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

computation of lickorish s three manifold invariant using chern simons theory it is well known that any three manifold can be obtained by surgery on a framed link in s 3 lickorish gave an elementary proof for the existence of the three manifold invariant of witten using a framed link description of the manifold and the formalisation of the bracket polynomial as the temperley lieb algebra kaul determined a three manifold invariant from link polynomials in su 2 chern simons theory lickorish s formula for the invariant involves computation of bracket polynomials of several cables of the link we describe an easier way of obtaining the bracket polynomial of a cable using representation theory of composite braiding in su 2 chern simons theory we prove that the cabling corresponds to taking tensor products of fundamental representations of su 2 this enables us to verify that the two apparently distinct three manifold invariants are equivalent for a specific relation of the polynomial variables

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

higher dimensional abelian chern simons theories and their link invariants the role played by deligne beilinson cohomology in establishing the relation between chern simons theory and link invariants in dimensions higher than three is investigated deligne beilinson cohomology classes provide a natural abelian chern simons action non trivial only in dimensions 4l 3 whose parameter k is quantized the generalized wilson 2l 1 loops are observables of the theory and their charges are quantized the chern simons action is then used to compute invariants for links of 2l 1 loops first on closed 4l 3 manifolds through a novel geometric computation then on r4l 3 through an unconventional field theoretic computation

a combinatorial approach to topological quantum field theories and invariants of graphs the combinatorial state sum of turaev and viro for a compact 3 manifold in terms of quantum 6j symbols is generalized by introducing observables in the form of coloured graphs they satisfy braiding relations and allow for surgeries and a discussion of cobordism theory application of these techniques give the dimension and an explicit basis for the vector space of the topological quantum field theory associated to any riemann surface with arbitrary coloured punctures

invariants of spin three manifolds from chern simons theory and finite dimensional hopf algebras abstract a version of kirby calculus for spin and framed three manifolds is given and is used to construct invariants of spin and framed three manifolds in two situations the first is ribbon categories which possess odd degenerate objects this case includes the quantum group situations corresponding to the half integer level chern simons theories conjectured to give spin tqfts by dijkgraaf and witten 1990 commun math phys 129 393 429 in particular the spin invariants constructed by kirby and melvin 1991 invent math 105 473 545 are shown to be identical to the invariants associated to so 3 second an invariant of spin manifolds analogous to the hennings invariant is constructed beginning with an arbitrary factorizable unimodular quasitriangular hopf algebra in particular a framed manifold invariant is associated to every finite dimensional hopf algebra via its quantum double and is conjectured to be identical to kuperberg s noninvolutory invariant of framed manifolds associated to that hopf algebra

3d topological models and heegaard splitting i partition function the aim of this article is twofold first we show how to recover the smooth deligne beilinson cohomology groups from a heegaard splitting of a closed oriented smooth 3 manifold by extending the usual cech de rham construction second thanks to the above and still relying on a heegaard splitting we explain how to compute the partition functions of the u 1 chern simons and bf theories

random attractors for the stochastic benjamin bona mahony equation on unbounded domains abstract we prove the existence of a compact random attractor for the stochastic benjamin bona mahony equation defined on an unbounded domain this random attractor is invariant and attracts every pulled back tempered random set under the forward flow the asymptotic compactness of the random dynamical system is established by a tail estimates method which shows that the solutions are uniformly asymptotically small when space and time variables approach infinity

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

long time dynamics for a damped benjamin bona mahony equation in low regularity spaces abstract long time behavior of solutions for a damped bbm equation in low regularity space is investigated by the i method and high low frequency decomposition we prove the existence of a global attractor in l 2 if the force belongs to l 2 moreover we show that the global attractor has finite fractal dimension in the sharp regularity space h 2

quantum geometry and quantum algorithms motivated by algorithmic problems arising in quantum field theories whose dynamical variables are geometric in nature we provide a quantum algorithm that efficiently approximates the coloured jones polynomial the construction is based on the complete solution of the chern simons topological quantum field theory and its connection to wess zumino witten conformal field theory the coloured jones polynomial is expressed as the expectation value of the evolution of the q deformed spin network quantum automaton a quantum circuit is constructed capable of simulating the automaton and hence of computing such an expectation value the latter is efficiently approximated using a standard sampling procedure in quantum computation

canonical structure in so 4 q and relations to e 3 q and so 3 1 q starting with the su 2 q su 2 q basis and using q cg coefficients the closest possible q deformation of the classical canonical gelfand zetlin formalism is constructed explicitly for so 4 q simple introduction of q brackets for each factor of the classical matrix elements valid for su n q is shown to need significant and unexpected modifications for so n q even when n 4 contraction to e 3 q and continuation to so 3 1 q are studied making explicit the possibilities and the problems that arise periodic representations for q a root of unity are commented upon briefly

on the decay of solutions of the generalized benjamin bona mahony equation on etudie le comportement des solutions du probleme aux valeurs initiales u t u x u p u x u xxt 0 x r t o u x 0 x x r p 0 entier on suppose p 4 alors il existe 0 tel que si 1 1 5 0 et x r ou a ne depend pas de x ou t

almost periodic and almost automorphic solutions of linear differential difference equations without favard s separation condition ii abstract the well known favard s theorem states that the linear differential equation 1 x a t x f t with bohr almost periodic coefficients admits at least one bohr almost periodic solution if it has a bounded solution the main assumption in this theorem is the separation among bounded solutions of homogeneous equations 2 x b t x where b h a b b t lim n a t t n if there are bounded solutions which are non separated sometimes almost periodic solutions do not exist r johnson r ortega and m tarallo v zhikov and b levitan in this paper we prove that linear differential equation 1 with levitan almost periodic coefficients has a unique levitan almost periodic solution if it has at least one bounded solution and the bounded solutions of the homogeneous equation 3 x a t x are homoclinic to zero i e lim t t 0 for all bounded solutions of 3 if the coefficients of 1 are bohr almost periodic and all bounded solutions of all limiting equations 2 are homoclinic to zero then eq 1 admits a unique almost automorphic solution the analogue of this result for difference equations is also given we study the problem of existence of bohr levitan almost periodic solutions of eq 1 in the framework of general non autonomous dynamical systems cocycles

asymptotic smoothing and attractors for the generalized benjamin bona mahony equation on r 3 abstract we study the asymptotic behavior of the solutions of the benjamin bona mahony equation defined on r 3 we first provide a sufficient condition to verify the asymptotic compactness of an evolution equation defined in an unbounded domain which involves the littlewood paley projection operators we then prove the existence of an attractor for the benjamin bona mahony equation in the phase space h 1 r 3 by showing the solutions are point dissipative and asymptotic compact finally we establish the regularity of the attractor and show that the attractor is bounded in h 2 r 3

decay of solutions of some nonlinear wave equations abstract we study the large time behaviour of solutions to the initial value problem for the korteweg de vries equation and for the regularized long wave equation with a dissipative term appended using energy estimates a maximum principle and a transformation of cole hopf type sharp rates of temporal decay of certain norms of the solution are obtained

abelian turaev virelizier theorem and u 1 bf surgery formulas in this article we construct the reshetikhin turaev invariant associated with the drinfeld center of the spherical category arising from the u 1 bf theory on a closed 3 manifold m this invariant is shown to coincide with the turaev viro invariant of m thus providing an example of the turaev virelizier theorem finally we exhibit some surgery formulas for the abelian turaev viro invariant which are very similar to the surgery formulas of the abelian reshetikhin turaev invariant obtained in the u 1 chern simons context

existence of global attractors for the benjamin bona mahony equation in unbounded domains in this paper we investigate the asymptotic behavior of the benjamin bona mahony equation in unbounded domains we prove the existence of a global attractor when the equation is defined in a three dimensional channel the asymptotic compactness of the solution operator is obtained by the uniform estimates on the tails of solutions

invariants of 3 manifolds and projective representations of mapping class groups via quantum groups at roots of unity an example of a finite dimensional factorizable ribbon hopf algebra is given by a quotienth u q g of the quantized universal enveloping algebrau q g at a root of unityq of odd degree the mapping class groupm g 1 of a surface of genusg with one hole projectively acts by automorphisms in theh moduleh g ifh is endowed with the coadjointh module structure there exists a projective representation of the mapping class groupm g n of a surface of genusg withn holes labeled by finite dimensionalh modulesx 1 x n in the vector space hom h x 1 x n h g an invariant of closed oriented 3 manifolds is constructed modifications of these constructions for a class of ribbon hopf algebras satisfying weaker conditions than factorizability including most ofu q g at roots of unityq of even degree are described

almost periodic and almost automorphic solutions of linear differential equations the paper is dedicated to the study problem of existence of almost periodic respectively almost automorphic recurrent solutions of linear differential equations u t a t y u t f t y y y in banach space with almost periodic respectively almost automorphic recurrent coefficients where y r is a dynamical system on the metric space y in particular we prove that if the operator a y is positive and auto adjoint then for the equation 1 u t a t y u t 0 one of the following alternative is fulfilled i there exists a complete trajectory of 1 with constant positive norm ii the trivial solution of equation 1 is uniformly asymptotically stable this problem we investigate in the framework of general linear non autonomous dynamical system we apply our general results also for the functional differential equations and difference equations

three manifold invariants from chern simons field theory with arbitrary semi simple gauge groups invariants for framed links in s 3 obtained from chern simons gauge field theory based on an arbitrary gauge group semi simple have been used to construct a three manifold invariant this is a generalization of a similar construction developed earlier for su 2 chern simons theory the procedure exploits a theorem of lickorish and wallace and also those of kirby fenn and rourke which relate three manifolds to surgeries on framed unoriented links the invariant is an appropriate linear combination of framed link invariants which does not change under kirby calculus this combination does not see the relative orientation of the component knots the invariant is related to the partition function of chern simons theory this thus provides an efficient method of evaluating the partition function for these field theories as some examples explicit computations of these manifold invariants for a few three manifolds have been done

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

on two invariants of three manifolds from hopf algebras abstract we prove a 20 year old conjecture concerning two quantum invariants of three manifolds that are constructed from finite dimensional hopf algebras namely the kuperberg invariant and the hennings kauffman radford invariant the two invariants can be viewed as a non semisimple generalization of the turaev viro barrett westbury tvbw invariant and the witten reshetikhin turaev wrt invariant respectively by a classical result relating tvbw and wrt it follows that the kuperberg invariant for a semisimple hopf algebra is equal to the hennings kauffman radford invariant for the drinfeld double of the hopf algebra however whether the relation holds for non semisimple hopf algebras has remained open partly because the introduction of framings in this case makes the kuperberg invariant significantly more complicated to handle we give an affirmative answer to this question an important ingredient in the proof involves using a special heegaard diagram in which one family of circles gives the surgery link of the three manifold represented by the heegaard diagram

attractors for autonomous and nonautonomous 3d benjamin bona mahony equations in this paper we consider the long time behavior of the three dimensional benjamin bona mahony equations for the autonomous and nonautonomous cases a useful decomposition method is introduced to overcome the difficulties in proving the asymptotical regularity of the 3d benjamin bona mahony equations for the autonomous case we prove the existence of global attractor when the external forcing belongs to v for the nonautonomous case we only assume that f x t is translation bounded instead of translation compact by means of this useful decomposition methods we prove the asymptotic regularity of solutions of 3d benjamin bona mahony equations and also obtain the existence of the uniform attractor

almost periodic linear differential equations with non separated solutions a celebrated result by favard states that for certain almost periodic linear differential systems the existence of a bounded solution implies the existence of an almost periodic solution a key assumption in this result is the separation among bounded solutions here we prove a theorem of anti favard type if there are bounded solutions which are non separated in a strong sense sometimes almost periodic solutions do not exist strongly non separated solutions appear when the associated homogeneous system has homoclinic solutions this point of view unifies two fascinating examples by zhikov levitan and johnson for the scalar case our construction uses the ideas of zhikov levitan together with the theory of characters in topological groups

quantum deformations of certain simple modules over enveloping algebras soit f un corps soit f 0 un sous corps de f soit u l algebre enveloppante universelle sur f 0 de l algebre de lie de kac moody correspondant a la matrice a d entiers on montre qu un module integrable de poids le plus eleve v de u admet une deformation quantique

blow up of solutions of pseudoparabolic equations we obtain sufficient conditions for the blow up of solutions of the initial boundary value problem for nonlinear pseudoparabolic equation involving nonlinear convective term

induced module construction for highest weight representations of uq gl n at roots of unity a method is investigated for inducing highest weight representations for the quantum group uq gl n from the canonical subalgebra uq gl n 1 when q is a root of unity we classify the irreps into two types typical and atypical where the former is a generalization of the class of irreps with maximal dimensionality the structures of both the typical and atypical irreps are studied in particular a sufficiency condition is given for an irrep to be typical as examples we consider flat representations induced from a one dimensional representation of the canonical subalgebra and representations induced from vector representation

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

attractors for the generalized benjamin bona mahony equation abstract we consider the periodic initial boundary value problem for a multidimensional generalized benjamin bona mahony equation we show the existence of the global attractor with a finite fractal dimension and the existence of the exponential attractor for the corresponding semigroup

integrality and symmetry of quantum link invariants 0 introduction quantum invariants of framed links whose components are colored by modules of a simple lie algebra g are laurent polynomials in v1 d with integer coefficients where v is the quantum parameter and d an integer depending on g we show that quantum invariants with a suitable normalization are laurent polynomials in v2 we also establish two symmetry properties of quantum link invariants at roots of unity the first asserts that quantum link invariants at rth roots of unity are invariant under the action of the affine weyl group wr which acts on the weight lattice a fundamental domain of wr is the fundamental alcove cr a simplex let g be the center of the corresponding simply connected complex lie group there is a natural action of g on cr the second symmetry property in its simplest form asserts that quantum link invariants are invariant under the action ofg if the link has zero linking matrix the second symmetry property generalizes symmetry principles of kirby and melvin the sl2 case and kohno and takata the sln case to arbitrary simple lie algebra

regularity of attractors for the benjamin bona mahony equation this paper deals with the regularity of the global attractor for the benjamin bona mahony equation we prove that the global attractor is smooth if the forcing term is smooth

almost automorphic linear dynamics by favard theory abstract we prove that linear almost periodic systems always carry almost automorphic dynamics thus confirming in this case a more general conjecture by shen and yi the result is based on the accurate description of the breakdown of the favard separation condition

chern simons theory coloured oriented braids and link invariants a method to obtain explicit and complete topological solution of su 2 chern simons theory ons3 is developed to this effect the necessary aspects of the theory of coloured oriented braids and duality properties of conformal blocks for the correlators ofsu 2 k wess zumino conformal field theory are presented a large class of representations of the generators of the groupoid of coloured oriented braids are obtained these provide a whole lot of new link invariants of which jones polynomials are the simplest examples these new invariants are explicity calculated as illustrations for knots up to eight crossings and twocomponent multicoloured links up to seven crossings

analytic torsion and r torsion for unimodular representations let m be a closed c manifold of dimension n both r torsion and analytic torsion are smooth invariants of acyclic orthogonal or unitary representations p of the fundamental group 7i m the reidemeister franz torsion or r torsion tm p of p is defined in terms of the combinatorial structure of m given by its smooth triangulations the analytic torsion tm p was introduced by ray and singer rs as an analytic counterpart of r torsion in order to define the analytic torsion one has to choose a riemannian metric on m then tm p is a certain weighted alternating product of regularized determinants of the laplacians on differential qforms of m with values in the flat bundle e defined by p it was conjectured by ray and singer rs that tm p trm p for all acyclic orthogonal or unitary representations p this conjecture was proved independently by cheeger c and the author mu the restriction to orthogonal or unitary representations is certainly a limitation of the applicability of this result if 7i m is infinite because an infinite discrete group will have in general many nonorthogonal finite dimensional representations it is the purpose of the present paper to remove this limitation we call a representation p r m gl e on a finite dimensional real or complex vector space e unimodular if i det p y i 1 for all y e i m then we define r torsion and analytic torsion for unimodular representations and the main result is that for odd dimensional manifolds m the equality of the two torsions extends to all unimodular representations

quantum representations and monodromies of fibered links abstract andersen masbaum and ueno conjectured that certain quantum representations of surface mapping class groups should send pseudo anosov mapping classes to elements of infinite order for large enough level r in this paper we relate the amu conjecture to a question about the growth of the turaev viro invariants t v r of hyperbolic 3 manifolds we show that if the r growth of t v r m for a hyperbolic 3 manifold m that fibers over the circle is exponential then the monodromy of the fibration of m satisfies the amu conjecture building on earlier work 11 we give broad constructions of oriented hyperbolic fibered links of arbitrarily high genus whose s o 3 turaev viro invariants have exponential r growth as a result for any g n 2 we obtain infinite families of non conjugate pseudo anosov mapping classes acting on surfaces of genus g and n boundary components that satisfy the amu conjecture we also discuss integrality properties of the traces of quantum representations and we answer a question of chen and yang about turaev viro invariants of torus links

2 1 d topological quantum field theory and 2 d conformal field theory we describe the relation between three dimensional topological quantum field theory and two dimensional conformal field theory some applications to quantum knot invariants leading to the equivalence of chern simons witten and kohno s approaches are outlined

favard theory for the adjoint equation and fredholm alternative abstract fredholm alternative is a classical tool of periodic linear equations allowing to describe the existence of periodic solutions of an inhomogeneous equation in terms of the adjoint equation a few partial extensions have been proposed in the literature for recurrent equations our aim is to point out that they have a common root and discuss whether such a root gives rise to a general fredholm type alternative sacker sell spectral theory and favard theory are main ingredients in this discussion a considerable effort is devoted to understand how favard theory is affected by adjunction at least for planar equations

asymptotic soliton like solutions to the singularly perturbed benjamin bona mahony equation with variable coefficients the paper deals with a problem of asymptotic soliton like solutions to the benjamin bona mahony bbm equation with a small parameter at the highest derivative and variable coefficients depending on the variables x and t as well as a small parameter an algorithm for constructing solutions to the bbm equation has been proposed and theorems on accuracy of such solutions have been proved the paper deals with a problem of asymptotic soliton like solutions to the benjamin bona mahony bbm equation with a small parameter at the highest derivative and variable coefficients depending on the variables x and t as well as a small parameter an algorithm for constructing solutions to the bbm equation has been proposed and theorems on accuracy of such solutions have been proved

asymptotic behavior of a class of non autonomous degenerate parabolic equations abstract we investigate the asymptotic behavior of solutions of a class of non autonomous degenerate parabolic equations on r n with unbounded external forcing terms the existence of a pullback global attractor is proved in l 2 r n l p r n w 1 r r n for any p r 2

the yang baxter equation and invariants of links on considere le fait que l on peut construire des invariants d isotopie p et f pour des liens en utilisant certaines solutions des equations de yang baxter

finite dimensional behaviour for the benjamin bona mahony equation this paper deals with the asymptotic behaviour of solutions for the benjamin bona mahony equation we first show the existence of the global weak attractor for this equation in and then by an idea of ball we prove that the global weak attractor is actually the global strong attractor the finite dimensionality of the global attractor is also established

quantum groups at odd roots of unity and topological invariants of 3 manifolds topological invariants of three manifolds are constructed using quantum groups associated with thea b c andd series of lie algebras at odd roots of unity these invariants are also explicitly computed for the lens spaces