spectral flow is the integral of one forms on the banach manifold of self adjoint fredholm operators abstract one may trace the idea that spectral flow should be given as the integral of a one form back to the 1974 vancouver icm address of i m singer our main theorem gives analytic formulae for the spectral flow along a norm differentiable path of self adjoint bounded breuer fredholm operators in a semifinite von neumann algebra these formulae have a geometric interpretation which derives from the proof namely we define a family of banach submanifolds of all bounded self adjoint breuer fredholm operators and on each submanifold define global one forms whose integral on a norm differentiable path contained in the submanifold calculates the spectral flow along this path we emphasise that our methods do not give a single globally defined one form on the self adjoint breuer fredholms whose integral along all paths is spectral flow rather as the choice of the plural forms in the title suggests we need a family of such one forms in order to confirm singer s idea the original context for this result concerned paths of unbounded self adjoint fredholm operators we therefore prove analogous formulae for spectral flow in the unbounded case as well the proof is a synthesis of key contributions by previous authors whom we acknowledge in detail in the introduction combined with an additional important recent advance in the differential calculus of functions of non commuting operators

spectral shift functions and dirichlet to neumann maps the spectral shift function of a pair of self adjoint operators is expressed via an abstract operator valued titchmarsh weyl m function this general result is applied to different self adjoint realizations of second order elliptic partial differential operators on smooth domains with compact boundaries and schrodinger operators with compactly supported potentials in these applications the spectral shift function is determined in an explicit form with the help of energy parameter dependent dirichlet to neumann maps

injective hulls of simple modules over finite dimensional nilpotent complex lie superalgebras abstract we show that the finite dimensional nilpotent complex lie superalgebras g whose injective hulls of simple u g modules are locally artinian are precisely those whose even part g 0 is isomorphic to a nilpotent lie algebra with an abelian ideal of codimension 1 or to a direct product of an abelian lie algebra and a certain 5 dimensional or a certain 6 dimensional nilpotent lie algebra

on a spectral flow formula for the homological index abstract consider a selfadjoint unbounded operator d on a hilbert space h and a one parameter norm continuous family of selfadjoint bounded operators a t t r that converges in norm to asymptotes a at then under certain conditions 22 that include the assumption that the operators d t d a t t r all have discrete spectrum the spectral flow along the path d t can be shown to be equal to the index of t d t when the latter is an unbounded fredholm operator on l 2 r h in 16 an investigation of the index spectral flow question when the operators in the path may have some essential spectrum was started but under restrictive assumptions that rule out differential operators in general in 11 the question of what happens when the fredholm condition is dropped altogether was investigated in these circumstances the fredholm index is replaced by the witten index in this paper we take the investigation begun in 11 much further we show how to generalize a formula known from the setting of the l 2 index theorem to the non fredholm setting restricting back to the case of selfadjoint fredholm operators our formula extends the result of 22 in the sense of relaxing the discrete spectrum condition it also generalizes some other fredholm operator results of 21 16 11 that permit essential spectrum for the operators in the path our result may also apply however when the operators d t have essential spectrum equal to the whole real line our main theorem gives a trace formula relating the homological index of 7 to an integral formula that is known for a path of selfadjoint fredholms with compact resolvent and with unitarily equivalent endpoints to compute spectral flow our formula however applies to paths of selfadjoint non fredholm operators we interpret this as indicating there is a generalization of spectral flow to the non fredholm setting

relative index theorems and supersymmetric scattering theory we discuss supersymmetric scattering theory and employ krein s theory of spectral shift functions to investigate supersymmetric scattering systems this is the basis for the derivation of relative index theorems on some classes of open manifolds as an example we discuss the de rham complex for obstacles in n and asymptotically flat manifolds it is shown that the absolute or relative euler characteristic of an obstacle in n may be obtained from scattering data for the laplace operator on forms with absolute or relative boundary conditions respectively in the case of asymptotically flat manifolds we obtain the chern gauss bonnet theorem for thel2 euler characteristic

a nonembeddable noetherian ring abstract we provide an example of a left and right noetherian c algebra which cannot be embedded into any artinian ring

fredholm differential operators with unbounded coefficients abstract we prove that a first order linear differential operator g with unbounded operator coefficients is fredholm on spaces of functions on r with values in a reflexive banach space if and only if the corresponding strongly continuous evolution family has exponential dichotomies on both r and r and a pair of the ranges of the dichotomy projections is fredholm and that the fredholm index of g is equal to the fredholm index of the pair the operator g is the generator of the evolution semigroup associated with the evolution family in the case when the evolution family is the propagator of a well posed differential equation u t a t u t with generally unbounded operators a t t r the operator g is a closure of the operator d dt a t thus this paper provides a complete infinite dimensional generalization of well known finite dimensional results by palmer and by ben artzi and gohberg

families of minimal involutive surfaces in projective space abstract we present an algorithm that can be used to determine if a given surface in complex projective 3 space is minimal involutive and we apply it to the construction of explicit examples of families of minimal involutive surfaces of degrees 3 and 4

determinant bundles manifolds with boundary and surgery we define determinant bundles associated to the following data i a family of generalized dirac operators on even dimensional manifolds with boundary ii the choice of a spectral section for the family of dirac operators induced on the boundary under the assumption that the operators of the boundary family have null spaces of constant dimension we define through the notion ofb zeta function a quillen metric we also introduce the analogue of the bismut freed connection we prove that the curvature of a natural perturbation of the bismut freed connection equals the 2 form piece in the right hand side of the family index formula thus extending to manifolds with boundary results of quillen bismut and freed given a closed fibration we investigate the behaviour of the quillen metric and of the bismut freed connection under the operation of surgery along a fibering hypersurface we prove in particular additivity formulae for the curvature and for the logarithm of the holonomy

zeta determinants on manifolds with boundary abstract through a general theory for relative spectral invariants we study the determinant of global boundary problems of aps type in particular we compute the determinant ratio for dirac laplacian boundary problems in terms of a scattering fredholm determinant over the boundary

eta invariants and regularized determinants for odd dimensional hyperbolic manifolds with cusps we study eta invariants of dirac operators and regularized determinants of dirac lapla cians over hyperbolic manifolds with cusps and their relations with selberg zeta functions using the selberg trace formula and a detailed analysis of the unipotent orbital integral we show that the eta and zeta functions defined by the relative traces are regular at the origin so that we can define the eta invariant and the regularized determinant we also show that the selberg zeta function of odd type has a meromorphic extension over c prove a relation of the eta invariant and a certain value of the selberg zeta function of odd type and derive a corresponding functional equation these results generalize the earlier work of john millson to hyperbolic manifolds with cusps we also prove that the selberg zeta function of even type has a meromorphic extension over c relate it to the regularized determinant and obtain a corresponding functional equation

spectra of graph neighborhoods and scattering let g e e 0 be a family of e thin riemannian manifolds modeled on a finite metric graph g for example the e neighborhood of an embedding of g in some euclidean space with straight edges we study the asymptotic behavior of the spectrum of the laplace beltrami operator on g e as e 0 for various boundary conditions we obtain complete asymptotic expansions for the kth eigenvalue and the eigenfunctions uniformly for k ce 1 in terms of scattering data on a non compact limit space we then use this to determine the quantum graph which is to be regarded as the limit object in a spectral sense of the family g e our method is a direct construction of approximate eigenfunctions from the scattering and graph data and the use of a priori estimates to show that all eigenfunctions are obtained in this way

categorification of induced cell modules and the rough structure of generalised verma modules this paper presents categorifications of right cell modules and induced cell modules for hecke algebras of finite weyl groups fit type a we show that these categorifications depend only on the isomorphism class of the cell module not on the cell itself our main application is multiplicity formulas for parabolically induced modules over a reductive lie algebra of type a which finally determines the so called rough structure of generalised verma modules on the way we present several categorification results and give a positive answer to kostant s problem from a joseph kostant s problem goldie rank and the gelfand kirillov conjecture invent math 56 3 1980 191 213 in many cases we also present a general setup of decategorification precategorification and categorification

eta invariant and holonomy the even dimensional case abstract in previous work we introduced eta invariants for even dimensional manifolds it plays the same role as the eta invariant of atiyah patodi singer which is for odd dimensional manifolds it is associated to k 1 representatives on even dimensional manifolds and is closely related to the so called wzw theory in physics in fact it is an intrinsic interpretation of the wess zumino term without passing to the bounding 3 manifold spectrally the eta invariant is defined on a finite cylinder rather than on the manifold itself thus it is an interesting question to find an intrinsic spectral interpretation of this new invariant we address this issue here using adiabatic limit technique the general formulation relates the mod z reduction of eta invariant for even dimensional manifolds with the holonomy of the determinant line bundle of a natural family of dirac type operators in this sense our result might be thought of as an even dimensional analogue of witten s holonomy theorem proved by bismut freed and cheeger independently

krull dimension of generalized weyl algebras with noncommutative coefficients abstract we obtain formulae for the krull dimension of the generalized weyl algebra t r a where r is a left noetherian ring

finite subnormalizing extensions of rings the purpose of this paper is to provide proofs of or counterexamples to possible generalizations to a larger class of ring extensions of several basic results about finite normalizing extensions l 7 11 13 14 161 the situation we are concerned with involves rings

the ζ determinant and the additivity of the η invariant on the smooth self adjoint grassmannian in this paper we discuss the existence of the determinant of a dirac operator on a compact manifold with boundary we show that the determinant is well defined in the case of the operator with a domain determined by a boundary condition from the smooth self adjoint grassmannian discussed in the papers 5 13 29 we prove a generalization of a pasting formula for the invariant see 34 the results of the paper are used in the recent proof of the projective equality of the determinant and quillen determinant on see 30 31

on pairs of projections in a hilbert space we give an algebraic derivation of the canonical form of a generic pair of projections the result is used to determine the spectral shift of a pair of projections and various properties of fredholm pairs

commutation methods applied to the mkdv equation an explicit construction of solutions of the modified korteweg de vries equation given a solution of the ordinary korteweg de vries equation is provided our theory is based on commutation methods i e n 1 supersymmetry underlying miura s transformation that links solutions of the two evolution equations

the spectral shift function and the invariance principle abstract the new representation formula for the spectral shift function due to f gesztesy and k a makarov is considered this formula is extended to the case of relatively trace class perturbations the proof is based on the analysis of a certain new unitary invariant for a pair of self adjoint operators

adiabatic limits of the η invariants the odd dimensional atiyah patodi singer problem we study the invariant of boundary value problems of atiyah patodi singer type we prove the formula for the spectral flow of the families overs1 assuming a product structure in a collar neighbourhood of the boundary we show that the invariant behaves the same way as on a closed manifold we also study the adiabatic limit of the spectral invariants in nice cases we are able to split them into a contribution from the interior one from the cylinder and an error term then we show that the error term disappears with the increasing length of the cylinder

nonholonomic modules over enveloping algebras of semisimple lie algebras abstract we prove that the enveloping algebras of all semisimple lie algebras over the complex numbers which are not of type a 1 admit nonholonomic simple modules

the influence of ideal structure on representation theory the purpose of this paper is to explore the representation theoretic consequences of the two sided ideal structure of a noetherian ring with special emphasis on an additional combinatorial structure on the prime spectrum of the ring the graph of links we obtain existence theorems for extensions and for rings satisfying the strong second layer condition we obtain extensive information about the indecomposable injective modules particularly detailed results are obtained when the ring is polynormal

the burghelea friedlander kappeler gluing formula for zeta determinants on a warped product manifold and a product manifold the burghelea friedlander kappeler bfk gluing formula for the regularized zeta determinants of laplacians contains a constant which is expressed by the constant term in the asymptotic expansion of the regularized zeta determinants of a one parameter family of the dirichlet to neumann operators when the dimension of a cutting hypersurface is odd or the metric is a product one near a cutting hypersurface this constant is well known in this paper we discuss this constant in two cases one is when a warped product metric is given near a cutting hypersurface and the other is when a manifold is a product manifold especially in the first case we use the result of fucci and kirsten commun math phys 317 635 665 2013 in which the regularized zeta determinant of the laplacian defined on a warped product manifold is computed

the s eulerian polynomials have only real roots we study the roots of generalized eulerian polynomials via a novel approach we interpret eulerian polynomials as the generating polynomials of a statistic over inversion sequences inversion sequences also known as lehmer codes or subexcedant functions were recently generalized by savage and schuster to arbitrary sequences s of positive integers which they called s inversion sequences our object of study is the generating polynomial of the ascent statistic over the set of s inversion sequences of length n since this ascent statistic over inversion sequences is equidistributed with the descent statistic over permutations we call this generalized polynomial the s eulerian polynomial the main result of this paper is that for any sequence s of positive integers the s eulerian polynomial has only real roots this result is first shown to generalize several existing results about the real rootedness of various eulerian polynomials we then show that it can be used to settle a conjecture of brenti that eulerian polynomials for all finite coxeter groups have only real roots and partially settle a conjecture of dilks petersen stembridge on type b affine eulerian polynomials it is then extended to several q analogs we show that the macmahon carlitz q eulerian polynomial has only real roots whenever q is a positive real number confirming a conjecture of chow and gessel the same holds true for the hyperoctahedral group and the wreath product groups confirming further conjectures of chow and gessel and chow and mansour respectively our results have interesting geometric consequences as well

non holonomic modules over weyl algebras and enveloping algebras a number of counterexamples in the theory of enveloping algebras and noetherian rings are presented in particular we construct non holonomic simple modules over the weyl algebra a and the enveloping algebra u s1 z xs12 further we show that u s1 z x s12 is not weakly ideal invariant and that there exist simple s1 z x s12 modules m and e with e finite dimensional such that e is not artinian if m is a module over a ring r then the gelfand kirillov dimension of m written gk dim m is defined in 7 suppose that m is a simple module over a noetherian ring r and that g k d i m r oc then m is called holonomic if g k d i m m 8 9 the reason for the definition is that in the case when r is a weyl algebra or the enveloping algebra of a finite dimensional algebraic lie algebra holonomic modules have particularly nice properties

a gentle guide to the basics of two projections theory abstract this paper is a survey of the basics of the theory of two projections it contains in particular the theorem by halmos on two orthogonal projections and roch silbermann gohberg and krupnik s theorem on two idempotents in banach algebras these two theorems which deliver the desired results usually very quickly and comfortably are missing or wrongly cited in many recent publications on the topic the paper is intended as a gentle guide to the field the basic theorems are precisely stated some of them are accompanied by full proofs others not but precise references are given in each case and many examples illustrate how to work with the theorems

relative index pairing and odd index theorem for even dimensional manifolds abstract we prove an analogue for even dimensional manifolds of the atiyah patodi singer twisted index theorem for trivialized flat bundles we show that the eta invariant appearing in this result coincides with the eta invariant by dai and zhang up to an integer we also obtain the odd dimensional counterpart for manifolds with boundary of the relative index pairing by lesch moscovici and pflaum

hankel hyperdeterminants and selberg integrals we investigate the simplest class of hyperdeterminants defined by cayley in the case of hankel hypermatrices tensors of the form ai1i2 ik f i1 i2 ik it is found that many classical properties of hankel determinants can be generalized and a connection with selberg type integrals is established in particular selberg s original formula amounts to the evaluation of all hankel hyperdeterminants built from the moments of the jacobi polynomials many higher dimensional analogues of classical hankel determinants are evaluated in closed form the toeplitz case is also briefly discussed in physical terms both cases are related to the partition functions of one dimensional coulomb systems with logarithmic potential

spectral boundary conditions for generalizations of laplace and dirac operators spectral boundary conditions for laplace type operators on a compact manifold x with boundary are partly dirichlet partly oblique neumann conditions where the partitioning is provided by a pseudodifferential projection they have an interest in string and brane theory relying on pseudodifferential methods we give sufficient conditions for the existence of the associated resolvent and heat operator and show asymptotic expansions of their traces in powers and power log terms allowing a smearing function the leading log coefficient is identified as a non commutative residue which vanishes when 1 the study has new consequences for well posed spectral boundary problems for first order dirac like elliptic operators generalizing the atiyah patodi singer problem it is found e g that the zeta function is always regular at zero in the selfadjoint case there is a stability of the zeta function value and the eta function regularity at zero under perturbations of the boundary projection of order dim x

the resolvent expansion for second order regular singular operators abstract the paper gives a scheme for computing the asymptotics of tr e tl as t 0 where l is an elliptic operator of the form l d 2 x 2 a x and a x is a family of operators satisfying appropriate ellipticity and smoothness conditions a principle example is the laplace operator for a manifold with an asymptotically conic singularity the expansion has the usual terms away from the singularity appropriately regularized at x 0 plus singular contributions determined by the function of a 0 1 4 1 2 applications to index theorems are given in a subsequent paper

corners in m theory m theory can be defined on closed manifolds as well as on manifolds with boundary as an extension we show that manifolds with corners appear naturally in m theory we illustrate this with four situations the lift to bounding 12 dimensions of m theory on anti de sitter spaces ten dimensional heterotic string theory in relation to 12 dimensions and the two m branes within m theory in the presence of a boundary the m2 brane is taken with or as a boundary and the worldvolume of the m5 brane is viewed as a tubular neighborhood we then concentrate on the variant of the heterotic theory as a corner and explore analytical and geometric consequences in particular we formulate and study the phase of the partition function in this setting and identify the corrections due to the corner s the analysis involves considering m theory on disconnected manifolds and makes use of the extension of the atiyah patodi singer index theorem to manifolds with corners and the b calculus of melrose

once more on algebras generated by two projections we consider von neumann algebras generated by two arbitrary orthoprojections on a hilbert space a canonical decomposition is obtained for elements a of these algebras in terms of the operator angle between the ranges of the above mentioned projections this decomposition leads to explicit descriptions and formulas for kernels ranges spectra and essential spectra generalized inverses and other objects related to a

spectral geometry of singular riemannian spaces on considere la trace du noyau de la chaleur et l application de la methode de l equation de la chaleur au theoreme d indice pour la caracteristique d euler et des complexes signatures

dixmier s problem 5 for the weyl algebra abstract the purpose of this paper is to give a negative answer to problem 5 and a positive answer in the case of homogeneous elements to problem 4 of the article of j dixmier bull soc math france 96 1968 209 242 on the first weyl algebra

foliations of multiprojective spaces and a conjecture of bernstein and lunts we use foliations of multiprojective spaces defined by hamiltonian functions on the underlying affine space to prove the three dimensional case of a conjecture of bernstein and lunts according to which the symbol of a generic first order differential operator gives rise to a hypersurface of the cotangent bundle which does not contain involutive conical subvarieties apart from the zero section and fibres of the bundle

spectral determinants on mandelstam diagrams we study the regularized determinant of the laplacian as a functional on the space of mandelstam diagrams noncompact translation surfaces glued from finite and semi infinite cylinders a mandelstam diagram can be considered as a compact riemann surface equipped with a conformal flat singular metric omega 2 where omega is a meromorphic one form with simple poles such that all its periods are pure imaginary and all its residues are real the main result is an explicit formula for the determinant of the laplacian in terms of the basic objects on the underlying riemann surface the prime form theta functions the canonical meromorphic bidifferential and the divisor of the meromorphic form omega as an important intermediate result we prove a decomposition formula of the type of burghelea friedlander kappeler for the determinant of the laplacian for flat surfaces with cylindrical ends and conical singularities

l2 index formulae for perturbed dirac operators the callias index theorem is generalized from the euclidean case to certain spin manifolds with warped ends making use of certain index preserving deformations

spectral triples and manifolds with boundary abstract we investigate manifolds with boundary in noncommutative geometry spectral triples associated to a symmetric differential operator and a local boundary condition are constructed we show that there is no tadpole for classical dirac operators with a chiral boundary condition on spin manifolds

simple holonomic modules over the second weyl algebra a2 abstract for simple generalized weyl algebras of gelfand kirillov dimension 4 a class including the second weyl algebra a 2 some simple factor algebras of the universal enveloping algebra of the lie algebra sl 2 sl 2 and of u q sl 2 u q sl 2 etc the simple holonomic modules are classified up to pairs of irreducible elements of certain noncommutative euclidean ring

aps boundary conditions eta invariants and adiabatic limits we prove an adiabatic limit formula for the eta invariant of a manifold with boundary the eta invariant is defined using the atiyah patodisinger boundary condition and the underlying manifold is fibered over a manifold with boundary our result extends the work of bismut cheeger to manifolds with boundary

anomalies of dirac type operators on euclidean space we develop by example a type of index theory for non fredholm operators a general framework using cyclic homology for this notion of index was introduced in a separate article carev and kaad topological invariance of the homological index arxiv 1402 0475 math kt 2014 where it may be seen to generalise earlier ideas of carey pincus and gesztesy simon on this problem motivated by an example in two dimensions in bolle et al j math phys 28 1512 1525 1987 we introduce in this paper a class of examples of dirac type operators on mathbb r 2n that provide non trivial examples of our homological approach our examples may be seen as extending old ideas about the notion of anomaly introduced by physicists to handle topological terms in quantum action principles with an important difference namely we are dealing with purely geometric data that can be seen to arise from the continuous spectrum of our dirac type operators

spectral flow and the unbounded kasparov product abstract we present a fairly general construction of unbounded representatives for the interior kasparov product as a main tool we develop a theory of c 1 connections on operator modules we do not require any smoothness assumptions our unitality assumptions are minimal furthermore we use work of kucerovsky and our recent local global principle for regular operators in hilbert c modules as an application we show that the spectral flow theorem and more generally the index theory of dirac schrodinger operators can be nicely explained in terms of the interior kasparov product

differentiation of operator functions in non commutative lp spaces the principal results in this paper are concerned with the description of differentiable operator functions in the non commutative lp spaces 1 p associated with semifinite von neumann algebras for example it is established that if f r r is a lipschitz function then the operator function f is g teaux differentiable in l2 m for any semifinite von neumann algebra m if and only if it has a continuous derivative furthermore if f r r has a continuous derivative which is of bounded variation then the operator function f is g teaux differentiable in any lp m 1 p

the dichotomy theorem for evolution bi families we prove that the operator g the closure of the rst order dierential operator d dt d t on l2 r x is fredholm if and only if the not well posed equation u 0 t d t u t t 2 r has exponential dichotomies on r and r and the ranges of the dichotomy projections form a fredholm pair moreover the index of this pair is equal to the fredholm index of g here x is a hilbert space d t a b t a is the generator of a bi semigroup b is a bounded piecewise strongly continuous operator valued function also we prove some perturbations results and consider various examples of not well posed problems

relative index theory abstract we develop the framework for the heat equation method in the relative index theory we study pairs of dirac operators on complete manifolds and give an analytic interpretation of the difference of the integrals over their local index densities this yields a generalization of the relative index theorem of gromov and lawson under some assumptions on the curvature we show that supersymmetric scattering theories of borisov muller and schrader arise naturally the scattering index is related to the relative topological index

fredholm theory for pairs of closed subspaces of a banach space we study the fredholm theory for pairs of closed subspaces of a banach space developed by kato we define the strictly singular and the strictly cosingular pairs of subspaces and we show that some of the results of operator theory can be extended to this context however there are some notable differences on the one hand the perturbation classes problem has a positive answer in this context the upper and lower semi fredholm pairs are stable under strictly singular and strictly cosingular perturbations respectively and this stability characterizes the strictly singular and the strictly cosingular pairs note that it has been proved recently that the perturbation classes problem for continuous semi fredholm operators has a negative answer on the other hand unlike in the case of operators the fredholm pairs are not stable under perturbation by strictly singular or strictly cosingular pairs we also show the stability under composition of the compact the strictly singular and the strictly cosingular pairs of subspaces

topological invariance of the witten index abstract we discuss the witten index in terms of krein s spectral shift function and prove invariance of the witten index under suitable relative trace class hypotheses

ordinary differential operators in hilbert spaces and fredholm pairs let a t be a path of bounded operators on a real hilbert space hyperbolic at pm infty we study the fredholm theory of the operator f a d dt a t we relate the fredholm property of f a to the stable and unstable linear spaces of the associated system x prime a t x several examples are included to point out the differences with respect to the finite dimensional case in particular concerning the role of the spectral flow we define a general class of paths a for which many properties typical of the finite dimensional framework still hold our motivation is to develop the linear theory which is necessary for the set up of morse homology on hilbert manifolds

the index formula and the spectral shift function for relatively trace class perturbations we compute the fredholm index index da of the operator da d dt a on l2 r h associated with the operator path a t t where af t a t f t for a e t r and appropriate f l2 r h via the spectral shift function a a associated with the pair a a of asymptotic operators a a on the separable complex hilbert space h in the case when a t is generally an unbounded relatively trace class perturbation of the unbounded self adjoint operator a we derive a formula an extension of a formula due to pushnitski relating the spectral shift function a a for the pair a a and the corresponding spectral shift function h2 h1 for the pair of operators h2 h1 dada da da in this relative trace class context h2 h1 1 1 2 1 2 a a d 2 1 2for a e 0 this formula is then used to identify the fredholm index of da with 0 a a in addition we prove that index da coincides with the spectral flow spflow a t t of the family a t t r and also relate it to the fredholm perturbation determinant for the pair a a index da spflow a t t 0 a a 1lime 0im ln deth a iei a iei 1 0 h2 h1 with the choice of the branch of ln deth on c such that limim z ln deth a zi a zi 1 0 we also provide some applications in the context of supersymmetric quantum mechanics to zeta function and heat kernel regularized spectral asymmetries and the eta invariant

an index theorem for toeplitz operators on odd dimensional manifolds with boundary we establish an index theorem for toeplitz operators on odd dimensional spin manifolds with boundary it may be thought of as an odd dimensional analogue of the atiyah patodi singer index theorem for dirac operators on manifolds with boundary in particular there occurs naturally an invariant of type associated to k 1 representatives on even dimensional manifolds which should be of independent interests for example it gives an intrinsic interpretation of the so called wess zumino term in the wzw theory in physics

double operator integrals this paper is concerned with perturbation formulae of the form f a f b lp m k a b lp m with k 0 being a constant depending on p and f only where f is a real valued lipschitz function and a b are self adjoint measurable operators affiliated with a semifinite von neumann algebra m such that the difference a b belongs to lp m 1 p in order to treat the situation where the von neumann algebra m is not necessarily hyperfinite we first develop an integration theory with respect to finitely additive spectral measures in a banach space applied to product measures this integration theory may be considered as an abstract version of the double operator integrals due to birman and solomyak to describe the class of integrable functions we employ our recent study of multiplier theory in umd spaces our perturbation formulae extend those of davies and birman solomyak for the case when m is a hyperfinite i factor i e for the schatten p classes we also discuss analogous perturbation results in the setting of symmetric operator spaces associated with m

on boundary value problems for dirac type operators i regularity and self adjointness in a series of papers we will develop systematically the basic spectral theory of self adjoint boundary value problems for operators of dirac type we begin in this paper with the characterization of self adjoint boundary conditions with optimal regularity for which we will derive the heat asymptotics and index theorems in subsequent publications along with a number of new results we extend and simplify the proofs of many known theorems our point of departure is the simple structure which is displayed by dirac type operators near the boundary thus our proofs are given in an abstract functional analytic setting generalizing considerably the framework of compact manifolds with boundary the results of this paper have been announced previously by the authors j bruning and m lesch in geometric aspects of partial differential equations b booss bavnbek and k p wojciechowski eds contemporary mathematics vol 242 pp 203 205 amer math soc providence ri 1999

lecture hall tableaux abstract we introduce lecture hall tableaux which are fillings of a skew young diagram satisfying certain conditions lecture hall tableaux generalize both lecture hall partitions and anti lecture hall compositions and also contain reverse semistandard young tableaux as a limit case we show that the coefficients in the schur expansion of multivariate little q jacobi polynomials are generating functions for lecture hall tableaux using a selberg type integral we show that moments of multivariate little q jacobi polynomials which are equal to generating functions for lecture hall tableaux of a young diagram have a product formula we also explore various combinatorial properties of lecture hall tableaux

the importance of the selberg integral it has been remarked that a fair measure of the impact of atle selberg s work is the number of mathematical terms that bear his name one of these is the selberg integral an n dimensional generalization of the euler beta integral we trace its sudden rise to prominence initiated by a question to selberg from enrico bombieri more than thirty years after its initial publication in quick succession the selberg integral was used to prove an outstanding conjecture in random matrix theory and cases of the macdonald conjectures it further initiated the study of q analogues which in turn enriched the macdonald conjectures we review these developments and proceed to exhibit the sustained prominence of the selberg integral as evidenced by its central role in random matrix theory calogero sutherland quantum many body systems knizhnik zamolodchikov equations and multivariable orthogonal polynomial theory

the index of a pair of projections we discuss pairs of self adjoint projections with p q j 2n 1 the trace ideal and prove that for m n tr p q 2m 1 tr p q 2n 1 dim ker q ran p dim ker p ran q is an integer we also prove that there exists a unitary v interchanging p and q if and only if this integer is 0

poles of zeta and eta functions for perturbations of the atiyah patodi singer problem the zeta and eta functions of a differential operator of dirac type on a compact n dimensional manifold provided with a well posed pseudodifferential boundary condition have been shown in g99 to be meromorphic on with simple or double poles on the real axis extending results from g99 we show how perturbations of the boundary condition of order j affect the poles in particular they preserve a possible regularity of zeta at 0 and a possible simple pole of eta at 0 when j n this applies to perturbations of spectral boundary conditions also when the structure is non product and the problem is non selfadjoint

lipschitz continuity of the absolute value and riesz projections in symmetric operator spaces a principal result of the paper is that ifeis a symmetric banach function space on the positive half line with the fatou property then for all semifinite von neumann algebras m the absolute value mapping is lipschitz continuous on the associated symmetric operator spacee m with lipschitz constant depending only oneif and only ifehas non trivial boyd indices it follows that if m is any von neumann algebra then the absolute value map is lipschitz continuous on the corresponding haageruplp space provided 1 p

on involutive homogeneous varieties and representations of weyl algebras abstract this paper is concerned with the geometry of minimal involutive homogeneous varieties in complex affine 2 n space and its application to the study of the representation theory of the n th complex weyl algebra a n the main results are the existence of minimal involutive homogeneous varieties of any given codimension and of a n modules that have these varieties for characteristic variety we also determine conditions on the codimensions of a n modules m and n under which ext 1 m n is a finite dimensional vector space

relative zeta functions relative determinants and scattering theory we use the method of zeta function regularization to regularize the ratio det a det a0 of the determinants of two elliptic self adjoint operators a a0 satisfying certain natural assumptions this is of interest especially if the regularized determinants of the individual operators don t exist as for example in the case of elliptic operators on a noncompact manifold

combinatorics of the two species asep and koornwinder moments abstract in previous work 12 13 14 the first and third authors introduced staircase tableaux which they used to give combinatorial formulas for the stationary distribution of the asymmetric simple exclusion process asep and for the moments of the askey wilson weight function the fact that the asep and askey wilson moments are related at all is unexpected and is due to 45 the asep is a model of particles hopping on a one dimensional lattice of n sites with open boundaries particles can enter and exit at both left and right borders it was introduced around 1970 34 43 and is cited as a model for both traffic flow and translation in protein synthesis meanwhile the askey wilson polynomials are a family of orthogonal polynomials in one variable which sit at the top of the hierarchy of classical orthogonal polynomials so from this previous work we have the relationship asep staircase tableaux askey wilson moments the askey wilson polynomials can be viewed as the one variable case of the multivariate koornwinder polynomials which are also known as the macdonald polynomials attached to the non reduced affine root system c n c n it is natural then to ask whether one can generalize the relationships among the asep askey wilson moments and staircase tableaux in such a way that koornwinder moments replace askey wilson moments in 15 we made a precise link between koornwinder moments and the two species asep a generalization of the asep which has two species of particles with different weights in this article we introduce rhombic staircase tableaux and show that we have the relationship 2 species asep rhombic staircase tableaux koornwinder moments in particular we give formulas for the stationary distribution of the two species asep and for koornwinder moments in terms of rhombic staircase tableaux download download high res image 75kb download download full size image

grothendieck groups of invariant rings filtrations we investigate the grothendieck group ga r of finitely generated modules over the ring of invariants r s g of the action of a finite group g on an fbn ring s under the assumption that the trace map from s to r is surjective using a certain filtration of g0 r that is defined in terms of gabriel rentschler krull dimension properties of g0 r are derived to a large extent from the connections between the sets of prime ideals of 5 and r a crucial ingredient is an equivalence relation on spec that was introduced by montgomery 25 for example we show that rank go rank g0 s g q 1 a where q runs over the equivalence classes in spec r and c denotes g coinvariants the torsion subgroup of g0 r is also considered we apply our results to group actions on the weyl algebra in positive characteristics the quantum plane and the localized quantum plane for roots of unity

non commutative banach function spaces in this paper we survey some aspects of the theory of non commutative banach function spaces that is spaces of measurable operators associated with a semi finite von neumann algebra these spaces are also known as non commutative symmetric spaces the theory of such spaces emerged as a common generalization of the theory of classical commutative rearrangement invariant banach function spaces in the sense of w a j luxemburg and a c zaanen and of the theory of symmetrically normed ideals of bounded linear operators in hilbert space in the sense of i c gohberg and m g krein these two cases may be considered as the two extremes of the theory in the first case the underlying von neumann algebra is the commutative algebra l on some measure space with integration as trace in the second case the underlying von neumann algebra is b the algebra of all bounded linear operators on a hilbert space with standard trace important special cases of these non commutative spaces are the non commutative l p spaces which correspond in the commutative case with the usual l p spaces on a measure space and in the setting of symmetrically normed operator ideals they correspond to the schatten p classes mathfrak s p