a tqft from quantum teichmuller theory by using quantum teichmuller theory we construct a one parameter family of tqfts on the categroid of admissible leveled shaped 3 manifolds

invariants of 3 manifolds via link polynomials and quantum groups the aim of this paper is to construct new topological invariants of compact oriented 3 manifolds and of framed links in such manifolds our invariant of a link in a closed oriented 3 manifold is a sequence of complex numbers parametrized by complex roots of 1 for a framed link in s 3 the terms of the sequence are equale to the values of the suitably parametrized jones polynomial of the link in the corresponding roots of 1 in the case of manifolds with boundary our invariant is a sequence of finite dimensional complex linear operators this produces from each root of unity q a 3 dimensional topological quantum field theory

a local index theorem for families of overline partial operators on punctured riemann surfaces and a new kahler metric on their moduli spaces we prove a local index theorem for families of bar partial operators on riemann surfaces of type g n i e of genusg withn 0 punctures we calculate the first chern form of the determinant line bundle on the teichmuller spacetg n endowed with quillen s metric where the role of the determinant of the laplace operators is played by the values of the selberg zeta function at integer points the result differs from the case of compact riemann surfaces by an additional term which turns out to be the kahler form of a new kahler metric on the moduli space of punctured riemann surfaces as a corollary of this result we derive for instance an analog of mumford s isomorphism in the case of the universal curve

multiple solutions for asymptotically linear elliptic systems in this article we consider cooperative and noncooperative elliptic systems that are asymptotically linear at infinity we obtain infinitely many solutions with small energy if the potential is even if the noncooperative system is resonant both at zero and at infinity then the number of nontrivial solutions depends on the dimension of the eigenspaces between resonant values

solutions of systems of elliptic differential equations on circular domains abstract we study global bifurcation of weak solutions of systems of elliptic differential equations considered on so 2 invariant domains we formulate sufficient conditions for the existence of unbounded continua of nontrivial solutions branching from the trivial ones as the main tool we use the degree for so 2 equivariant gradient maps defined by the second author in rybicki nonlinear anal tma 23 1 1994 83

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

existence results for the positive solutions of nonlinear elliptic systems in this paper we discuss the existence of solutions for nonlinear elliptic systems the positive weak solution is sought by the leray schauder degree

a mathematical construction of the non abelian chern simons functional integral we construct rigorously an infinite dimensional distribution which corresponds to the chern simons cs functional integral associated with a principal fiber bundle over r 3 with structure group a compact connected lie group we determine the moments of the cs distribution and show that these coincide with those used in informal studies of the cs integral a locality property of the cs distribution is proven the complexified theory of frohlich and king is also discussed within our framework

an optimal partition problem related to nonlinear eigenvalues we extend to the case of many competing densities the results of the paper ann inst h poincare 6 2002 more precisely we are concerned with an optimal partition problem in n dimensional domains related to the method of nonlinear eigenvalues introduced by nehari acta math 105 1961 we prove existence of the minimal partition and some extremality conditions moreover in the case of two dimensional domains we give an asymptotic formula near the multiple intersection points finally we show some connections between the variational problem and the behavior of competing species systems with large interaction

on complete manifolds of nonnegative kth ricci curvature in this paper we establish some vanishing and finiteness theorems for the topological type of complete open riemannian manifolds under certain positivity conditions for curvature key tools are comparison techniques and morse theory of busemann and distance functions

a construction of pseudo anosov homeomorphisms we describe a generalization of thurston s original construction of pseudo anosov maps on a surface f of negative euler characteristic in fact we construct whole semigroups of pseudo anosov maps by taking appropriate compositions of dehn twists along certain families of curves our arguments furthermore apply to give examples of pseudo anosov maps on nonorientable surfaces for each self map f f f arising from our recipe we construct an invariant bigon track a slight generalization of train track whose incidence matrix is perron frobenius standard arguments produce a projective measured foliation invariant by f to finally prove that f is pseudo anosov we directly produce a transverse invariant projective measured foliation using tangential measures on bigon tracks as a consequence of our argument we derive a simple criterion for a surface automorphism to be pseudo anosov introduction a homeomorphism p of a surface f is said to be pseudo anoosov if no iterate of p fixes any essential nonboundaryor puncture parallel free homotopy class of simple curves in f examples of these homeomorphisms date back to the work of nielsen see n and gi but a systematic study of these maps was not undertaken until the work of thurston t1 anosov a studied maps of the torus which preserve two foliations of the torus by lines of irrational slope and pseudo anosov maps on f similarly preserve a pair of foliations with singularities pseudo anosov maps are by no means special indeed the monodromy of any nontorus fibred knot which is not a satellite is pseudo anosov t4 note that being pseudo anosov is a conjugacy invariant moreover these maps play an important role in the geometrization of three manifolds indeed a mapping torus has hyperbolic structure if and only if the monodromy is pseudo anosov t4 in the original preprint t1 there is described a construction of pseudo anosov maps which we will recall later in this paper we generalize thurston s construction and give a recipe for constructing whole semigroups of pseudo anosov maps many of which do not arise from thurston s construction our recipe is also applicable to nonorientable surfaces and we give examples of pseudo anosov maps in this setting in t3 thurston proved the existence of such and ay gave the first explicit examples this paper is organized as follows in 1 we review the basic terminology and results on train tracks in surfaces and indicate the connection between measured train tracks and measured foliations 2 is devoted to tangential measure on bigon received by the editors march 7 1986 and in revised form june 17 1987 1980 mathematics subject cassification 1985 revision primary 57n06 57n50

ergodic theory on moduli spaces let m be a compact surface with x m 0 and let g be a compact lie group whose levi factor is a product of groups locally isomorphic to su 2 for example su 2 itself then the mapping class group rm of m acts on the moduli space x m of flat g bundles over m possibly twisted by a fixed central element of g when m is closed then fm preserves a symplectic structure on x m which has finite total volume on m more generally the subspace of x m corresponding to flat bundles with fixed behavior over am carries a rm invariant symplectic structure the main result is that rm acts ergodically on x m with respect to the measure induced by the symplectic structure

on chern simons and wzw partition functions direct analysis of the path integral reduces partition functions in chern simons theory on a three manifold m with group g to partition functions in a wzw model of maps from a riemann surface to g in particular chern simons theory on s 3 s 1 b 3 and the solid torus correspond respectively to the wzw model of maps from s 2 to g the g g model for and witten s gauged wzw path integral ansatz for chern simons states using maps from s 2 and from the torus to g the reduction hinges on the characterization of the space of connections modulo those gauge transformations which are the identity at a point n as itself a principal fiber bundle with affine linear fiber

flat connections and geometric quantization using the space of holomorphic symmetric tensors on the moduli space of stable bundles over a riemann surface we construct a projectively flat connection on a vector bundle over teichmuller space the fibre of the vector bundle consists of the global sections of a power of the determinant bundle on the moduli space both dolbeault and cech techniques are used

the chern simons theory and knot polynomials the chern simons gauge theory is studied using a functional integral quantization this leads to a differential equation for expectations of wilson lines the solution of this differential equation is shown to be simply related to the two variable jones polynomial of the corresponding link in the case where the gauge group issu n a similar equation has been used before to get the jones polynomial from a braid representation of the link the main novelty of our approach is that we get the jones polynomial from a plat representation of the link

from simplicial chern simons theory to the shadow invariant ii this is the second of a series of papers in which we introduce and study a rigorous simplicial realization of the non abelian chern simons path integral for manifolds m of the form m s1 and arbitrary simply connected compact structure groups g more precisely we introduce for general links l in m a rigorous simplicial version wlorig l of the corresponding wilson loop observable wlo l in the so called torus gauge by blau and thompson nucl phys b 408 2 345 390 1993 for a simple class of links l we then evaluate wlorig l explicitly in a non perturbative way finding agreement with turaev s shadow invariant l

asymptotic estimates for the spatial segregation of competitive systems abstract for a class of population models of competitive type we study the asymptotic behavior of the positive solutions as the competition rate tends to infinity we show that the limiting problem is a remarkable system of differential inequalities which defines the functional class s in 2 by exploiting the regularity theory recently developed in conti et al indiana univ math j to appear for the elements of functional classes of the form s we provide some qualitative and regularity property of the limiting configurations besides for the case of two competing species we obtain a full description of the limiting states and we prove some quantitative estimates for the rate of convergence finally we prove some new liouville type results which allow to have uniform regularity estimates of the solutions

combinatorial expression for universal vassiliev link invariant a general model similar to r matrix type models for link invariants is constructed it contains all r matrix invariants and is a generating function for universal vassiliev link invariants this expression is simpler than kontsevich s expression for the same quantity because it is defined combinatorially and does not contain any integrals except for an expression for the universal drinfeld s associator

hyperasymptotics for integrals with saddles integrals involving exp kf z where k is a large parameter and the contour passes through a saddle of f z are approximated by refining the method of steepest descent to include exponentially small contributions from the other saddles through which the contour does not pass these contributions are responsible for the divergence of the asymptotic expansion generated by the method of steepest descent the refinement is achieved by means of an exact resurgence relation expressing the original integral as its truncated saddle point asymptotic expansion plus a remainder involving the integrals through certain adjacent saddles determined by a topological rule iteration of the resurgence relation and choice of truncation near the least term of the original series leads to a representation of the integral as a sum of contributions associated with multiple scattering paths among the saddles no resummation of divergent series is involved each path gives a hyperseries depending on the terms in the asymptotic expansions for each saddle these depend on the particular integral being studied and so are non universal and certain hyperterminant functions defined by integrals these are always the same and hence universal successive hyperseries get shorter so the scheme naturally halts for two saddles the ultimate error is approximately varepsilon 2 386 where varepsilon proportional to exp a k where a is a positive constant is the error in optimal truncation of the original series as a numerical example an integral with three saddles is computed hyperasymptotically

on diagonalization in map m g motivated by some questions in the path integral approach to topological gauge theories we are led to address the following question given a smooth map from a manifoldm to a compact groupg is it possible to smoothly diagonalize it i e conjugate it into a map to a maximal torust ofg

multiplicity and concentration of solutions for elliptic systems with vanishing potentials abstract in this article we use variational methods to study a strongly coupled elliptic system depending on a positive parameter we suppose that the potentials are nonnegative and the intersection of the sets where they vanish has positive measure a technical condition imposed on the product of the potentials allows us to consider a setting where we do not assume any positive lower bound for the potentials considering the associated functional defined on an appropriated subspace of d 1 2 r n d 1 2 r n we are able to establish results on the existence and multiplicity of solutions for the system when the parameter is sufficiently large we also study the asymptotic behavior of these solutions when

bifurcation for an elliptic system coupled in the linear part we study linearly coupled elliptic systems without any assumption on the signs of the coefficients of the linear part we derive first the leray schauder degree of the associated linear system then we obtain sufficient conditions on the nonlinear terms for having bifurcation to coexistence states our results include the sublinear case keys words linearly coupled elliptic systems bifurcation to positive solutions coexistence states leray schauder degree email jfleck cict fr rpardo sunma4 mat ucm es

classical bv theories on manifolds with boundary in this paper we extend the classical bv framework to gauge theories on spacetime manifolds with boundary in particular we connect the bv construction in the bulk with the bfv construction on the boundary and we develop its extension to strata of higher codimension in the case of manifolds with corners we present several examples including electrodynamics yang mills theory and topological field theories coming from the aksz construction in particular the chern simons theory the bf theory and the poisson sigma model this paper is the first step towards developing the perturbative quantization of such theories on manifolds with boundary in a way consistent with gluing

on the existence of many solutions for a class of superlinear elliptic systems abstract we seek multiple solutions to systems of semilinear elliptic equations of the type u x h x u x x u x 0 x where r n n 1 is a bounded regular domain and u u1 u2 r 2 for a class of potentials where the variables are strongly coupled as in the model h x u1 u2 u1 1 u2 2 with i 1 we provide the existence of nine nontrivial solutions characterized by sign properties of each component

standing waves in the fitzhugh nagumo system and a problem in combinatorial geometry we show that there is a close relation between standing wave solutions for the fitzhugh nagumo system

construction of the witten reshetikhin turaev tqft from conformal field theory in andersen and ueno j knot theory ramif 16 127 202 2007 we constructed the vacua modular functor based on the sheaf of vacua theory developed in tsuchiya et al adv stud pure math 19 459 566 1989 and the abelian analog in andersen and ueno int j math 18 919 993 2007 we here provide an explicit isomorphism from the modular functor underlying the skein theoretic model for the witten reshetikhin turaev tqft due to blanchet habbeger masbaum and vogel to the vacua modular functor this thus provides a geometric construction of the tqft first proposed by witten and constructed first by reshetikhin turaev from the quantum group u q text sl n

existence results for some nonlinear elliptic systems abstract the paper is concerned with some nonlinear elliptic systems by using variational methods we obtain the existence of weak solutions for the semilinear elliptic system and the quasilinear elliptic system

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

toeplitz quantization of kahler manifolds ang gl n n limits for general compact kahler manifolds it is shown that both toeplitz quantization and geometric quantization lead to a well defined by operator norm estimates classical limit this generalizes earlier results of teh authors and klimek and lesniewski obtained for the tours and higher genus riemann surfaces respectively we thereby arrive at an approximation of the poisson algebra by a sequence of finitedimensional matrix algebrasgl n n

minimal coexistence configurations for multispecies systems abstract we deal with strongly competing multispecies systems of lotka volterra type with homogeneous neumann boundary conditions in dumbbell like domains under suitable non degeneracy assumptions we show that as the competition rate grows indefinitely the system reaches a state of coexistence of all the species in spatial segregation furthermore the limit configuration is a local minimizer for the associated free energy

examples of manifolds of positive ricci curvature on presente de nouveaux exemples de varietes de riemann simplement connexes de dimension 7 a courbure de ricci positive qui n admettent pas de metrique de courbure sectionnelle non negative dans les cas compacts et non compacts

vassiliev knot invariants and chern simons perturbation theory to all orders at any order the perturbative expansion of the expectation values of wilson lines in chern simons theory gives certain integral expressions we show that they all lead to knot invariants moreover these are finite type invariants whose order coincides with the order in the perturbative expansion together they combine to give a universal vassiliev invariant

on the coupled systems of second and fourth order elliptic equations in this paper by the variational method we discuss the existence and uniqueness of solutions for a coupled system of second and fourth order elliptic equations in the cases where the perturbation is zero sublinear and superlinear respectively

asymptotic faithfulness of the quantum su n representations of the mapping class groups we prove that the sequence of projective quantum su n representations of the mapping class group of a closed oriented surface obtained from the projective flat su n verlinde bundles over teichm uller space is asymptotically faithful that is the intersection over all levels of the kernels of these representations is trivial whenever the genus is at least 3 for the genus 2 case this intersection is exactly the order 2 subgroup generated by the hyper elliptic involution in the case of even degree and n 2 otherwise the intersection is also trivial in the genus 2 case

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

the witten reshetikhin turaev invariant for links in finite order mapping tori i abstract we state asymptotic expansion and growth rate conjectures for the witten reshetikhin turaev invariants of arbitrary framed links in 3 manifolds and we prove these conjectures for the natural links in mapping tori of finite order automorphisms of marked surfaces our approach is based upon geometric quantisation of the moduli space of parabolic bundles on the surface which we show coincides with the construction of the witten reshetikhin turaev invariants using conformal field theory as was recently completed by andersen and ueno

identification of berezin toeplitz deformation quantization we give a complete identification of the deformation quantization which was obtained from the berezin toeplitz quantization on an arbitrary compact kahler manifold the deformation quantization with the opposite star product proves to be a differential deformation quantization with separation of variables whose dassifying form is explicitly calculated its characteristic dass which dassifies star products up to equivalence is obtained the proof is based on the microlocal description of the szego kernel of a strictly pseudoconvex domain given by boutet de monvel and sjostrand

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

mountain pass solutions for the coupled systems of second and fourth order elliptic equations abstract in this paper by the mountain pass theorem we give an existence theorem of nontrivial solutions for a coupled system of second and fourth order elliptic equations

jones witten invariants for nonsimply connected lie groups and the geometry of the weyl alcove abstract the quotient process of muger and bruguieres is used to construct modular categories and tqfts out of closed subsets of the weyl alcove of a simple lie algebra in particular it is determined at which levels closed subsets associated to nonsimply connected groups lead to tqfts many of these tqfts are shown to decompose into a tensor product of tqfts coming from smaller subsets the prime subsets among these are classified and apart from some giving tqfts depending on homology as described by murakami ohtsuki and okada they are shown to be in one to one correspondence with the tqfts predicted by dijkgraaf and witten to be associated to chern simons theory with a nonsimply connected lie group thus in particular a rigorous construction of the dijkgraaf witten tqfts is given as a byproduct a purely quantum groups proof of the modularity of the full weyl alcove for arbitrary quantum groups at arbitrary levels is given

asymptotic properties of the quantum representations of the mapping class group we establish two results on the large level limit of projective quantum representations of surface mapping class groups obtained by quantizing moduli spaces of flat su n bundle with hitchin s connection first we prove that these projective representations lift to asymptotic representations second we show that under an infinitesimal rigidity assumption the characters of these representations have an asymptotic expansion the leading order term of this expansion agrees with the formula derived heuristically by e witten in quantum field theory and the jones polynomial

metrics of positive ricci curvature on quotient spaces one of the classical problems in differential geometry is the investigation of closed manifolds which admit riemannian metrics with given lower bounds for the sectional or the ricci curvature and the study of relations between the existence of such metrics and the topology and geometry of the underlying manifold despite many efforts during the past decades this problem is still far from being understood for example so far the only obstructions to the existence of a metric with positive ricci curvature come from the obstructions to the existence of metrics with positive scalar curvature li hi ro schy ta and the bonnet myers theorem which implies that the fundamental group of a closed manifold with positive ricci curvature must be finite fruitful constructions of metrics with positive ricci curvature on closed manifolds have so far been established by techniques that include deformation of metrics au eh we kahler geometry yau1 yau2 bundles and warping po na bb gpt special kinds of surgery sy wr metrical glueing gz2 and sasakian geometry bgn particularly large classes of examples of manifolds with positive ricci curvature are given by all compact homogeneous spaces with finite fundamental group na and all closed cohomogeneity one manifolds with finite fundamental group gz2 in this article we present several new classes of manifolds which admit metrics of positive or nonnegative ricci curvature the idea is to consider quotients of manifolds m g of positive or nonnegative ricci curvature by a free isometric action while taking such a quotient non decreases the sectional curvature it may well happen in general that the ricci curvature of the quotient is inferior to the one of m our first results however state that the quotient of m does admit metrics of positive ricci curvature if m belongs to one of the aforementioned classes of homogeneous spaces or spaces with a cohomogeneity one action more precisely we prove

global asymptotics for multiple integrals with boundaries under convenient geometric assumptions the saddle point method for multidimensional laplace integrals is extended to the case where the contours of integration have boundaries the asymptotics are studied in the case of nondegenerate and of degenerate isolated critical points the incidence of the stokes phenomenon is related to the monodromy of the homology via generalized picard lefschetz formulae and is quantified in terms of geometric indices of intersection exact remainder terms and the hyperasymptotics are then derived a direct consequence is a numerical algorithm to determine the stokes constants and indices of intersections examples are provided

on the self linking of knots this note describes a subcomplex f of the de rham complex of parametrized knot space which is combinatorial over a number of universal anomaly integrals the self linking integrals of guadaguini martellini and mintchev perturbative aspects of chern simons field theory phys lett b 227 111 1989 and bar natan perturbative aspects of the chern simons topological quantum field theory ph d thesis princeton university 1991 also on the vassiliev knot invariants to appear in topology are seen to represent the first nontrivial element in h0 f occurring at level 4 and are anomaly free however already at the next level an anomalous term is possible

hyperasymptotics for multidimensional integrals exact remainder terms and the global connection problem the method of steepest descents for single dimensional laplace type integrals involving an asymptotic parameter k was extended by berry howls in 1991 to provide exact remainder terms for truncated asymptotic expansions in terms of contributions from certain non local saddlepoints this led to an improved asymptotic expansion hyperasymptotics which gave exponentially accurate numerical and analytic results based on the topography of the saddle distribution in the single complex plane of the integrand in this paper we generalize these results to similar well behaved multidimensional integrands with quadratic critical points integrated over infinite complex domains as previously pointed out the extra complex dimensions give rise to interesting problems and phenomena first the conventionally defined surfaces of steepest descent are no longer unique second the stokes s phenomenon whereby contributions from subdominant saddles enter the asymptotic representation is of codimension one third we can collapse the representation of the integral onto a single complex plane with branch cuts at the images of critical points the new results here demonstrate that dimensionality only trivially affects the form of the exact multidimensional remainder thus the growth of the late terms in the expansion can be identified and a hyperasymptotic scheme implemented we show by a purely algebraic method how to determine which critical points contribute to the remainder and hence resolve the global connection problem riemann sheet structure and homology associated with the multidimensional topography of the integrand