopoles and geodesics using the holomorphic geometry of the space of straight lines in euclidean 3 space it is shown that every static monopole of chargek may be constructed canonically from an algebraic curve by means of the atiyah ward ansatz a k

connections with l p bounds on curvature we show by means of the implicit function theorem that coulomb gauges exist for fields over a ball inr n when the integrall n 2 field norm is sufficiently small we then are able to prove a weak compactness theorem for fields on compact manifolds withl p integral norms bounded p n 2

resource bounded randomness and weakly complete problems we introduce and study resource bounded random sets based on lutz s concept of resource bounded measure 5 6 we concentrate on nc randomness c 2 which corresponds to the polynomial time bounded p measure of lutz and which is adequate for studying the internal and quantative structure of e dtime 2lin first we show that the class of nc random sets has p measure 1 this provides a new simplified approach to p measure 1 results next we compare randomness with genericity in the sense of 1 2 and we show that nc 1 random sets are nc generic whereas the converse fails from the former we conclude thatnc random sets are not p btt complete for e our technical main results describe the distribution of the nc random sets under p m reducibility we show that every nc random set in e has nk random predecessors in e for any k 1 whereas the amount of randomness of the successors is bounded we apply this result to answer a question raised by lutz 8 we show that the class of weakly complete sets has measure 1 in e and that there are weakly complete problems which are not p btt complete for e

bi immunity separates strong np completeness notions we prove that if for some e 0 np contains a set that is dtime 2ne bi immune then np contains a set that is 2 turing complete for np hence 3 truth table complete but not 1 truth table complete for np thus this hypothesis implies a strong separation of completeness notions for np lutz and mayordomo theor comput sci 164 1996 141 163 and ambos spies and bentzien j comput syst sci 61 3 2000 335 361 previously obtained the same consequence using strong hypotheses involving resource bounded measure and or category theory our hypothesis is weaker and involves no assumptions about stochastic properties of np

cook versus karp levin separating completeness notions if np is not small under the hypothesis that np does not have p measure 0 roughly that np contains more than a negligible subset of exponential time it is shown that there is a language that is t p complete cook complete but not m p complete karp levin complete for np this conclusion widely believed to be true is not known to follow from p np or other traditional complexity theoretic hypotheses

twisted harmonic maps and the self duality equations si m est une surface de riemann compacte on utilise des resultats de la theorie des applications harmoniques pour trouver une solution aux equations d auto dualite associees a une representation irreductible p 1 m psl 2 c