the classification of differential structures on quantum 2 spheres exterior algebras of differential forms on quantum 2 spheress qc 2 q 1 1 0 c 0 c 0 forq 1 are classified in the definition of exterior algebras we assume the invariance w r t the action of the quantumsu 2 group and dimensionality conditions which imply that we deal with two dimensional manifolds the exterior algebras exist only forc 0 and are unique in that case the corresponding generalized directional derivatives are provided

uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory we consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn x dx e nv x dx on the line as n the potentials v are assumed to be real analytic with sufficient growth at infinity the principle results concern plancherel rotach type asymptotics for the orthogonal polynomials down to the axis using these asymptotics we then prove universality for a variety of statistical quantities arising in the theory of random matrix models some of which have been considered recently in 31 and also in 4 an additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials see also 4 the orthogonal polynomial problem is formulated as a riemann hilbert problem following 19 20 the riemann hilbert problem is analyzed in turn using the steepest descent method introduced in 12 and further developed in 11 13 a critical role in our method is played by the equilibrium measure d v for v as analyzed in 8 1999 john wiley sons inc

semidirect products of c quantum groups multiplicative unitaries approach c quantum groups with projection are the noncommutative analogues of semidirect products of groups radford s theorem about hopf algebras with projection suggests that any c quantum group with projection decomposes uniquely into an ordinary c quantum group and a braided c quantum group we establish this on the level of manageable multiplicative unitaries

the asymptotics of monotone subsequences of involutions we compute the limiting distributions of the lengths of the longest monotone subsequences of random signed involutions with or without conditions on the number of fixed points and negated points as the sizes of the involutions tend to infinity the resulting distributions are depending on the number of fixed points 1 the tracy widom distributions for the largest eigenvalues of random goe gue gse matrices 2 the normal distribution or 3 new classes of distributions which interpolate between pairs of the tracy widom distributions we also consider the second rows of the corresponding young diagrams in each case the convergence of moments is also shown the proof is based on the algebraic work of j baik and e rains in 7 which establishes a connection between the statistics of random involutions and a family of orthogonal polynomials and an asymptotic analysis of the orthogonal polynomials which is obtained by extending the riemann hilbert analysis for the orthogonal polynomials by p deift k johansson and baik in 3

quantization of the poisson su 2 and its poisson homogeneous space the 2 sphere we show that deformation quantizations of the poisson structures on the poisson lie groupsu 2 and its homogeneous space the 2 sphere are compatible with woronowicz s deformation quantization ofsu 2 s group structure and podles deformation quantization of 2 sphere s homogeneous structure respectively so in a certain sense the multiplicativity of the lie poisson structure onsu 2 at the classical level is preserved under quantization

shape fluctuations and random matrices we study a certain random growth model in two dimensions closely related to the one dimensional totally asymmetric exclusion process the results show that the shape fluctuations appropriately scaled converges in distribution to the tracy widom largest eigenvalue distribution for the gaussian unitary ensemble gue

on the distribution of the length of the longest increasing subsequence of random permutations let sn be the group of permutations of 1 2 n if 7r e sn we say that 7 i1 7f ik is an increasing subsequence in 7r if il i2 ik and 7r ii 7r i2 7r ik let 1n r be the length of the longest increasing subsequence for example if n 5 and 7r is the permutation 5 1 3 2 4 in one line notation thus 7r 1 5 7r 2 1 then the longest increasing subsequences are 1 2 4 and 1 3 4 and n 3 equip sn with uniform distribution

random vicious walks and random matrices a lock step walk is a one dimensional integer lattice walk in discrete time suppose that initially there are infinitely many walkers on the nonnegative even integer sites at each moment of time every walker moves either to its left or to its right with equal probability the only constraint is that no two walkers can occupy the same site at the same time hence we describe the walk as vicious it is proved that as time tends to infinity a certain limiting conditional distribution of the displacement of the leftmost walker is identical to the limiting distribution of the scaled largest eigenvalue of a random goe matrix goe tracy widom distribution the proof is based on the bijection between path configurations and semistandard young tableaux established recently by guttmann owczarek and viennot the distribution of semistandard young tableaux is analyzed using the hankel determinant expression for the probability obtained from the work of rains and the author the asymptotics of the hankel determinant are then obtained by applying the deift zhou steepest descent method to the riemann hilbert problem for the related orthogonal polynomials 2000 john wiley sons inc

limit theorems for height fluctuations in a class of discrete space and time growth models we introduce a class of one dimensional discrete space discrete time stochastic growth models described by a height function ht x with corner initialization we prove with one exception that the limiting distribution function of ht x suitably centered and normalized equals a fredholm determinant previously encountered in random matrix theory in particular in the universal regime of large x and large t the limiting distribution is the fredholm determinant with airy kernel in the exceptional case called the critical regime the limiting distribution seems not to have previously occurred the proofs use the dual rsk algorithm gessel s theorem the borodin okounkov identity and a novel rigorous saddle point analysis in the fixed x large t regime we find a brownian motion representation this model is equilvalent to the seppalainen johansson model hence some of our results are not new but the proofs are

unbounded elements affiliated with c algebras and non compact quantum groups the affiliation relation that allows to include unbounded elements operators into thec algebra framework is introduced investigated and applied to the quantum group theory the quantum deformation of the two fold covering of the group of motions of euclidean plane is constructed a remarkable radius quantization is discovered it is also shown that the quantumsu 1 1 group does not exist on thec algebra level for real value of the deformation parameter

random walks and random permutations a connection is made between the random turns model of vicious walkers and random permutations indexed by their increasing subsequences consequently the scaled distribution of the maximum displacements in a particular asymmeteric version of the model can be determined to be the same as the scaled distribution of the eigenvalues at the soft edge of the gue random hermitian matrices the scaling of the distribution gives the maximum mean displacement after t time steps as 2t 1 2 with standard deviation proportional to 1 3 the exponent 1 3 is typical of a large class of two dimensional growth problems

longest increasing subsequences from patience sorting to the baik deift johansson theorem we describe a simple one person card game patience sorting its analysis leads to a broad circle of ideas linking young tableaux with the longest increasing subsequence of a random permutation via the schensted correspondence a recent highlight of this area is the work of baik deift johansson which yields limiting probability laws via hard analysis of toeplitz determinants

symmetries of quantum spaces subgroups and quotient spaces of quantum su 2 and so 3 groups we prove that each action of a compact matrix quantum group on a compact quantum space can be decomposed into irreducible representations of the group we give the formula for the corresponding multiplicities in the case of the quotient quantum spaces we describe the subgroups and the quotient spaces of quantumsu 2 andso 3 groups

random words toeplitz determinants and integrable systems ii this paper connects the analysis of the length of the longest weakly increasing subsequence of inhomogeneous random words to a riemnn hilbert problem and an associated system of integrable partial differential equations in particular we show that the poissonization of the distribution function of this length can be identified as the jimbo miwa ueno tau function

podleś spheres for the braided quantum su 2 abstract starting with the braided quantum group su q 2 for a complex deformation parameter q we perform the construction of the quotient su q 2 t which serves as a model of a quantum sphere then we follow the reasoning of podle who for real q classified quantum spaces with the action of su q 2 with appropriate spectral properties these properties can also be expressed in the context of the braided quantum su q 2 with complex q and we find that they lead to precisely the same family of quantum spaces as found by podle for the real parameter q

quantum spheres and projective spaces as graph algebras the c algebras of continuous functions on quantum spheres quantum real projective spaces and quantum complex projective spaces are realized as cuntz krieger algebras corresponding to suitable directed graphs structural results about these quantum spaces especially about their ideals and k theory are then derived from the general theory of graph algebras it is shown that the quantum even and odd dimensional spheres are produced by repeated application of a quantum double suspension to two points and the circle respectively

asymptotic values for degrees associated with strips of young diagrams abstract the sums s l n occur in the representations of the symmetric and the general linear groups in combinatorics and in pi algebras we give asymptotic values for these sums

a priori bounds for co dimension one isometric embeddings let x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i g inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i be a c 4 isometric embedding of a c4 metric g of nonnegative sectional curvature on inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i into the euclidean space inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 04i we prove a priori bounds for the trace of the second fundamental form h in terms of the scalar curvature r of g and the diameter d of the space inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 05i g these estimates give a bound on the extrinsic geometry in terms of intrinsic quantities they generalize estimates originally obtained by weyl for the case n 2 and positive curvature and then by p guan and the first author for nonnegative curvature and n 2 using c 2 interior estimates of evans and krylov for concave fully nonlinear elliptic partial differential equations these bounds allow us to obtain the following convergence theorem for any 0 the set of metrics of nonnegative sectional curvature and scalar curvature bounded below by which are isometrically embedable in euclidean space inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 06i is closed in the holder space c 4 0 suppose that g is a smooth metric of nonnegative sectional curvature and positive scalar curvature on inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 07i which admits locally convex isometric embedding into inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 08i does inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 09i g then admit a smooth global isometric embedding x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 10i g inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 11i

examples of braided groups and braided matrices matrix braided groups are developed as an analog of the coordinate functions on a group or supergroup the 1 in the super case is replaced by braid statistics there are braided group analogs of all the classical simple lie groups as well as braided matrix groups and braided matrices b r for every regular solution r of the quantum yang baxter equations a direct verification of b r is provided and some of the simplest examples are computed in detail

on the solvability of painleve ii and iv we introduce a rigorous methodology for studying the riemann hilbert problems associated with certain integrable nonlinear ordinary differential equations for concreteness we investigate the painleve ii and painleve iv equations we show that the cauchy problems for these equations admit in general global meromorphic in t solutions furthermore for special relations among the monodromy data and for t on stokes lines these solutions are bounded for finite t

closed quantum subgroups of locally compact quantum groups abstract we investigate the fundamental concept of a closed quantum subgroup of a locally compact quantum group two definitions one due to s vaes and one due to s l woronowicz are analyzed and relations between them discussed among many reformulations we prove that the former definition can be phrased in terms of quasi equivalence of representations of quantum groups while the latter can be related to an old definition of podle from the theory of compact quantum groups the cases of classical groups duals of classical groups compact and discrete quantum groups are singled out and equivalence of the two definitions is proved in the relevant context a deep relationship with the quantum group generalization of the herz restriction theorem from classical harmonic analysis is also established in particular in the course of our analysis we give a new proof of the herz restriction theorem

quantum groups with projection on von neumann algebra level abstract we introduce an axiomatization of the notion of a semidirect product of locally compact quantum groups and study their properties our approach is slightly different from the one introduced in the thesis of s roy 16 and unlike the investigations of roy we work within the von neumann algebraic picture this allows the use of powerful techniques related to crossed products by actions of locally compact quantum groups in particular we show existence of a braided comultiplication on the algebra spanned by slices of braided multiplicative unitary

quantum and braided linear algebra quantum matrices a r are known for every r matrix obeying the quantum yang baxter equations it is also known that these act on vectors given by the corresponding zamalodchikov algebra this interpretation is developed in detail distinguishing between two forms of this algebra v r vectors and v r covectors a r v r21 v r is an algebra homomorphism i e quantum matrices are realized by the tensor product of a quantum vector with a quantum covector while the inner product of a quantum covector with a quantum vector transforms as a scaler it is shown that if v r and v r are endowed with the necessary braid statistics then their braided tensor product v r v r is a realization of the braided matrices b r introduced previously while their inner product leads to an invariant quantum trace introducing braid statistics in this way leads to a fully covariant quantum braided linear algebra the braided groups obtained from b r act on themselves by conjugation in a way impossible

examples of non compact quantum group actions abstract we present two examples of actions of non regular locally compact quantum groups on their homogeneous spaces the homogeneous spaces are defined in a way specific to these examples but the definitions we use have the advantage of being expressed in purely c algebraic language we also discuss continuity of the obtained actions finally we describe in detail a general construction of quantum homogeneous spaces obtained as quotients by compact quantum subgroups

asymptotics of plancherel measures for symmetric groups 1 1 plancherel measures given a finite group g by the corresponding plancherel measure we mean the probability measure on the set g of irreducible representations of g which assigns to a representation g the weight dim g for the symmetric group s n the set s n is the set of partitions of the number n which we shall identify with young diagrams with n squares throughout this paper the plancherel measure on partitions arises naturally in representation theoretic combinatorial and probabilistic problems for example the plancherel distribution of the first part of a partition coincides with the distribution of the longest increasing subsequence of a uniformly distributed random permutation 31 we denote the plancherel measure on partitions of n by mn

random unitary matrices permutations and painleve this paper is concerned with certain connections between the ensemble of n n unitary matrices specifically the characteristic function of the random variable tr u and combinatorics specifically ulam s problem concerning the distribution of the length of the longest increasing subsequence in permutation groups and the appearance of painleve functions in the answers to apparently unrelated questions among the results is a representation in terms of a painleve v function for the characteristic function of tr u and using recent results of baik deift and johansson an expression in terms of a painleve ii function for the limiting distribution of the length of the longest increasing subsequence in the hyperoctahedral groups