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

radon nikodým property and thick families of geodesics abstract banach spaces without the radon nikod m property are characterized as spaces containing bilipschitz images of thick families of geodesics defined as follows a family t of geodesics joining points u and v in a metric space is called thick if there is 0 such that for every g t and for any finite collection of points r 1 r n in the image of g there is another u v geodesic g t satisfying the conditions g also passes through r 1 r n and possibly has some more common points with g on the other hand there is a finite collection of common points of g and g which contains r 1 r n and is such that the sum of maximal deviations of the geodesics between these common points is at least

isoparametric functions on exotic spheres abstract this paper extends widely the work in 11 existence and non existence results of isoparametric functions on exotic spheres and eells kuiper projective planes are established in particular every homotopy n sphere n 4 carries an isoparametric function with certain metric with 2 points as the focal set in strong contrast to the classification of cohomogeneity one actions on homotopy spheres 26 only exotic kervaire spheres admit cohomogeneity one actions besides the standard spheres as an application we improve a beautiful result of berard bergery 2 see also pp 234 235 of 3

a characterization of the radon nikodym property abstract it is well known that every bounded below and non increasing sequence in the real line converges we give a version of this result valid in banach spaces with the radon nikodym property thus extending a former result of a prochazka

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

η invariant and a problem of berard bergery on the existence of closed geodesics we use the invariant of atiyah patodi singer to compute the eells kuiper invariant for the eells kuiper quaternionic projective plane by combining with a known result of berard bergery it shows that every eells kuiper quaternionic projective plane carries a riemannian metric such that all geodesics passing through a certain point are simply closed and of the same length

on tensor categories of lie type en n 9 let v be the minuscule representation of the lie algebra g en n 6 7 we show that the centralizer of the action of the quantum group uqg en on v n is generated by the r matrices and one additional element appearing in the n 1 st tensor power a similar description can be given for the classical limit an analogous statement is true for a certain direct summand of v n for uqg en with n 5 n 9 here v is the module whose highest weight is the fundamental weight labeled by the vertex furthest from the triple point of the graph en moreover we obtain a 2 parameter family of braid representations generalizing a q version of the brauer centralizer algebras

a ribbon hopf algebra approach to the irreducible representations of centralizer algebras the brauer birman wenzl and type a iwahori hecke algebras abstract we show how the ribbon hopf algebra structure on the drinfel d jimbo quantum groups of types a b c and d can be used to derive formulas giving explicit realizations of the irreducible representations of the iwahori hecke algebras of type a and the birman wenzl algebras we use this derivation to give explicit realizations of the irreducible representations of the brauer algebras as well the derivation is accomplished by way of a combination of techniques from operator algebras quantum groups and the theory of 3 manifold invariants although our applications are in the cases of the quantum groups of types a b c and d most of the aspects of our approach apply in the general setting of ribbon hopf algebras

classical invariant theory for the quantum symplectic group in this paper we establish the quantum analogues of the first and second fundamental theorem for vector invariants for the symplectic group let us briefly recall this result consider a vector space v endowed with a non degenerate antisymmetric bilinear form let sp v denote the group of isometries the first and second fundamental theorems for vector invariants for sp v tells us that any invariant polynomial function on

ricci curvature of double manifolds via isoparametric foliations abstract given a closed manifold m and a vector bundle of rank n over m by gluing two copies of the disc bundle of we can obtain a closed manifold d m the so called double manifold in this paper we firstly prove that each sphere bundle s r of radius r 0 is an isoparametric hypersurface in the total space of equipped with a connection metric and for r 0 small enough the induced metric of s r has positive ricci curvature under the additional assumptions that m has a metric with positive ricci curvature and n 3 as an application if m admits a metric with positive ricci curvature and n 2 then we construct a metric with positive ricci curvature on d m moreover under the same metric d m admits a natural isoparametric foliation for a compact minimal isoparametric hypersurface y n in s n 1 1 which separates s n 1 1 into s n 1 and s n 1 one can get double manifolds d s n 1 and d s n 1 inspired by tang xie and yan s work on scalar curvature of such manifolds with isoparametric foliations cf 25 we study ricci curvature of them with isoparametric foliations in the last part

isoparametric foliation and yau conjecture on the first eigenvalue a well known conjecture of yau states that the first eigenvalue of every closed minimal hypersurface m n in the unit sphere s n 1 1 is just its dimension n the present paper shows that yau conjecture is true for minimal isoparametric hypersurfaces moreover the more fascinating result of this paper is that the first eigenvalues of the focal submanifolds are equal to their dimensions in the non stable range

linear subsets of nonlinear sets in topological vector spaces for the last decade there has been a generalized trend in mathematics on the search for large algebraic structures linear spaces closed subspaces or infinitely generated algebras composed of mathematical objects enjoying certain special properties this trend has caught the eye of many researchers and has also had a remarkable influence in real and complex analysis operator theory summability theory polynomials in banach spaces hypercyclicity and chaos and general functional analysis this expository paper is devoted to providing an account on the advances and on the state of the art of this trend nowadays known as lineability and spaceability

transcending classical invariant theory let sp2 r sp2 sp be the real symplectic group of rank n let sp denote the two fold cover of sp there is a unitary representation constructed by shale sh and weil wa of sp that is of considerable interest in the theory of automorphic forms we denote this representation by co and call it the oscillator representation the purpose of this paper is to establish about c a fact that should be useful for clarifying the structure of spaces of automorphic forms constructed using co we begin by formulating the result let e be a reductive subgroup of sp and let e denote the inverse image of e in sp denote by r e the set of infinitesimal equivalence classes of continuous irreducible admissible representations of e on locally convex spaces let co be the smooth representation of sp associated to co let co be realized on a hilbert space and let the subspace of smooth vectors on which space co is defined be written denote by r9 e co the set of elements of e which are realized as quotients by co e invariant closed subspaces of t g consider a reductive dual pair g g c sp h2 it is not hard to show that g and g commute with one another hence we may regard co i g g as a representation of gx g it is well known f that r gx g s9 g xr9 g the identification associates to p e r g and p e i g the tensor product p x p we note that p x p is not defined as a topological vector space nevertheless the infinitesimal equivalence class of p op is well defined select p x p e g x g suppose that in fact p p e 9 g g c then clearly p e r g c and p e 9 g c hence g co defines the graph of a correspondence between certain subsets of r9 g cv and j g co in fact the situation is quite precise

infinite games banach space geometry and the eikonal equation we study a class of infinite games which turns out to be closely related to banach space geometry using one of these games we construct a bounded differentiable almost everywhere solution of the eikonal equation u 1 onr d w ithd 2

isoparametric hypersurfaces with g m 6 2 we prove that isoparametric hypersurfaces with g m 6 2 are homogeneous which answers dorfmeister neher s conjecture armatively and solves yau s problem in the case g 6

viscosity solutions to second order partial differential equations on riemannian manifolds we prove comparison uniqueness and existence results for viscosity solutions to a wide class of fully nonlinear second order partial differential equations f x u du d 2 u 0d ef ined on afi nite dimensional riemannian manifold m finest results with hypothesis that require the function f to be degenerate elliptic that is nonincreasing in the second order derivative variable and uniformly continuous with respect to the variable x are obtained under the assumption that m has nonnegative sectional curvature while if one additionally requires f to depend on d 2 u in a uniformly continuous manner then comparison results are established with no restrictive assumptions on curvature

isoparametric hypersurfaces with four principal curvatures let m be an isoparametric hypersurface in the sphere s n with four distinct principal curvatures munzner showed that the four principal curvatures can have at most two distinct multiplicities m1 m 2 and stolz showed that the pair m1 m 2 must either be 2 2 4 5 or be equal to the multiplicities of an isoparametric hypersurface of fkm type constructed by ferus karcher and munzner from orthogonal representations of clifford algebras in this paper we prove that if the multiplicities satisfy m2 2m1 1 then the isoparametric hypersurface m must be of fkm type together with known results of takagi for the case m1 1 and ozeki and takeuchi for m1 2 this handles all possible pairs of multiplicities except for four cases for which the classification problem remains open

isoparametric foliation and a problem of besse on generalizations of einstein condition abstract the focal sets of isoparametric hypersurfaces in spheres with g 4 are all willmore submanifolds being minimal but mostly non einstein 25 22 inspired by a gray s view the present paper shows that these focal sets are all a manifolds but rarely ricci parallel except possibly for the only unclassified case as a byproduct it gives infinitely many simply connected examples to the problem 16 56 i of besse concerning generalizations of the einstein condition

a proof of the smale conjecture diff s3 0 4 the smale conjecture 9 is the assertion that the inclusion of the orthogonal group 0 4 into diff s3 the diffeomorphism group of the 3 sphere with the cw topology is a homotopy equivalence there are many equivalent forms of this conjecture some of which are listed in the appendix to this paper we shall prove

isoparametric foliation and yau conjecture on the first eigenvalue ii abstract this is a continuation of 23 which investigated the first eigenvalues of minimal isoparametric hypersurfaces with g 4 distinct principal curvatures and focal submanifolds in unit spheres for the focal submanifolds with g 6 the present paper obtains estimates on all the eigenvalues among others giving an affirmative answer in one case to the problem posed in 23 which may be regarded as a generalization of yau s conjecture in two of the four unsettled cases in 23 for focal submanifolds m 1 of ot fkm type we prove the first eigenvalues to be their respective dimensions

structure and representations of the quantum general linear supergroup the structure and representations of the quantum general linear supergroup gl q m n are studied systematically by investigating the hopf superalgebra g q of its representative functions g q is factorized into g q pi g q bar pi and a peter weyl basis is constructed for each factor parabolic induction for the quantum supergroup is developed the underlying geometry of induced representations is discussed and an analog of frobenius reciprocity is obtained a quantum borel weil theorem is proven for the irreducible covariant and contravariant tensorial representations and explicit realizations are given for classes of irredducible tensorial representatins in terms of sections of quantum super vector bundles over quantum projective superspaces

differentiable classification of 4 manifolds with singular riemannian foliations in this paper we first prove that any closed simply connected 4 manifold that admits a decomposition into two disk bundles of rank greater than 1 is diffeomorphic to one of the standard elliptic 4 manifolds mathbb s 4 mathbb cp 2 mathbb s 2 times mathbb s 2 or mathbb cp 2 pm mathbb cp 2 as an application we prove that any closed simply connected 4 manifold admitting a nontrivial singular riemannian foliation is diffeomorphic to a connected sum of copies of standard mathbb s 4 pm mathbb cp 2 and mathbb s 2 times mathbb s 2 a classification of singular riemannian foliations of codimension 1 on all closed simply connected 4 manifolds is obtained as a byproduct in particular there are exactly 3 non homogeneous singular riemannian foliations of codimension 1 complementing the list of cohomogeneity one 4 manifolds

exotic spheres and curvature since their discovery by milnor in 1956 exotic spheres have provided a fascinating object of study for geometers in this article we survey what is known about the curvature of exotic spheres

isoparametric foliations diffeomorphism groups and exotic smooth structures abstract in this paper we are concerned with interactions between isoparametric theory and differential topology two foliations are called equivalent if there exists a diffeomorphism between the foliated manifolds mapping leaves to leaves using differential topology we obtain several results towards the classification problem of isoparametric foliations up to equivalence in particular we show that each homotopy n sphere has the same isoparametric foliations as the standard sphere s n has except for n 4 reducing the classification problem on homotopy spheres to that on the standard sphere moreover we prove the uniqueness up to equivalence of isoparametric foliations with two points as the focal submanifolds on each sphere s n except for n 5 besides we show that the uniqueness holds on s 5 if and only if 0 diff s 4 z 2 i e pseudo isotopy implies isotopy for diffeomorphisms on s 4 at last some ideas behind the proofs enable us to discover new exotic smooth structures on certain manifolds

errata of isoparametric hypersurfaces with g m 6 2 we give a correction of the proof of the homogeneity of isoparametric hypersurfaces with g m 6 2

strongly multiplicity free modules for lie algebras and quantum groups abstract let u be either the universal enveloping algebra of a complex semisimple lie algebra g or its drinfel d jimbo quantisation over the field c z of rational functions in the indeterminate z we define the notion of strongly multiplicity free smf for a finite dimensional u module v and prove that for such modules the endomorphism algebras end u v r are generic in the sense that in the classical unquantised case they are quotients of kohno s infinitesimal braid algebra t r while in the quantum case they are quotients of the group ring c z b r of the r string braid group b r in the classical case the generators are generalisations of the quadratic casimir operator c of u while in the quantum case they arise from r matrices which may be thought of as square roots of a quantum analogue of c in a completion of u r this unifies many known results and brings some new cases into their context these include the irreducible 7 dimensional module in type g 2 and arbitrary irreducibles for sl 2 the work leads naturally to questions concerning non semisimple deformations of the relevant endomorphism algebras which arise when the ground rings are varied

a diffeomorphism classification of manifolds which are like projective planes we give a complete diffeomorphism classification of 1 connected closed manifolds m with integral homology h m z z z provided that dim m 6 4

a quantum analogue of the first fundamental theorem of classical invariant theory we establish a noncommutative analogue of the first fundamental theorem of classical invariant theory for each quantum group associated with a classical lie algebra we construct a noncommutative associative algebra whose underlying vector space forms a module for the quantum group and whose algebraic structure is preserved by the quantum group action the subspace of invariants is shown to form a subalgebra which is finitely generated we determine generators of this subalgebra of invariants and determine their commutation relations in each case considered the noncommutative modules we construct are flat deformations of their classical commutative analogues our results are therefore noncommutative generalisations of the first fundamental theorem of classical invariant theory which follows from our results by taking the limit as q 1 our method similarly leads to a definition of quantum spheres which is a noncommutative generalisation of the classical case with orthogonal quantum group symmetry

moment maps of the spin action and the cartan munzner polynomials of degree four we express all the known cartan munzner polynomials of degree four in terms of the moment map of certain group actions