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

factorization homology i higher categories abstract we construct a pairing which we call factorization homology between framed manifolds and higher categories the essential geometric notion is that of a vari framing of a stratified manifold which is a framing on each stratum together with a coherent system of compatibilities of framings along links between strata our main result constructs labeling systems on disk stratified vari framed n manifolds from n categories these n categories in contrast with the literature to date are not required to have adjoints this allows the following conceptual definition the factorization homology m c of a framed n manifold m with coefficients in an n category c is the classifying space of c labeled disk stratifications over m the core calculation underlying our main result is the following for any disk stratified manifold the space of conically smooth diffeomorphisms which preserve a vari framing is discrete

topological gauge theories and group cohomology we show that three dimensional chern simons gauge theories with a compact gauge groupg not necessarily connected or simply connected can be classified by the integer cohomology grouph4 bg z in a similar way possible wess zumino interactions of such a groupg are classified byh3 g z the relation between three dimensional chern simons gauge theory and two dimensional sigma models involves a certain natural map fromh4 bg z toh3 g z we generalize this correspondence to topological spin theories which are defined on three manifolds with spin structure and are related to what might be calledz2 graded chiral algebras or chiral superalgebras in two dimensions finally we discuss in some detail the formulation of these topological gauge theories for the special case of a finite group establishing links with two dimensional holomorphic orbifold models

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

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

generating functional in cft on riemann surfaces ii homological aspects we revisit and generalize our previous algebraic construction of the chiral effective action for conformal field theory on higher genus riemann surfaces we show that the action functional can be obtained by evaluating a certain deligne cohomology class over the fundamental class of the underlying topological surface this deligne class is constructed by applying a descent procedure with respect to a cech resolution of any covering map of a riemann surface detailed calculations are presented in the two cases of an ordinary cech cover and of the universal covering map which was used in our previous approach we also establish a dictionary that allows to use the same formalism for different covering morphisms

relative quantum field theory we highlight the general notion of a relative quantum field theory which occurs in several contexts one is in gauge theory based on a compact lie algebra rather than a compact lie group this is relevant to the maximal superconformal theory in six dimensions

more on gauge theory and geometric langlands abstract the geometric langlands correspondence was described some years ago in terms of s duality of n 4 super yang mills theory some additional matters relevant to this story are described here the main goal is to explain directly why an a brane of a certain simple kind can be an eigenbrane for the action of t hooft operators to set the stage we review some facts about higgs bundles and the hitchin fibration we consider only the simplest examples in which many technical questions can be avoided

weak and strong convergence theorems for strict pseudo contractions in hilbert spaces abstract let c be a closed convex subset of a real hilbert space h and assume that t is a strict pseudo contraction on c with a fixed point for some 0 1 given an initial guess x 0 c and given also a real sequence n in 0 1 the mann s algorithm generates a sequence x n by the formula x n 1 n x n 1 n t x n n 0 it is proved that if the control sequence n is chosen so that n 1 and n 0 n 1 n then x n converges weakly to a fixed point of t however this convergence is in general not strong we then modify mann s algorithm by applying projections onto suitably constructed closed convex sets to get an algorithm which generates a strong convergent sequence this result extends a recent result of nakajo and takahashi k nakajo w takahashi strong convergence theorems for nonexpansive mappings and nonexpansive semigroups j math anal appl 279 2003 372 379 from nonexpansive mappings to strict pseudo contractions

categorified symplectic geometry and the classical string a lie 2 algebra is a categorified version of a lie algebra that is a category equipped with structures analogous to those of a lie algebra for which the usual laws hold up to isomorphism in the classical mechanics of point particles the phase space is often a symplectic manifold and the poisson bracket of functions on this space gives a lie algebra of observables multisymplectic geometry describes an n dimensional field theory using a phase space that is an n plectic manifold a finite dimensional manifold equipped with a closed nondegenerate n 1 form here we consider the case n 2 for any 2 plectic manifold we construct a lie 2 algebra of observables we then explain how this lie 2 algebra can be used to describe the dynamics of a classical bosonic string just as the presence of an electromagnetic field affects the symplectic structure for a charged point particle the presence of a b field affects the 2 plectic structure for the string

the fusion algebra of an extraspecial p group one can remark that the study of twisted fusion algebras for finite groups is closely related to the modular data associated with finite groups coste et al finite group modular data nucl phys b 521 679 undergo the study of this finite group modular data and write down the modular s and t matrices of the group g z p 3 for a particular twist here we show that the modular data of g for a particular twist are the same as the modular data of the extraspecial p group h of order p3 with exponent p this result implies that the twisted fusion algebra of g is the same as the nontwisted fusion algebra of h

strong convergence to common fixed points of families of nonexpansive mappings abstract we introduce iteration schemes for families of nonexpansive mappings in hilbert spaces and prove that the iterates converge strongly to common fixed points of the mappings

delocalized chern character for stringy orbifold k theory in this paper we define a stringy product on k orb x c the orbifold k theory of any almost complex presentable orbifold x we establish that under this stringy product the delocali zed chern character chdeloc k x c h cr x after a canonical modification is a ring isomorphism here h cr x is the chen ruan cohomology of x the proof relies on an intrinsic description of the obstruc tion bundles in the construction of the chen ruan product as an application we investigate this stringy product on the equivariant k theory k g g of a finite group g with the conjugation action it turns out that the stringy pr oduct is different from the pontryagin product the latter is also called the fusion pro duct in string theory

inner products of 2 representations abstract we define and calculate inner products of 2 representations along the way we prove that the categorical trace t r of 24 sec 3 is multiplicative with respect to various notions of categorical tensor product and we identify the center of the category of equivariant objects we discuss applications ranging from schur s result about the number of irreducible projective representations to a formula for the hochschild cohomology of a global quotient orbifold

chern simons theory with finite gauge group we construct in detail a 2 1 dimensional gauge field theory with finite gauge group in this case the path integral reduces to a finite sum so there are no analytic problems with the quantization the theory was originally introduced by dijkgraaf and witten without details the point of working it out carefully is to focus on the algebraic structure and particularly the construction of quantum hilbert spaces on closed surfaces by cutting and pasting this includes the verlinde formula the careful development may serve as a model for dealing with similar issues in more complicated cases

viscosity approximation methods for pseudo contractive semigroups in banach spaces abstract we study the strong convergence of two viscosity iteration processes for pseudo contractive semigroup and for strongly pseudo contractive mapping in uniformly convex banach spaces with uniformly g teaux differentiable norm as special cases we get strong convergence of two viscosity iteration processes for approximating common fixed points of nonexpansive semigroups in certain banach spaces the results presented in this paper extend and generalize previous results

op lax natural transformations twisted quantum field theories and even higher morita categories abstract motivated by the challenge of defining twisted quantum field theories in the context of higher categories we develop a general framework for lax and oplax transformations and their higher analogs between strong n functors we construct a double n category built out of the target n category governing the desired diagrammatics we define op lax transformations as functors into parts thereof and an op lax twisted field theory to be a symmetric monoidal op lax natural transformation between field theories we verify that lax trivially twisted relative field theories are the same as absolute field theories as a second application we extend the higher morita category of e d algebras in a symmetric monoidal n category c to an n d category using the higher morphisms in c

aqft from n functorial qft there are essentially two different approaches to the axiomatization of quantum field theory qft algebraic qft going back to haag and kastler and functorial qft going back to atiyah and segal more recently based on ideas by baez and dolan the latter is being refined to extended functorial qft by freed hopkins lurie and others the first approach uses local nets of operator algebras which assign to each patch an algebra of observables the latter uses n functors which assign to each patch a propagator of states

bundle gerbes for chern simons and wess zumino witten theories we develop the theory of chern simons bundle 2 gerbes and multiplicative bundle gerbes associated to any principal g bundle with connection and a class in h4 bg for a compact semi simple lie group g the chern simons bundle 2 gerbe realises differential geometrically the cheeger simons invariant we apply these notions to refine the dijkgraaf witten correspondence between three dimensional chern simons functionals and wess zumino witten models associated to the group g we do this by introducing a lifting to the level of bundle gerbes of the natural map from h4 bg to h3 g the notion of a multiplicative bundle gerbe accounts geometrically for the subtleties in this correspondence for non simply connected lie groups the implications for wess zumino witten models are also discussed

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

strong convergence of the cq method for fixed point iteration processes strong convergence theorems are obtained for the cq method for an ishikawa iteration process a contractive type iteration process for nonexpansive mappings and the proximal point algorithm for maximal monotone operators in hilbert spaces

iterated spans and classical topological field theories we construct higher categories of iterated spans possibly equipped with extra structure in the form of higher categorical local systems and classify their fully dualizable objects by the cobordism hypothesis these give rise to framed topological quantum field theories which are the framed versions of the classical topological quantum field theories considered in the quantization programme of freed hopkins lurie teleman using this machinery we also construct an infty 1 category of symplectic derived algebraic stacks and lagrangian correspondences and show that all its objects are dualizable

higher algebraic structures and quantization we derive quasi quantum groups in 2 1 dimensional topological field theory directly from the classical action and the path integral detailed computations are carried out for the chern simons theory with finite gauge group the principles behind our computations are presumably more general we extend the classical action in ad 1 dimensional topological theory to manifolds of dimension less thand 1 we then construct a generalized path integral which ind 1 dimensions reduces to the standard one and ind dimensions reproduces the quantum hilbert space in a 2 1 dimensional topological theory the path integral over the circle is the category of representations of a quasi quantum group in this paper we only consider finite theories in which the generalized path integral reduces to a finite sum new ideas are needed to extend beyond the finite theories treated here

higher dimensional algebra ii 2 hilbert spaces abstract a 2 hilbert space is a category with structures and properties analogous to those of a hilbert space more precisely we define a 2 hilbert space to be an abelian category enriched over hilb with a structure conjugate linear on the hom sets satisfying fg h g f h f hg we also define monoidal braided monoidal and symmetric monoidal versions of 2 hilbert spaces which we call 2 h algbebras braided 2 h algebras and symmetric 2 h algbebras and we describe the relation between these and tangles in two three and four dimensions respectively we prove a generalized doplicher roberts theorem starting that every symmetric 2 h algebra is equivalent to the category rep g of continuous unitary finite dimensional representations of some compact supergroupoid g the equivalence is given by a categorified version of the gelfand transform we also construct a categorified version of the fourier transform when g is a compact abelian group finally we characterize rep g by its universal properties when g is a compact classical group for example rep u n is the free connected symmetric 2 h algebra on one even object of dimension n

strong convergence theorems for nonexpansive mappings and nonexpansive semigroups in this paper we show strong convergence theorems for nonexpansive mappings and nonexpansive semigroups in hilbert spaces by the hybrid method in the mathematical programming

hybrid ishikawa iterative methods for a nonexpansive semigroup in hilbert space in this paper on the base of the ishikawa iteration method and the hybrid method in mathematical programming we give two new strong convergence methods for finding a point in the common fixed point set of a nonexpansive semigroup in hilbert space

strong convergence theorems of total quasi φ asymptotically nonexpansive semi groups in banach spaces the purpose of this article is first to introduce the concept of total quasi f asymptotically nonexpansive semigroups which contains many kinds of semi groups as its special cases and then to modify the halpern mann type iteration algorithm for this kind of semi groups to have the strong convergence under a limit condition only in the framework of banach spaces the results presented in the paper improve and extend the corresponding recent results announced by many authors

filtrations and test configurations we introduce a strengthening of k stability based on filtrations of the homogeneous coordinate ring this allows for considering certain limits of families of test configurations which arise naturally in several settings we prove that if a manifold with no automorphisms admits a csck metric then it satisfies this stronger stability notion we also discuss the relation with the birational transformations in the definition of b stability

differential geometry of gerbes abstract we define in a global manner the notion of a connective structure for a gerbe on a space x when the gerbe is endowed with trivializing data with respect to an open cover of x we describe this connective structure in two separate ways which extend from abelian to general gerbes the corresponding descriptions due to j l brylinski and n hitchin we give a global definition of the 3 curvature of this connective structure as a 3 form on x with values in the lie stack of the gauge stack of the gerbe we also study this notion locally in terms of more traditional lie algebra valued 3 forms the bianchi identity which the curvature of a connection on a principal bundle satisfies is replaced here by a more elaborate equation