superintegrable systems on the two dimensional sphere s2 and the hyperbolic plane h2 the existence of superintegrable systems with n 2 degrees of freedom possessing three independent globally defined constants of motion which are quadratic in the velocities is studied on the two dimensional sphere s2 and on the hyperbolic plane h2 the approach used is based on enforcing the conditions for the existence of two independent integrals further than the energy this is done in a way which allows us to discuss at once the cases of the sphere s2 and the hyperbolical plane h2 by considering the curvature as a parameter different superintegrable potentials are obtained as the solutions of certain systems of two dependent second order partial differential equations the euclidean results are directly recovered for 0 and the superintegrable potentials on either the standard unit sphere radius r 1 or the unit lobachewski plane radius r 1 appear as the particular values of the dependent superintegrable potentials for the values 1 and 1 some new superintegrable potentials are f

counting representations of quivers over finite fields abstract by counting the numbers of isomorphism classes of representations indecomposable or absolutely indecomposable of quivers over finite fields with fixed dimension vectors we obtain a multi variable formal identity if the quiver has no edge loops this identity turns out to be a q analogue of the kac denominator identity modulus a conjecture of kac

arithmetic harmonic analysis on character and quiver varieties ii we study connections between the topology of generic character varieties of fundamental groups of punctured riemann surfaces macdonald polynomials quiver representations hilbert schemes on c x x c x modular forms and multiplicities in tensor products of irreducible characters of finite general linear groups c 2012 elsevier inc all rights reserved

exact solvability of superintegrable systems it is shown that all four superintegrable quantum systems on the euclidean plane possess the same underlying hidden algebra sl 3 the gauge rotated hamiltonians as well as their integrals of motion once rewritten in appropriate coordinates preserve a flag of polynomials this flag corresponds to highest weight finite dimensional representations of the sl 3 algebra realized by first order differential operators

decoupling inequalities for the tail probabilities of multivariate u statistics in this paper we present a decoupling inequality that shows that multivariate u statistics can be studied as sums of conditionally independent random variables this result has important implications in several areas of probability and statistics including the study of random graphs and multiple stochastic integration more precisely we get the following result let x j be a sequence of independent random variables on a measurable space mathscr j s and let x j i j 1 ldots k be k independent copies of x i let f i 1i 2 ldots i k be families of functions of k variables taking s times cdots times s into a banach space b cdots then for all n geq k geq 2 t 0 there exist numerical constants c k depending on k only so that p bigg big sum 1 leq i 1 neq i 2 neq cdots neq i k leq n f i 1 cdots i k x 1 i 1 x 1 i 2 ldots x 1 i k big geq t bigg leq c kp bigg c k big sum 1 leq i 1 neq i 2 neq cdots neq i k leq n f i 1 cdots i k x 1 i 1 x 2 i 2 ldots x k i k big geq t bigg the reverse bound holds if in addition the following symmetry condition holds almost surely f i 1i 2 cdots i k x i 1 x i 2 ldots x i k f i pi 1 i pi 2 cdots i pi k x i pi 1 x i pi 2 ldots x i pi k for all permutations pi of 1 ldots k

high precision calculation of the eigenvalues for the x2 λx2 1 gx2 potential an analytic continuation procedure using a taylor series is used to produce very accurate wavefunctions and eigenvalues for the one dimensional anharmonic oscillator governed by the potential v x x2 x2 1 gx2

virtual classes of parabolic sl2 c character varieties abstract in this paper we compute the virtual classes in the grothendieck ring of algebraic varieties of sl 2 c character varieties over compact orientable surfaces with parabolic points of semi simple type when the parabolic punctures are chosen to be semi simple non generic we show that a new interaction phenomenon appears generating a recursive pattern

generalized quantum nonlinear oscillators exact solutions and rational extensions we construct exact solutions and rational extensions to quantum systems of generalized nonlinear oscillators our method is based on a connection between nonlinear oscillator systems and schrodinger models for certain hyperbolic potentials the rationally extended models admit discrete spectrums and corresponding closed form solutions are expressed through jacobi type xm exceptional orthogonal polynomials

stochastic calculus with anticipating integrands we study the stochastic integral defined by skorohod in 24 of a possibly anticipating integrand as a function of its upper limit and establish an extended ito formula we also introduce an extension of stratonovich s integral and establish the associated chain rule in all the results the adaptedness of the integrand is replaced by a certain smoothness requirement

exact solutions of the schrodinger equation d dx2 x2 λx2 1 gx2 ψ x eψ x the authors prove the existence of a class of exact eigenvalues and eigenfunctions of the schrodinger equation for the potential x2 x2 1 gx2 when certain algebraic relations between and g hold some of the properties of these solutions are discussed it is shown that in a certain sense they may be regarded as sturmians for the schrodinger equation with the potential x2 g g2x2

the schrodinger equation for the x2 λx2 1 gx2 interaction certain exact odd and even wavefunctions for the potential x2 x2 1 gx2 x in are obtained the coefficients of the wavefunctions and expressions for the energies are obtained using the symbolic manipulation code reduce the chief use of these results is to provide a check for the effectiveness of numerical algorithms the results supplement those of flessas 1982 for the same potential on xe 0

stochastic evolution equations in umd banach spaces we discuss existence uniqueness and space time holder regular ity for solutions of the parabolic stochastic evolution equation du t au t f t u t dt b t u t dwh t t 2 0 t0 u 0 u0 where a generates an analytic c0 semigroup on a umd banach space e and wh is a cylindrical brownian motion with values in a hilbert space h we prove that if the mappings f 0 t e e and b 0 t e l h e satisfy suitable lipschitz conditions and u0 is f0 measurable and bounded then this problem has a unique mild solution which has trajectories in c 0 t d a provided 0 and 0 satisfy 1 2 vari ous extensions of this result are given and the results are applied to parabolic stochastic partial dierential equations

quasi sure analysis and stratonovich anticipative stochastic differential equations consider a stochastic differential equation on d with smooth and bounded coefficients we apply the techniques of the quasi sure analysis to show that this equation can be solved pathwise out of a slim set furthermore we can restrict the equation to the level sets of a nondegenerate and smooth random variable and this provides a method to construct the solution to an anticipating stochastic differential equation with smooth and nondegenerate initial condition

families of quasi exactly solvable extensions of the quantum oscillator in curved spaces we introduce two new families of quasi exactly solvable qes extensions of the oscillator in a d dimensional constant curvature space for the first three members of each family we obtain closed form expressions of the energies and wavefunctions for some allowed values of the potential parameters using the bethe ansatz method we prove that the first member of each family has a hidden sl 2 symmetry and is connected with a qes equation of the first or second type respectively one dimensional results are also derived from the d dimensional ones with d 2 thereby getting qes extensions of the mathews lakshmanan nonlinear oscillator

holomorphic semi groups and the geometry of banach spaces according to a fundamental theorem of dvoretzky cf 24 25 any infinite dimensional banach space x contains for each integer n and each e 0 a subspace xn which is 1 e isomorphic to 12n in the classical banach spaces lp and 1p with 1 p x it is well known that this result can be improved these spaces contain uniformly complemented ln 7 s this means that we can find subspaces xn as above with the additional property that there exist projections p x xn such that supn 11 pn 11 cx moreover it is also well known that this property no longer holds for 11 or lo this motivated the question cf 13 to determine which infinite dimensional banach spaces contain uniformly complemented 1 n s we will prove below that all uniformly convex spaces have this property moreover we will show that a banach space x verifies this property locally see definition 2 10 for more precision if and only if x does not contain l n s uniformly progressively the work of previous authors cf 14 12 and particularly 5 has reduced such so called geometric questions to a purely analytic problem consisting of verifying in x a certain inequality referred to as k convexity in this paper we connect this inequality with the holomorphy of a certain semi group of operators on lp x and we show how this leads to the results already mentioned

decoupling inequalities for polynomial chaos let x x 1 ldots x d be a sequence of independent symmetric identically distributed random vectors with independent components the main subject of this paper is the so called decoupling inequalities i e inequalities of the form begin align e phi cq x x ldots x leq e phi q x 1 x 2 ldots x d leq e phi cq x x ldots x end align where q is a symmetric multilinear form with values in a vector space f with all diagonal terms equal to zero and phi is a convex function on f

malliavin calculus for infinite dimensional systems with additive noise abstract we consider an infinite dimensional dynamical system with polynomial nonlinearity and additive noise given by a finite number of wiener processes by studying how randomness is spread by the dynamics we develop in this setting a partial counterpart of hormander s classical theory of hypoelliptic operators we study the distributions of finite dimensional projections of the solutions and give conditions that provide existence and smoothness of densities of these distributions with respect to the lebesgue measure we also apply our results to concrete spdes such as a stochastic reaction diffusion equation and the stochastic 2d navier stokes system

sobolev spaces of banach valued functions associated with a markov process we discuss sobolev spaces of banach valued functions they are extensions of sobolev spaces of scalar functions we use a gamma transform of a semigroup associated with a markov process a typical example is the ornstein uhlenbeck process on the wiener space

heisenberg algebra and hilbert schemes of points on projective surfaces the purpose of this paper is to throw a bridge between two seemingly unrelated subjects one is the hilbert scheme of points on projective surfaces which has been intensively studied by various people see e g i br es g61 g62 the other is the infinite dimensional heisenberg algebra which is closely related to affine lie algebras see e g k we shall construct a representation of the heisenberg algebra on the homology group of the hilbert scheme in other words the homology group will become a fock space the basic idea is to introduce certain correspondences in the product of the hilbert scheme then they define operators on the homology group by a well known procedure they give generators of the heisenberg algebra and the only thing we must check is that they satisfy the defining relation here we remark that the components of the hilbert scheme are parameterized by numbers of points and our representation will be constructed on the direct sum of homology groups of all components our correspondences live in the product of the different components thus it is quite essential to study all components together our construction has the same spirit as the author s construction nal na4 of representations of affine lie algebras on homology groups of moduli spaces of instantons 1 on ale spaces which are minimal resolutions of simple singularities certain correspondences called hecke correspondences were used to define operators these twist instantons along curves irreducible components of the exceptional set while ours twist ideals around points in fact the hilbert scheme of points can be considered as the moduli space of rank 1 vector bundles or more precisely torsion free sheaves our construction should be considered as a first step to extend nal na4 to general 4 manifolds the same program was also proposed by ginzburg kapranov and vasserot gkv

an accurate method for numerical calculations in quantum mechanics an accurate method for numerical calculations of matrix elements and for solving the eigenvalue problem in quantum mechanics is presented methods for numerical interpolation differentiation and integration provide 15 digit accuracy with double precision arithmetic operations a method for solution of the eigenvalue problem of an ordinary differential equation by using discretization and matrix eigenvalue methods provides 13 to 15 digit accuracy the efficiency of the proposed methods is demonstrated by the applications to bound states for the linear harmonic oscillator anharmonic oscillators the morse potential and the modified poschl teller potential

on the divisibility of hom γ g by g abstract we extend and reformulate a result of solomon on the divisibility of the title we show for example that if is a finitely generated group then g divides hom g for every finite group g if and only if has infinite abelianization as a consequence we obtain some arithmetic properties of the number of subgroups of a given index in such a group

quasi sure analysis of stochastic flows and banach space valued smooth functionals on the wiener space abstract we prove meyer s inequalities for functionals on the wiener space that take values on a banach space belonging to the burkbolder u m d class as an application we analyse the quasi sure regularity in time of the stochastic flow of diffeomorphisms generated by a stochastic differential equation with smooth coefficients

the mckay correspondence as an equivalence of derived categories the classical mckay correspondence relates representations of a finite subgroup g sl 2 c to the cohomology of the well known minimal resolution of the kleinian singularity c2 g gonzalez sprinberg and verdier 10 interpreted the mckay correspondence as an isomorphism on k theory observing that the representation ring of g is equal to the g equivariant k theory of c2 more precisely they identify a basis of the k theory of the resolution consisting of the classes of certain tautological sheaves associated to the irreducible representations of g

counting quiver representations over finite fields via graph enumeration let be a quiver on n vertices v1 v2 vn with gij edges between vi and vj and let nn hua gave a formula for a q the number of isomorphism classes of absolutely indecomposable representations of over the finite field fq with dimension vector kac showed that a q is a polynomial in q with integer coefficients using hua s formula we show that for each integer s 0 the sth derivative of a q with respect to q when evaluated at q 1 is a polynomial in the variables gij and we compute the highest degree terms in this polynomial our formulas for these coefficients depend on the enumeration of certain families of connected graphs

generalized double affine hecke algebras of rank 1 and quantized del pezzo surfaces abstract let d be a simply laced dynkin diagram of rank r whose affinization has the shape of a star i e d 4 e 6 e 7 e 8 to such a diagram one can attach a group g whose generators correspond to the legs of the affinization have orders equal to the leg lengths plus 1 and the product of the generators is 1 the group g is then a 2 dimensional crystallographic group g z l z 2 where l is 2 3 4 and 6 respectively in this paper we define a flat deformation h t q of the group algebra c g of this group by replacing the relations saying that the generators have prescribed orders by their deformations saying that the generators satisfy monic polynomial equations of these orders with arbitrary roots which are deformation parameters the algebra h t q for d 4 is the cherednik algebra of type c c 1 which was studied by noumi sahi and stokman and controls askey wilson polynomials we prove that h t q is the universal deformation of the twisted group algebra of g and that this deformation is compatible with certain filtrations on c g we also show that if q is a root of unity then for generic t the algebra h t q is an azumaya algebra and its center is the function algebra on an affine del pezzo surface for generic q the spherical subalgebra e h t q e provides a quantization of such surfaces we also discuss connections of h t q with preprojective algebras and painleve vi

a geometric approach to a generalized virial theorem the virial theorem introduced by clausius in the field of statistical mechanics and later applied in both classical mechanics and quantum mechanics is studied by making use of symplectic formalism as an approach in the case of both the hamiltonian and lagrangian systems the possibility of establishing virial like theorems from one parameter groups of non strictly canonical transformations is analysed the case of systems with a position dependent mass is also discussed using the modern symplectic approach to quantum mechanics we arrive at the quantum virial theorem in full analogy with the classical case

potential r2 λr2 1 gr2 and the analytic continued fractions the author constructs the exact wavefunctions and an analytic continued fractional green function for the one and three dimensional schrodinger equation with the potential v r h r2 r2 lambda r2 1 gr2 h 1 4 lambda g 0 in the numerical application of the formulae the energy and norm of the boundary state psi may be approximated by both their lower and upper estimates

generalized nonlinear oscillators with quasi harmonic behaviour classical solutions the classical nonlinear oscillator proposed by mathews and lakshmanan q appl math 32 215 1974 and including a position dependent mass in the kinetic energy term is generalized in two different ways by adding an extra term to the potential the solutions of the euler lagrange equation are shown to exhibit richer behaviour patterns than those of the original nonlinear oscillator

exact and quasiexact solvability of second order superintegrable quantum systems ii relation to separation of variables we make explicit the intimate relationship between quasiexact solvability as expounded for example by ushveridze quasi exactly solvable models in quantum mechanics iop bristol 1993 and the technique of separation of variables as it applies to specific superintegrable quantum hamiltonians it is the multiseparability of superintegrable systems that forces the existence of interesting families of polynomial solutions characteristic of quasiexact solvability that enables us to solve these systems in distinct ways and that gives us the basis of a classification theory this connection is generalized in terms of the understanding of the role of finite solutions of quantum hamiltonians

torus orbits on homogeneous varieties and kac polynomials of quivers in this paper we prove that the counting polynomials of certain torus orbits in products of partial flag varieties coincides with the kac polynomials of supernova quivers which arise in the study of the moduli spaces of certain irregular meromorphic connections on trivial bundles over the projective line we also prove that these polynomials can be expressed as a specialization of tutte polynomials of certain graphs providing a new way to compute them we also discuss connections with the gel fand macpherson correspondence

multiplicativity of perverse filtration for hilbert schemes of fibered surfaces abstract let s c be a smooth projective surface with numerically trivial canonical bundle fibered onto a curve we prove the multiplicativity of the perverse filtration with respect to the cup product on h s n q for the natural morphism s n c n we also prove the multiplicativity for five families of hitchin systems obtained in a similar way and compute the perverse numbers of the hitchin moduli spaces we show that for small values of n the perverse numbers match the predictions of the numerical version of the de cataldo hausel migliorini p w conjecture and of the conjecture by hausel letellier and rodriguez villegas

eigenenergies of the r2 lambda r2 1 gr2 and 1 2 r2 or gr4 1 g alpha r2 potentials obtained by the hypervirial method the energy levels of the schrodinger equation involving the potentials v r r2 lambda r2 1 gr2 and v or r 1 2 r2 or gr4 1 g alpha r2 are calculated by using hypervirial and pade approximant methods

indecomposable vector bundles and stable higgs bundles over smooth projective curves we prove that the number of geometrically indecomposable vector bundles of xed rank r and degree d over a smooth projective curve x dened over a nite eld is given by a polynomial depending only on r d and the genus g of x in the weil numbers of x we provide a closed formula expressed in terms of generating series for this polynomial we also show that the same polynomial computes the number of points of the moduli space of stable higgs bundles of rank r and degree d over x this entails a closed formula for the poincar e polynomial of the moduli spaces of stable higgs bundles over a compact riemann surface and hence also for the poincar e polynomials of the twisted character varieties for the groups gl r

the deligne simpson problem a survey abstract the deligne simpson problem dsp respectively the weak dsp is formulated like this give necessary and sufficient conditions for the choice of the conjugacy classes c j gl n c or c j gl n c so that there exist irreducible respectively with trivial centralizer p 1 tuples of matrices m j c j or a j c j satisfying the equality m 1 m p 1 i or a 1 a p 1 0 the matrices m j and a j are interpreted as monodromy operators of regular linear systems and as matrices residua of fuchsian ones on riemann s sphere the present paper offers a survey of the results known up to now concerning the dsp

kac s conjecture from nakajima quiver varieties we prove a generating function formula for the betti numbers of nakajima quiver varieties we prove that it is a q deformation of the weyl kac character formula in particular this implies that the constant term of the polynomial counting the number of absolutely indecomposable representations of a quiver equals the multiplicity of a certain weight in the corresponding kac moody algebra which was conjectured by kac in 1982

mixed hodge polynomials of character varieties we calculate the e polynomials of certain twisted gl n character varieties mathcal m n of riemann surfaces by counting points over finite fields using the character table of the finite group of lie type text gl n mathbb f q and a theorem proved in the appendix by n katz we deduce from this calculation several geometric results for example the value of the topological euler characteristic of the associated pgl n character variety the calculation also leads to several conjectures about the cohomology of mathcal m n an explicit conjecture for its mixed hodge polynomial a conjectured curious hard lefschetz theorem and a conjecture relating the pure part to absolutely indecomposable representations of a certain quiver we prove these conjectures for n 2

vertex operators and character varieties abstract we prove some combinatorial conjectures extending those proposed in 13 14 the proof uses a vertex operator due to nekrasov okounkov and the first author 4 to obtain a gluing formula for the relevant generating series essentially reducing the computation to the case of cp 1 with three punctures

decoupling inequalities for multilinear forms in independent symmetric random variables on considere x 1 x 2 des copies independantes d une suite x x 1 x 2 de variables aleatoires symetriques independantes et m une forme multilineaire symetrique de rang s sur r n dont les composantes a i1 a is relatives a la base standard de r n satisfont a i1 i s 0

stochastic integration in umd banach spaces in this paper we construct a theory of stochastic integration of processes with values in l h e where h is a separable hilbert space and e is a umd banach space i e a space in which martingale differences are unconditional the integrator is an h cylindrical brownian motion our approach is based on a two sided l p decoupling inequality for umd spaces due to garling which is combined with the theory of stochastic integration of l h e valued functions introduced recently by two of the authors we obtain various characterizations of the stochastic integral and prove versions of the ito isometry the burkholder davis gundy inequalities and the representation theorem for brownian martingales

integrable potentials on spaces with curvature from quantum groups a family of classical integrable systems defined on a deformation of the two dimensional sphere hyperbolic and anti de sitter spaces is constructed through hamiltonians defined on the non standard quantum deformation of a sl 2 poisson coalgebra all these spaces have a non constant curvature that depends on the deformation parameter z as particular cases the analogues of the harmonic oscillator and kepler coulomb potentials on such spaces are proposed another deformed hamiltonian is also shown to provide superintegrable systems on the usual sphere hyperbolic and anti de sitter spaces with a constant curvature that exactly coincides with z according to each specific space the resulting potential is interpreted as the superposition of a central harmonic oscillator with either two more oscillators or centrifugal barriers the non deformed limit z 0 of all these hamiltonians can then be regarded as the zero curvature limit contraction which leads to the corresponding super integrable systems on the flat euclidean and minkowskian spaces

study of a class of non polynomial oscillator potentials we develop a variational method to obtain accurate bounds for the eigenenergies of h v in arbitrary dimensions n 1 where v r is the nonpolynomial oscillator potential the variational bounds are compared with results previously obtained in the literature an infinite set of exact solutions is also obtained and used as a source of comparison eigenvalues

exact analytical eigenfunctions for the x2 λ x2 1 gx2 interaction the author discusses an ansatz for the eigenfunctions of the x2 x2 1 gx2 interaction introduced by blecher and leach 1987 the author shows that the ansatz provides pairs of solutions in five cases having and g connected by 6g2 4g and conjectures the possible existence of an infinite number of such solution pairs

parabolic refined invariants and macdonald polynomials a string theoretic derivation is given for the conjecture of hausel letellier and rodriguez villegas on the cohomology of character varieties with marked points their formula is identified with a refined bps expansion in the stable pair theory of a local root stack generalizing previous work of the first two authors in collaboration with pan haiman s geometric construction for macdonald polynomials is shown to emerge naturally in this context via geometric engineering in particular this yields a new conjectural relation between macdonald polynomials and refined local orbifold curve counting invariants the string theoretic approach also leads to a new spectral cover construction for parabolic higgs bundles in terms of holomorphic symplectic orbifolds

on the x2 λx2 1 gx2 interaction presents an infinite set of exact solutions of odd parity for the one dimensional schrodinger equation with the interaction x2 lambda x2 1 gx2 these complement the exact solutions of even parity that have been obtained recently by flessas phys lett a vol 84 p 121 1981

cranks and t cores new statistics on partitions calledcranks are defined which combinatorially prove ramanujan s congruences for the partition function modulo 5 7 11 and 25 explicit bijections are given for the equinumerous crank classes the cranks are closely related to thet core of a partition usingq series some explicit formulas are given for the number of partitions which aret cores some related questions for self conjugate and distinct partitions are discussed

poincare polynomials of moduli spaces of higgs bundles and character varieties no punctures using our earlier results on polynomiality properties of plethystic logarithms of generating series of certain type we show that schiffmann s formulas for various counts of higgs bundles over finite fields can be reduced to much simpler formulas conjectured by mozgovoy in particular our result implies the conjecture of hausel and rodriguez villegas on the poincare polynomials of twisted character varieties and the conjecture of hausel and thaddeus on independence of e polynomials on the degree

deformed shape invariance symmetry and potentials in curved space with two known eigenstates we consider two families of extensions of the oscillator in a d dimensional constant curvature space and analyze them in a deformed supersymmetric framework wherein the starting oscillator is known to exhibit a deformed shape invariance property we show that the first two members of each extension family are also endowed with such a property provided some constraint conditions relating the potential parameters are satisfied in other words they are conditionally deformed shape invariant since in the second step of the construction of a partner potential hierarchy the constraint conditions change we impose compatibility conditions between the two sets to build potentials with known ground and first excited states to extend such results to any members of the two families we devise a general method wherein the first two superpotentials the first two partner potentials and the first two eigenstates of the starting potential are built from some generating function w r and its accompanying function w r we consider two families of extensions of the oscillator in a d dimensional constant curvature space and analyze them in a deformed supersymmetric framework wherein the starting oscillator is known to exhibit a deformed shape invariance property we show that the first two members of each extension family are also endowed with such a property provided some constraint conditions relating the potential parameters are satisfied in other words they are conditionally deformed shape invariant since in the second step of the construction of a partner potential hierarchy the constraint conditions change we impose compatibility conditions between the two sets to build potentials with known ground and first excited states to extend such results to any members of the two families we devise a general method wherein the first two superpotentials the first two partner potentials and the first two eigenstates of the starting potential are built from some generating function w r and its accompanying functio

decoupling of banach valued multilinear forms in independent symmetric banach valued random variables let e be a banach space and e be symmetric continuous and convex let ui and ri be independent sequences of random variables having respectively u 0 1 and symmetric bernoulli distributions and let u i j and r i j for j 1 2 d be independent copies of these sequences we prove two sided inequalities between the quantities e phi sum limits underset raise0 3em hbox smash scriptscriptstyle i in mathbb z d r i 1 r i d f underset raise0 3em hbox smash scriptscriptstyle i u i 1 u i d and their decoupled versions e phi sum limits underset raise0 3em hbox smash scriptscriptstyle iota in mathbb z d r i 1 1 r i d d f underset raise0 3em hbox smash scriptscriptstyle iota u i 1 1 u i d d for bochner integrable f i 0 1 d e this generalizes results of kwapien and of zinn

the divergence of banach space valued random variables on wiener space the domain of definition of the divergence operator on an abstract wiener space w h is extended to include w valued and open image in new window valued integrands the main properties and characterizations of this extension are derived and it is shown that in some sense the added elements in s extended domain have divergence zero these results are then applied to the analysis of quasiinvariant flows induced by w valued vector fields and among other results it turns out that these divergence free vector fields are responsible for generating measure preserving flows

on harmonic oscillators on the two dimensional sphere s2 and the hyperbolic plane h2 ii two harmonic oscillators isotropic and nonisotropic 2 1 are studied on the two dimensional sphere s2 and the hyperbolic plane h2 both systems are integrable and super integrable with constants of motion quadratic in the momenta these properties are shown to derive from a complex factorization for the constants of motion which holds for arbitrary values of the curvature and the dynamics of the euclidean harmonic 1 1 and 2 1 oscillators is directly recovered for 0 the harmonic oscillators on either the standard unit sphere radius r 1 or the unit lobachewski plane radius r 1 appear as the particular values of the dependent potentials for the values 1 and 1 finally a particular potential is proposed for representing the general spherical hyperbolic n 1 anisotropic harmonic oscillator on a two dimensional manifold of constant curvature

malliavin calculus and decoupling inequalities in banach spaces abstract we develop a theory of malliavin calculus for banach space valued random variables using radonifying operators instead of symmetric tensor products we extend the wiener ito isometry to banach spaces in the white noise case we obtain two sided l p estimates for multiple stochastic integrals in arbitrary banach spaces it is shown that the malliavin derivative is bounded on vector valued wiener ito chaoses our main tools are decoupling inequalities for vector valued random variables in the opposite direction we use meyer s inequalities to give a new proof of a decoupling result for gaussian chaoses in umd banach spaces

positivity for kac polynomials and dt invariants of quivers we give a cohomological interpretation of both the kac polynomial and the rened donaldson thomas invariants of quivers this interpretation yields a proof of a conjecture of kac from 1982 and gives a new perspective on recent work of kontsevich soibelman this is achieved by computing via an arithmetic fourier transform the dimensions of the isotypical components of the cohomology of associated nakajima quiver varieties under the action of a weyl group the generating function of the corresponding poincar e polynomials is an extension of hua s formula for kac polynomials of quivers involving hall littlewood symmetric functions the resulting formulae contain a wide range of information on the geometry of the quiver varieties

the quantum harmonic oscillator on the sphere and the hyperbolic plane κ dependent formalism polar coordinates and hypergeometric functions a nonlinear model representing the quantum harmonic oscillator on the sphere and the hyperbolic plane is solved in polar coordinates r by making use of a curvature dependent formalism the curvature is considered as a parameter and then the radial schrodinger equation becomes a dependent gauss hypergeometric equation the energy spectrum and the wave functions are exactly obtained in both the sphere s2 0 and the hyperbolic plane h2 0 a comparative study between the spherical and the hyperbolic quantum results is presented

superintegrable quantum u 3 systems and higher rank factorizations a class of two dimensional superintegrable systems on a constant curvature surface is considered as the natural generalization of some well known one dimensional factorized systems by using standard methods to find the shape invariant intertwining operators we arrive at a so 6 dynamical algebra and its hamiltonian hierarchies we pay attention to those associated to certain unitary irreducible representations that can be displayed by means of three dimensional polyhedral lattices we also discuss the role of superpotentials in this new context

quantum superintegrability and exact solvability in n dimensions a family of maximally superintegrable systems containing the coulomb atom as a special case is constructed in n dimensional euclidean space two different sets of n commuting second order operators are found overlapping in the hamiltonian alone the system is separable in several coordinate systems and is shown to be exactly solvable it is solved in terms of classical orthogonal polynomials the hamiltonian and n further operators are shown to lie in the enveloping algebra of a hidden affine lie algebra

mirror symmetry langlands duality and the hitchin system among the major mathematical approaches to mirror symmetry are those of batyrev borisov and strominger yau zaslow syz the first is explicit and amenable to computation but is not clearly related to the physical motivation the second is the opposite furthermore it is far from obvious that mirror partners in one sense will also be mirror partners in the other this paper concerns a class of examples that can be shown to satisfy the requirements of syz but whose hodge numbers are also equal this provides significant evidence in support of syz moreover the examples are of great interest in their own right they are spaces of flat slr connections on a smooth curve the mirror is the corresponding space for the langlands dual group pglr these examples therefore throw a bridge from mirror symmetry to the duality theory of lie groups and more broadly to the geometric langlands program

spectrum generating algebra for the continuous spectrum of a free particle in lobachevski space in this paper we construct a spectrum generating algebra sga for a quantum system with purely continuous spectrum the quantum free particle in a lobachevski space with constant negative curvature the sga contains the geometrical symmetry algebra of the system plus a subalgebra of operators that give the spectrum of the system and connects the eigenfunctions of the hamiltonian among themselves in our case the geometrical symmetry algebra is so 3 1 and the sga is so 4 2 we start with a representation of so 4 2 by functions on a realization of the lobachevski space given by a two sheeted hyperboloid where the lie algebra commutators are the usual poisson dirac brackets then we introduce a quantized version of the representation in which functions are replaced by operators on a hilbert space and poisson dirac brackets by commutators eigenfunctions of the hamiltonian are given and naive ladder operators are identified the previously defined naive ladder operators shift the eigenvalues by a

bps states torus links and wild character varieties a string theoretic framework is constructed relating the cohomology of wild character varieties to refined stable pair theory and torus link invariants explicit conjectural formulas are derived for wild character varieties with a unique irregular point on the projective line for this case this leads to a conjectural colored generalization of existing results of hausel mereb and wong as well as shende treumann and zaslow

absolutely indecomposable representations and kac moody lie algebras a conjecture of kac states that the polynomial counting the number of absolutely indecomposable representations of a quiver over a finite field with given dimension vector has positive coefficients and furthermore that its constant term is equal to the multiplicity of the corresponding root in the associated kac moody lie algebra in this paper we prove these conjectures for indivisible dimension vectors