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

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

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

strong integrality of quantum invariants of 3 manifolds we prove that the quantum so 3 invariant of an arbitrary 3 manifold m is always an algebraic integer if the order of the quantum parameter is co prime with the order of the torsion part of h 1 m z an even stronger integrality known as cyclotomic integrality was established by habiro for integral homology 3 spheres here we also generalize habiro s result to all rational homology 3 spheres

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

representations of the kauffman bracket skein algebra i invariants and miraculous cancellations we study finite dimensional representations of the kauffman bracket skein algebra of a surface s in particular we construct invariants of such irreducible representations when the underlying parameter q mathrm e 2 pi mathrm i hbar is a root of unity the main one of these invariants is a point in the character variety consisting of group homomorphisms from the fundamental group pi 1 s to mathrm sl 2 mathbb c or in a twisted version of this character variety the proof relies on certain miraculous cancellations that occur for the quantum trace homomorphism constructed by the authors these miraculous cancellations also play a fundamental role in subsequent work of the authors where novel examples of representations of the skein algebra are constructed

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

positivity of the universal pairing in 3 dimensions associated to a closed oriented surface s is the complex vector space with basis the set of all compact oriented 3 manifolds which it bounds gluing along s defines a hermitian pairing on this space with values in the complex vector space with basis all closed oriented 3 manifolds the main result in this paper is that this pairing is positive i e that the result of pairing a nonzero vector with itself is nonzero this has bearing on the question of what kinds of topological information can be extracted in principle from unitary 2 1 dimensional tqfts the proof involves the construction of a suitable complexity function c on all closed 3 manifolds satisfying a gluing axiom which we call the topological cauchy schwarz inequality namely that c ab max c aa c bb for all a b which bound s with equality if and only if a b the complexity function c involves input from many aspects of 3 manifold topology and in the process of establishing its key properties we obtain a number of results of independent interest for example we show that when two finite volume hyperbolic 3 manifolds are glued along an incompressible acylindrical surface the resulting hyperbolic 3 manifold has minimal volume only when the gluing can be done along a totally geodesic surface this generalizes a similar theorem for closed hyperbolic 3 manifolds due to agol storm thurston

idempotent completion of triangulated categories we show that the idempotent completion of a triangulated category has a natural structure of a triangulated category the idempotent completion of the bounded derived category of an exact category gives the derived category of the idempotent completion in particular the derived category of an idempotent complete exact category is idempotent complete

the geometry of tensor calculus i this paper defines and proves the correctness of the appropriate string diagrams for various kinds of monoidal categories with duals mathematics subject classifications 1991 18d10 52b11 53a45 57m25 68q10 82b23

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

integrality for tqfts we discuss ways that the ring of coefficients for a tqft can be reduced if one restricts somewhat the allowed cobordisms when we apply these methods to a tqft associated to so 3 at an odd prime p we obtain a functor from a somewhat restricted cobordism category to the category of free finitely generated modules over a ring of cyclotomic integers z 2p if p 1 mod 4 and z 4p if p 1 mod 4 where k is a primitive kth root of unity we study the quantum invariants of prime power order simple cyclic covers of 3 manifolds we define new invariants arising from strong shift equivalence and integrality similar results are obtained for some other tqfts but the modules are only guaranteed to be projective

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

p dissipative operator in this paper the authors prove that the generalized positive p selfadjoint gpps operators in banach space satisfy the generalized schwarz inequality solve the maximal dissipative extension representation of p dissipative operators in banach space by using the inequality and introducing the generalized indefinite inner product giip space and apply the result to a certain type of schrodinger operator

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

wavelet approximate inertial manifold in nonlinear solitary wave equation this paper studies the dynamical behavior of a weakly damped forced korteweg de vries kdv equation in a wavelet basis and introduces the wavelet approximate inertial manifold and wavelet galerkin solution of weakly damped forced kdv equation the results includes theorems that for the kdv equation the wavelet approximate inertial manifold waim exists and sets up the wavelet galerkin method of the equation error estimates are given by using the waim

symplectic fermions and a quasi hopf algebra structure on bar u _i sl 2 abstract we consider the finite dimensional restricted quantum group u q s l 2 at q i we show that u i s l 2 does not allow for a universal r matrix even though u v v u holds for all finite dimensional representations u v of u i s l 2 we then give an explicit coassociator and a universal r matrix r such that u i s l 2 becomes a quasi triangular quasi hopf algebra our construction is motivated by the two dimensional chiral conformal field theory of symplectic fermions with central charge c 2 there a braided monoidal category sf has been computed from the factorisation and monodromy properties of conformal blocks and we prove that rep u i s l 2 r is braided monoidally equivalent to sf

the coloured jones function the invariantsjk k of a framed knotk coloured by the irreduciblesu 2 q module of dimensionk are studied as a function ofk by means of the universalr matrix it is shown that whenjk k is written as a power series inh withq eh the coefficient ofhd is an odd polynomial ink of degree at most 2d 1 this coefficient is a vassiliev invariant ofk in the second part of the paper it is shown that ask varies these invariants span ad dimensional subspace of the space of all vassiliev invariants of degreed for framed knots the analogous questions for unframed knots are also studied

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

quantum invariants modular forms and lattice points ii we study the su 2 witten reshetikhin turaev wrt invariant for the seifert fibered homology spheres with m exceptional fibers we show that the wrt invariant can be written in terms of differential of the eichler integrals of modular forms with weight 1 2 and 3 2 by use of nearly modular property of the eichler integrals we shall obtain asymptotic expansions of the wrt invariant in the large n limit we further reveal that the number of the gauge equivalent classes of flat connections which dominate the asymptotics of the wrt invariant in n is related to the number of integral lattice points inside the m dimensional tetrahedron

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

a formula for casson s invariant suppose h is a homology 3 sphere obtained by dehn surgery on a link l in a homology 3 sphere m if every pair of components of l has zero linking number in m then we give a formula for the casson invariant x h in terms of a m the surgery coefficients of l and a certain coefficient from each of the conway polynomials of l and all its sublinks a few consequences of this formula are given introduction andrew casson has discovered an integral invariant of oriented homology 3 spheres which reduces mod 2 to the rochlin or n invariant 1 in this paper we give a formula for casson s invariant in the case where the homology sphere h is obtained by dehn surgery on a link l in a homology sphere m and furthermore the linking number between every pair of components of l is zero note that every homology 3 sphere can be obtained in this way with m s3 in this case the casson invariant x h can be expressed in terms of x m the surgery coefficients of l and a certain coefficient from each of the conway polynomials of l and all its sublinks to illustrate the utility of this formula we compute the casson invariant of homology spheres that arise as cyclic branched covers branched along untwisted double knots we also describe how the x invariant changes when a crossing of the framed link l between two strands of the same component is changed this provides at least in this setting an effective method for computing the invariant an interesting question is whether it is possible to show directly that the formula we give is preserved by the kirby rolfsen calculus and thus give an alternative proof of the invariance of x if l1 and l2 are two rationally framed links representing the same manifold then l1 can be transformed into l2 by a sequence of three types of moves homeomorphism twisting and trivial insertion or deletion 8 if the linking number between every pair of components of l1 is zero then the first two types of moves will preserve this property while the third move may not however if l1 can be transformed into l2 through a sequence of links all of whose linking numbers are all zero then we show that the formula for x is preserved we do not know if such a transformation is in general possible in order to state casson s theorem and the formula for x h we first establish some notation suppose l k1 kn is a framed link in a 3 manifold m with received by the editors october 22 1985 1980 mathematics subject classification 1985 revision primary 57m25 supported in part by the nsf under grant no dms 8502940 x 1986 american mathematical society 0002 9947 86 1 00 25 per page

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

on the quantum invariants for the spherical seifert manifolds we study the witten reshetikhin turaev su 2 invariant for the seifert manifolds s3 gamma where is a finite subgroup of su 2 we show that the wrt invariants can be written in terms of the eichler integral of modular forms with half integral weight and we give an exact asymptotic expansion of the invariants by use of the nearly modular property of the eichler integral we further discuss that those modular forms have a direct connection with the polyhedral group by showing that the invariant polynomials of modular forms satisfy the polyhedral equations associated to

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

quantum gl1 1 and tangle floer homology abstract we identify the grothendieck group of the tangle floer dg algebra with a tensor product of certain u q gl 1 1 representations under this identification up to a scalar factor the map on the grothendieck group induced by the tangle floer dg bimodule associated to a tangle agrees with the reshetikhin turaev homomorphism for that tangle we also introduce dg bimodules which act on the grothendieck group as the generators e and f of u q gl 1 1

broken symmetries in a location invariant word recognition network we studied the feedforward network proposed by dandurand et al 2010 which maps location specific letter inputs to location invariant word outputs probing the hidden layer to determine the nature of the code hidden patterns for words were densely distributed and k means clustering on single letter patterns produced evidence that the network had formed semi location invariant letter representations during training the possible confound with superseding bigram representations was ruled out and linear regressions showed that any word pattern was well approximated by a linear combination of its constituent letter patterns emulating this code using overlapping holographic representations plate 1995 uncovered a surprisingly acute and useful correspondence with the network stemming from a broken symmetry in the connection weight matrix and related to the group invariance theorem minsky papert 1969 these results also explain how the network can reproduce relative and transposition priming effects found in humans

deterministic boltzmann learning performs steepest descent in weight space the boltzmann machine learning procedure has been successfully applied in deterministic networks of analog units that use a mean field approximation to efficiently simulate a truly stochastic system peterson and anderson 1987 this type of deterministic boltzmann machine dbm learns much faster than the equivalent stochastic boltzmann machine sbm but since the learning procedure for dbm s is only based on an analogy with sbm s there is no existing proof that it performs gradient descent in any function and it has only been justified by simulations by using the appropriate interpretation for the way in which a dbm represents the probability of an output vector given an input vector it is shown that the dbm performs steepest descent in the same function as the original sbm except at rare discontinuities a very simple way of forcing the weights to become symmetrical is also described and this makes the dbm more biologically plausible than back propagation werbos 1974 parker 1985 rumelhart

a unified witten reshetikhin turaev invariant for integral homology spheres we construct an invariant j m of integral homology spheres m with values in a completion widehat mathbb z q of the polynomial ring q such that the evaluation at each root of unity gives the the su 2 witten reshetikhin turaev invariant m of m at thus j m unifies all the su 2 witten reshetikhin turaev invariants of m it also follows that m as a function on behaves like an analytic function defined on the set of roots of unity

on the melvin morton rozansky conjecture we prove a conjecture stated by melvin and morton and elucidated further by rozansky saying that the alexander conway polynomial of a knot can be read from some of the coefficients of the jones polynomials of cables of that knot i e coefficients of the colored jones polynomial we first reduce the problem to the level of weight systems using a general principle which may be of some independent interest and which sometimes allows to deduce equality of vassiliev invariants from the equality of their weight systems we then prove the conjecture combinatorially on the level of weight systems finally we prove a generalization of the melvin morton rozansky mmr conjecture to knot invariants coming from arbitrary semi simple lie algebras as side benefits we discuss a relation between the conway polynomial and immanants and a curious formula for the weight system of the colored jones polynomial

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

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

wavelet projections of the kuramoto sivashinsky equation i heteroclinic cycles and modulated traveling waves for short systems abstract we study fairly low dimensional projections of the kuramoto sivashinsky partial differential equation with periodic boundary conditions on a short interval onto bases spanned by periodic wavelets such projections break the translation reflection symmetry o 2 replacing it by a finite dihedral group dk however we show that for the perrier basdevant wavelets used here the loss of symmetry is sufficiently mild that key global features of the dynamics are preserved in particular we observe heteroclinic cycles and modulated travelling waves arising from interactions of unstable modes on a four dimensional subspace spanned by appropriate combinations of wavelets we use invariant manifold reductions in our analysis and pay particular attention to symmetries and the relation between periodic wavelets and fourier modes which preserve full o 2 symmetry and are also optimal in that fourier truncations maximise the energy l2 norm among all finite dimensional models this study provides a foundation for current and future work in which we use wavelet bases to extract local models of evolution equations in large space domains

representation theory of lattice current algebras lattice current algebras were introduced as a regularization of the left and right moving degrees of freedom in the wznw model they provide examples of lattice theories with a local quantum symmetry their representation theory is studied in detail in particular we construct all irreducible representations along with a lattice analogue of the fusion product for representations of the lattice current algebra it is shown that for an arbitrary number of lattice sites the representation categories of the lattice current algebras agree with their continuum counterparts

quantum groups constructed from the non standard braid group representations in the faddeev reshetikhin takhtajan approach for pt i see ibid vol 24 p l725 1991 from the nonstandard braid group representation for the fundamental representation of slq n the authors obtain the corresponding algebra in the faddeev reshetikhin takhtajan approach the main features of the algebra consist of x or k 2 0 k 1 2 n 1 which obliterates half of the generalized serre relations an unusual scalar product normalization of the underlying root vectors besides the presence of two parameters one of which is a root of unity explicit detail is given for the n 5 case an alternative interpretation suggests that such nonstandard solutions are closely connected with the quantum superalgebras

the universalr matrix burau representation and the melvin morton expansion of the colored jones polynomial abstract p melvin and h morton 9 studied the expansion of the colored jones polynomial of a knot in powers of q 1 and color they conjectured an upper bound on the power of color versus the power of 1 they also conjectured that the bounding line in their expansion generated the inverse alexander conway polynomial these conjectures were proved by d bar natan and s garoufalidis 1 we have conjectured 12 that other lines in the melvin morton expansion are generated by rational functions with integer coefficients whose denominators are powers of the alexander conway polynomial here we prove this conjecture by using the r matrix formula for the colored jones polynomial and presenting the universal r matrix as a perturbed burau matrix

weakly damped forced korteweg de vries equations behave as a finite dimensional dynamical system in the long time abstract it is shown that a weak dissipation of order zero is sufficient to impose a finite dimensional behavior for korteweg de vries equations with periodic boundary conditions a universal attractor which captures all the trajectories is constructed its fractal dimension is proved to be finite

bridged links and tangle presentations of cobordism categories abstract we develop a calculus of surgery data called bridged links which involves besides links also pairs of balls that describe one handle attachments as opposed to the usual link calculi of kirby and others this description uses only elementary local moves namely modifications and isolated cancellations and it is valid also on non simply connected and disconnected manifolds in particular it allows us to give a presentation of a 3 manifold by doing surgery on any other 3 manifold with the same boundary bridged link presentations on unions of handlebodies are used to give a cerf theoretical derivation of presentations of 2 1 dimensional cobordisms categories in terms of planar ribbon tangles and their composition rules as an application we give a different more natural proof of the matveev polyak presentations of the mapping class groups and furthermore find systematically surgery presentations of general mapping tori we discuss a natural extension of the reshetikhin turaev invariant to the calculus of bridged links invariance follows now similar as for knot invariants from simple identifications of the elementary moves with elementary categorical relations for invariances or cointegrals respectively hence we avoid the lengthy computations and the unnatural fenn rourke reduction of the original proofs moreover we are able to start from a much weaker modularity condition which implies the one of turaev generalizations of the presentation to cobordisms of surfaces with boundaries are outlined

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

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

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

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

structures and diagrammatics of four dimensional topological lattice field theories abstract crane and frenkel proposed a state sum invariant for triangulated 4 manifolds they sketched the definition of a hopf category that was to be used in their construction crane and yetter studied hopf categories and gave some examples using group cocycles that are associated to the drinfeld double of a finite group in this paper we define a state sum invariant of triangulated 4 manifolds using crane yetter cocycles as boltzmann weights our invariant is analogous to the 3 dimensional invariants defined by dijkgraaf and witten and the invariants that are defined via hopf algebras we present diagrammatic methods for the study of such invariants that illustrate connections between hopf categories and moves to triangulations

quasi quantum groups knots three manifolds and topological field theory we show how to construct starting from a quasi hopf algebra or quasiquantum group invariants of knots and links in some cases these invariants give rise to invariants of the three manifolds obtained by surgery along these links this happens for a finite dimensional quasi quantum group whose definition involves a finite groupg and a 3 cocycle which was first studied by dijkgraaf pasquier and roche we treat this example in more detail and argue that in this case the invariants agree with the partition function of the topological field theory of dijkgraaf and witten depending on the same datag

three manifold invariants associated with restricted quantum groups we show a simple relation between witten reshetikhin turaev su 2 invariant and the hennings invariant associated with the restricted quantum mathfrak sl 2 these invariants are defined in very different methods the former uses the representation theory of quantum mathfrak sl 2 while the latter uses the integral of the hopf algebra but they turn out to be the same for most rational homology 3 spheres up to a sign this relation can be used to prove the integrality of the former invariant

double construction for crossed hopf coalgebras abstract we provide an analog of the drinfeld quantum double construction in the context of crossed hopf group coalgebras introduced by turaev we prove that provided the base group is finite the double of a semisimple crossed hopf group coalgebra is both modular and unimodular

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

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

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

a note on the strong convergence towards attractors of damped forced kdv equations abstract this note is a continuation of a previous article which appeared in this journal 1988 74 369 390 in the previous work we showed that a weak dissipation is suffiient to impose a finite dimensional behavior for korteweg de vries equations with space periodic boundary conditions more precisely it was shown that all the trajectories are captured by a finite dimensional attractor however we were only able to show that the convergence of the trajectories holds good in the sense of weak topology the goal of this note is to show that in fact this convergence holds true in the sense of strong topology

convergence estimates for the wavelet galerkin method this paper presents an analysis of the galerkin approximation of a time dependent initial value problem correctly posed in the petrovskii sense the approximating spaces mathcal v h are spanned by translations and dilations of a single function phi and fourier techniques are used to analyze the accuracy of the method this kind of procedure has already been applied in the literature for spline approximations our purpose here is to point out that the same methodology can be used for wavelet based methods since the hypotheses required are automatically satisfied in the context of wavelet analysis for instance the basic function phi is supposed to be regular which means that phi and all its derivatives up to a certain order should have fast decay at infinity phi also must satisfy the so called strang and fix condition which guarantees that smooth functions can be approximated from mathcal v h with good accuracy this class of functions includes not only the b splines but al

quantum koszul complexes for majid s braided lie algebras a yang baxter operator was canonically associated to any cocommutative braided lie algebra as defined by majid applying previous results we construct quantum koszul complexes for braided lie algebras with respect to this operator in the usual case the complexes reduce to the chevalley eilenberg complexes for classical lie algebras

a categorification of the jones polynomial author s khovanov mikhail abstract we construct a bigraded cohomology theory of links whose euler characteristic is the jones polynomial

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

the colored jones polynomials and the simplicial volume of a knot we show that the set of colored jones polynomials and the set of generalized alexander polynomials defined by akutsu deguchi and ohtsuki intersect non trivially moreover it is shown that the intersection is at least includes the set of kashaev s quantum dilogarithm invariants for links therefore kashaev s conjecture can be restated as follows the colored jones polynomials determine the hyperbolic volume for a hyperbolic knot modifying this we propose a stronger conjecture the colored jones polynomials determine the simplicial volume for any knot if our conjecture is true then we can prove that a knot is trivial if and only if all of its vassiliev invariants are trivial in 13 r m kashaev defined a family of complex valued link invariants indexed by integers n 2 using the quantum dilogarithm later he calculated the asymptotic behavior of his invariant and observed that for the three simplest hyperbolic knots it grows as exp vol k n 2 when n goes to the infinity where vol k is the hyperbolic volume of the complement of a knotk 14 this amazing result and his conjecture that the same also holds for any hyperbolic knot have been almost ignored by mathematicians since his definition of the invariant is too complicated though it uses only elementary tools the aim of this paper is to reveal his mysterious definition and to show that his invariant is nothing but a specialization of the colored jones polynomial the colored jones polynomial is defined for colored links each component is decorated with an irreducible representation of the lie algebra sl 2 c the original jones polynomial corresponds to the case that all the colors are identical to the 2 dimensional fundamental representation we show that kashaev s invariant with parameter n coincides with the colored jones polynomial in a certain normalization with every color the n dimensional representation evaluated at the primitive n th root of unity we have to normalize the colored jones polynomial so that the value for the trivial knot is one for otherwise it always vanishes on the other hand there are other colored polynomial invariants the generalized multivariable alexander polynomial defined by y akutsu t deguchi and t ohtsuki 1 they used the same lie algebra sl 2 c but a different hierarchy of representations their invariants are parameterized by c 1 parameters an integer date february 1 2008 1991 mathematics subject classification 57m25 57m50 17b37 81r50

quantum invariants at the sixth root of unity a general topological formula is given for thesu 2 quantum invariant of a 3 manifoldm at the sixth root of unity it is expressed in terms of the homology witt invariants and signature defects of the various 2 fold covers ofm and thus ties in with basic 4 dimensional invariants a discussion of the range of values of these quantum invariants is included and explicit evaluations are made for lens spaces

equivalence of backpropagation and contrastive hebbian learning in a layered network backpropagation and contrastive hebbian learning are two methods of training networks with hidden neurons backpropagation computes an error signal for the output neurons and spreads it over the hidden neurons contrastive hebbian learning involves clamping the output neurons at desired values and letting the effect spread through feedback connections over the entire network to investigate the relationship between these two forms of learning we consider a special case in which they are identical a multilayer perceptron with linear output units to which weak feedback connections have been added in this case the change in network state caused by clamping the output neurons turns out to be the same as the error signal spread by backpropagation except for a scalar prefactor this suggests that the functionality of backpropagation can be realized alternatively by a hebbian type learning algorithm which is suitable for implementation in biological networks

a unified quantum so 3 invariant for rational homology 3 spheres given a rational homology 3 sphere m with h1 m z b and a link l inside m colored by odd numbers we construct a unified invariant im l belonging to a modification of the habiro ring where b is inverted our unified invariant dominates the whole set of the so 3 witten reshetikhin turaev invariants of the pair m l i fb 1a ndl im coincides with habiro s invariant of integral homology 3 spheres for b 1 the unified in variant defined by the third author is determined by im important applica tions are the new ohtsuki series perturbative expansions of im dominating quantum so 3 invariants at roots of unity whose order is not a power of a prime these series are not known to be determined by the lmo invariant

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