the cohomology of the cotangent bundle of heisenberg groups abstract given a parabolic subalgebra g1 n of a semisimple lie algebra kostant ann math 1963 and griffiths acta math 1963 independently computed the g1 invariants in the cohomology group of n with exterior adjoint coefficients by a theorem of bott ann math 1957 this is the cohomology of the associated compact homogeneous space with coefficients in the sheaf of local holomorphic forms in this paper we determine explicitly the full module structure over the symplectic group of the cohomology group of the heisenberg lie algebra with exterior adjoint coefficients this is the cohomology of the cotangent bundle of the heisenberg group

depth and the local cohomology of fi_g modules abstract in this paper we describe a machinery for homological calculations of representations of fi g and use it to develop a local cohomology theory over any commutative noetherian ring as an application we show that the depth introduced by the second author in 16 coincides with a more classical invariant from commutative algebra and obtain upper bounds of a few important invariants of fi g modules in terms of torsion degrees of their local cohomology groups

gl equivariant modules over polynomial rings in infinitely many variables consider the polynomial ring in countably infinitely many variables over a field of characteristic zero together with its natural action of the infinite general linear group g we study the algebraic and homological properties of finitely generated modules over this ring that are equipped with a compatible g action we define and prove finiteness properties for analogues of hilbert series systems of parameters depth local cohomology koszul duality and regularity we also show that this category is built out of a simpler more combinatorial quiver category which we describe explicitly our work is motivated by recent papers in the literature which study finiteness properties of infinite polynomial rings equipped with group actions for example the paper by church ellen berg and farb on the category of fi modules which is equivalent to our category along the way we see several connections with the character polynomials from the representation theory of the symmetric groups several examples are given to illustrate that the invariants we introduce are explicit and computable

a novel method for solving the inverse scattering problem for time harmonic acoustic waves in the resonance region ii in a previous paper siam j appl math 1985 pp 1039 1053 we presented a new method for determining the shape of an acoustically soft obstacle from a knowledge of the time harmonic incident wave and the far field pattern of the scattered wave the method given there was based on knowing the far field pattern for an interval of values of the square of the wave number k such that this interval contained the first eigenvalue lambda 1 of the interior dirichlet problem the purpose of this paper is to extend the methods of our earlier paper to treat the case when the far field pattern is only known for a single value of the wave number such that k 2 is not equal to lambda 1 in addition we show how to combine the results of this paper with the first to obtain a method that is valid for any fixed value of k independent of whether or not k 2 is an eigenvalue we also extend our analysis to include the case of the impedance boundary value problem numerical examples are given showing the pra

dimension of the torelli group for out fn let mathcal t n be the kernel of the natural map out fn gln we use combinatorial morse theory to prove that mathcal t n has an eilenberg maclane space which is 2n 4 dimensional and that h 2n 4 mathcal t n mathbb z is not finitely generated n 3 in particular this shows that the cohomological dimension of mathcal t n is equal to 2n 4 and recovers the result of krstic mccool that mathcal t 3 is not finitely presented we also give a new proof of the fact due to magnus that mathcal t n is finitely generated

grobner methods for representations of combinatorial categories given a category c of a combinatorial nature we study the following fundamental question how does the combinatorial behavior of c affect the algebraic behavior of representations of c we prove two general results the first gives a combinatorial criterion for representations of c to admit a theory of grobner bases from this we obtain a criterion for noetherianity of representations the second gives a combinatorial criterion for a general rationality result for hilbert series of representations of c this criterion connects to the theory of formal languages and makes essential use of results on the generating functions of languages such as the transfer matrix method and the chomsky schutzenberger theorem our work is motivated by recent work in the literature on representations of various specific categories our general criteria recover many of the results on these categories that had been proved by ad hoc means and often yield cleaner proofs and stronger statements for example we give a new more robust proof that fi modules originally introduced by church ellenberg farb and a family of natural generalizations are noetherian we give an easy proof of a generalization of the lannes schwartz artinian conjecture from the study of generic representation theory of finite fields we significantly improve the theory of modules introduced by snowden in connection to syzygies of segre embeddings and we establish fundamental properties of twisted commutative algebras in positive characteristic

permutation representations on schubert varieties this paper defines and studies permutation representations on the equivariant cohomology of schubert varieties as representations both over bbb c and over bbb c t 1 t 2 ldots t n we show these group actions are the same as an action of simple transpositions studied geometrically by m brion and give topological meaning to the divided difference operators of berstein gelfand gelfand demazure kostant kumar and others we analyze these representations using the combinatorial approach to equivariant cohomology introduced by goresky kottwitz macpherson we find that each permutation representation on equivariant cohomology produces a representation on ordinary cohomology that is trivial though the equivariant representation is not

castelnuovo mumford regularity of representations of certain product categories abstract we show in this paper that representations of a finite product of categories satisfying certain combinatorial conditions have finite castelnuovo mumford regularity if and only if they are presented in finite degrees and hence the category consisting of them is abelian these results apply to examples such as the categories fi m and fi g m

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

homological invariants of fi modules and fi_g modules abstract we explore a theory of depth for finitely presented fi g modules using this theory we prove results about the regularity and provide novel bounds on stable ranges of fi modules making effective a theorem of nagpal and thereby refining the stable range in results of church ellenberg and farb

a hybrid quantitative method for inverse scattering of multiple dielectric objects a hybrid method is proposed to locate the boundaries and determine the dielectric constants of multiple homogeneous scatterers by postprocessing the results of the linear sampling method lsm the hybrid method is composed of three steps combining the capabilities of the lsm the multiple signal classification music algorithm and the generalized multipole technique gmt initially a new measure of the noise level of the multistatic response msr matrix is proposed subsequently the lsm is applied using the tikhonov regularization then the scattered field is back propagated into the investigation domain using the gmt where in order to ensure the stability a music based approach is exploited to determine the optimum reconstructing multipole sources the case of limited scattered field data is also addressed by introducing a modification of the papoulis gerchberg algorithm pga finally the boundary and the dielectric constant of each scatterer are obtained simultaneously by solving an inverse boundary value problem through an optimization the proposed hybrid approach is capable of efficiently approximating the dielectric constant of multiple homogeneous dielectric scatterers in a parallel manner the effectiveness of the proposed method and the accuracy of the solutions are demonstrated by applying the method to both numerical and measured data

recent developments in inverse acoustic scattering theory we survey some of the highlights of inverse scattering theory as it has developed over the last 15 years with emphasis on uniqueness theorems and reconstruction algorithms for time harmonic acoustic waves included in our presentation are numerical experiments using real data and numerical examples of the use of inverse scattering methods to detect buried objects

a new non iterative singular sources method for the reconstruction of piecewise constant media we describe a novel non iterative method for the reconstruction of a piecewise constant inhomogeneous medium in acoustic scattering which we call the singular sources method the basic idea of the method is to use the behaviour of the scattered field for singular incident fields multipoles to calculate the size of the refractive index n at some point z0 on the boundary of the support of the scatterer and then eliminate this value from the data by subtracting a known piecewise constant background medium the paper includes the theory for the singular sources method to locate the unknown support of an inhomogeneous medium for a known inhomogeneous background medium also we give a new uniqueness proof for the reconstruction of a piecewise constant medium in two or three dimensions using techniques that differ from those used to prove previous well known results

characterization of su q l 1 equivariant spectral triples for the odd dimensional quantum spheres the quantum group suq l 1 has a canonical action on the odd dimensional sphere s 2l 1 q all odd spectral triples acting on the l2 space of s 2l 1 q and equivariant under this action have been characterized this characterization then leads to the construction of an optimum family of equivariant spectral triples having nontrivial k homology class these generalize the results of chakraborty pal for suq 2 ams subject classification no 58b34 46l87 19k33

recovery of singularities for formally determined inverse problems in this paper we considered several formally determined problems in two dimensions there are no global identifiability results for these problems however we can recover an important feature of these functions namely their singularities more precisely we prove that one can determine the location and strength of singularities of anl compactly supported potential by knowing the associated scattering amplitude at a fixed energy also we prove that one can determine the location and strength of the singularities of the sound speed of a medium by making measurements just on the boundary of the medium

on the chern character of θ summable fredholm modules we show that the entire cyclic cohomology class given by the jaffe lesniewski osterwalder formula is the same as the class we had constructed earlier as the chern character of summable fredholm modules

symmetries of levy processes on compact quantum groups their markov semigroups and potential theory quantum markov semigroups qms i e strongly continuous semigroups of unital completely positive maps on compact quantum groups are studied we show that translation invariant qmss on the universal or reduced c algebra of a compact quantum group are in one to one correspondence with levy processes on its hopf algebra we use the theory of levy processes on involutive bialgebras to characterize symmetry properties of the associated qms it turns out that the qms is self adjoint resp kms symmetric if and only if the generating functional of the levy process is invariant under the antipode resp the unitary antipode furthermore we study levy processes whose marginal states are invariant under the adjoint action finally some related aspects of the potential theory as dirichlet form derivation and spectral triple are investigated we discuss how the above results apply to compact groups group c algebras of discrete groups free orthogonal quantum groups on and twisted suq 2 quantum group

stability estimates and reconstructions in inverse acoustic scattering using singular sources abstract the problem of stability for the reconstruction of the scattered field us in the exterior of a scatterer d from far field data u and for the reconstruction of a sound soft or sound hard scatterer d from the far field pattern u x d x d for all incident plane waves is investigated in particular we show how stability estimates are linked to the behaviour of the scattered field s for incident point sources and to special minimum norm solutions for the herglotz wave operator the results can also be used to formulate a method of singular sources for the reconstruction of d which is new in inverse scattering the reconstruction method is described and numerical examples are provided

quantum group of orientation preserving riemannian isometries we formulate a quantum group analogue of the group of orientation preserving riemannian isometries of a compact riemannian spin manifold more generally of a possibly r twisted and of compact type spectral triple the main advantage of this formulation which is directly in terms of the dirac operator is that it does not need the existence of any good laplacian as in our previous works on quantum isometry groups several interesting examples including those coming from rieffel type deformation as well as the equivariant spectral triples on su 2 and s c2 are discussed

stability of schur functors abstract symplectic resp orthogonal schur functors are defined on a vector spacevwith a non degenerate skew symmetric resp symmetric bilinear form each of these functors determines an irreducible representation of s p v resp s o v we prove that if dim vis sufficiently large then the splitting into irreducible components of the composition of a symplectic orthogonal schur functor and an ordinary schur functor applied tovis independent of dim v we also prove the stability of the branching for the symplectic orthogonal schur functors

filtrations and homological degrees of fi modules abstract let k be a commutative noetherian ring in this paper we consider filtered modules of the category fi firstly introduced in 12 we show that a finitely generated fi module v is filtered if and only if its higher homologies all vanish and if and only if a certain homology vanishes using this homological characterization we characterize finitely generated c modules v whose projective dimension pd v is finite and describe an upper bound for pd v furthermore we give a new proof for the fact that v induces a finite complex of filtered modules and use it as well as a result of church and ellenberg in 1 to obtain another upper bound for homological degrees of v

supersymmetric quantum theory and non commutative geometry classical differential geometry can be encoded in spectral data such as connes spectral triples involving supersymmetry algebras in this paper we formulate non commutative geometry in terms of supersymmetric spectral data this leads to generalizations of connes non commutative spin geometry encompassing non commutative riemannian symplectic complex hermitian and hyper kahler geometry a general framework for non commutative geometry is developed from the point of view of supersymmetry and illustrated in terms of examples in particular the non commutative torus and the non commutative 3 sphere are studied in some detail

the minimal model of the complement of an arrangement of hyperplanes in this paper the methods of rational homotopy theory are applied to a family of examples from singularity theory let a be a finite collection of hyperplanes in cl and let m cl uhea h we say a is a rational k w 1 arrangement if the rational completion of m is aspherical for these arrangements an identity the lcs formula is established relating the lower central series of 7r1 m to the cohomology of m this identity was established by group theoretic means for the class of fiber type arrangements in previous work we reproduce this result by showing that the class of rational k w 1 arrangements contains all fiber type arrangements this class includes the reflection arrangements of types al and bl there is much interest in arrangements for which m is a k w 1 space the methods developed here do not apply directly because m is rarely a nilpotent space we give examples of k w 1 arrangements which are not rational k w 1 for which the lcs formula fails and k w 1 arrangements which are not rational k w 1 where the lcs formula holds it remains an open question whether rational k w 1 arrangements are necessarily k w 1 l introduction an arrangement of hyperplanes is a finite collection of clinear subspaces of dimension 1 l in c1 to such an arrangement a is associated an open 21 manifold the complement m cl u hih e a the connections between the topology of m and the combinatorial geometry of a are the source of much current research in this area the most successful investigations concern the cohomology of m 13 15 whereas the most difficult unsolved problems involve the homotopy groups of m 5 in this paper we study the link between cohomology and homotopy provided by sullivan s theory of minimal models 14 in 6 a numerical relationship was established between 1 m and h m for the class of fiber type arrangements this lcs formula reads as follows ri 1 tn y7n m pm t n l where the n m are the ranks of successive quotients in the lower central series of 1 m and pm t is the poincare polynomial of h m because the sequence n m is related to the l minimal model y of m we conjecture in 5 that the lds formula holds precisely when r determines h m with the methods developed in this paper we can resolve this conjecture specifically we show corollary 3 8 that the lcs formula holds when h 5 is isomorphic to h m arrangements satisfying the latter condition are called rational k 7r l corollary 3 8 may be received by the editors june 1 1987 1980 mathematicts subject clatstsification 1985 revitsion primary 55p62 57m05 secondary 06c10 r 1988 american mathematical society 0002 9947 88 1 00 25 per page

fiw modules and stability criteria for representations of classical weyl groups abstract in this paper we develop machinery for studying sequences of representations of any of the three families of classical weyl groups extending work of church ellenberg farb and nagpal 7 8 on the symmetric groups s n to the signed permutation groups b n and the even signed permutation groups d n for each family w n we present an algebraic framework where a sequence v n of w n representations is encoded into a single object we call an fi w module we prove that if an fi w module v satisfies a simple finite generation condition then the structure of the sequence is highly constrained one consequence is that the sequence is uniformly representation stable in the sense of church farb that is the pattern of irreducible representations in the decomposition of each v n eventually stabilizes in a precise sense using the theory developed here we obtain new results about the cohomology of generalized flag varieties associated to the classical weyl groups and more generally the r diagonal coinvariant algebras we analyze the algebraic structure of the category of fi w modules and introduce restriction and induction operations that enable us to study interactions between the three families of groups we use this theory to prove analogues of murnaghan s 1938 stability theorem for kronecker coefficients for the families b n and d n the theory of fi w modules gives a conceptual framework for stability results such as these

the stable moduli space of riemann surfaces mumford s conjecture d mumford conjectured in 30 that the rational cohomology of the stable moduli space of riemann surfaces is a polynomial algebra generated by certain classes i of di mension 2i for the purpose of calculating rational cohomology one may replace the stable moduli space of riemann surfaces by b 1 where 1 is the group of isotopy classes of automorphisms of a smooth oriented connected surface of large genus tillmann s insight 41 that the plus construction makes b 1 into an infinite loop space led to a stable homo topy version of mumford s conjecture stronger than the original 22 we prove the stronger version relying on harer s stability theorem 15 vassiliev s theorem concerning spaces of functions with moderate singularities 43 42 and methods from homotopy theory

automorphisms and cohomology of discrete groups let g denote a discrete group of finite cohomological dimension calculating the cohomology of such groups is notoriously difficult involving complicated geometric information attached to the group for example the cohomology of torsion free arithmetic groups involves delicate questions about symmetric spaces and number theory perhaps the key difficulty lies in that there is no practical method for building up the cohomology of g from that of its subgroups such as one can do for finite groups in this paper we outline a method for constructing non trivial classes in u h g f where f denotes a field with p elements based on the use of p p finite automorphism groups of g given an explicit presentation for g aut g can often be approached and in particular its finite subgroups are sometimes accessible furthermore it is an elementary observation that many interesting classes of groups admit numerous finite symmetries an obvious but very important example is given by a normal torsion free subgroup g in an arithmetic group u any finite subgroup k u will act on it via conjugation given g aut g we can form the semidirect product g s g g w let h g denote a finite subgroup mapping onto g under the natural projection map then c h s c h l g g and in the particular case g g 4 when h s 1 x x g g we have that

the no response test a sampling method for inverse scattering problems we describe a novel technique which we call the no response test to locate the support of a scatterer from knowledge of a far field pattern of a scattered acoustic wave the method uses a set of sampling surfaces and a special test response to detect the support of a scatterer without a priori knowledge of the physical properties of the scatterer specifically the method does not depend on information about whether the scatterer is penetrable or impenetrable nor does it depend on any knowledge of the nature of the scatterer absorbing reflecting etc in contrast to previous sampling algorithms the techniques described here enable one to locate obstacles or inhomogeneities from the far field pattern of only one incident field the no response test is a one wave method we investigate the theoretical basis for the no response test and derive a one wave uniqueness proof for a region containing the scatterer we show how to find the object within this region we demonstrate the applicability of the method by reconstructing sound soft sound hard impedance and inhomogeneous medium scatterers in two dimensions from one wave with full and limited aperture far field data

fi modules and stability for representations of symmetric groups in this paper we introduce and develop the theory of fi modules we apply this theory to obtain new theorems about the cohomology of the configuration space of n distinct ordered points on an arbitrary connected oriented manifold the diagonal coinvariant algebra on r sets of n variables the cohomology and tautological ring of the moduli space of n pointed curves the space of polynomials on rank varieties of n n matrices the subalgebra of the cohomology of the genus n torelli group generated by h1 and more the symmetric group sn acts on each of these vector spaces in most cases almost nothing is known about the characters of these representations or even their dimensions we prove that in each fixed degree the character is given for n large enough by a polynomial in the cycle counting functions that is independent of n in particular the dimension is eventually a polynomial in n in this framework representation stability in the sense of church farb for a sequence of sn representations is converted to a finite generation property for a single fi module

representation theory and homological stability abstract we introduce the idea of representation stability and several variations for a sequence of representations v n of groups g n a central application of the new viewpoint we introduce here is the importation of representation theory into the study of homological stability this makes it possible to extend classical theorems of homological stability to a much broader variety of examples representation stability also provides a framework in which to find and to predict patterns from classical representation theory littlewood richardson and murnaghan rules stability of schur functors to cohomology of groups pure braid torelli and congruence groups to lie algebras and their homology to the equivariant cohomology of flag and schubert varieties to combinatorics the n 1 n 1 conjecture the majority of this paper is devoted to exposing this phenomenon through examples in doing this we obtain applications theorems and conjectures beyond the discovery of new phenomena the viewpoint of representation stability can be useful in solving problems outside the theory in addition to the applications given in this paper it is applied by church ellenberg farb in preparation 20 to counting problems in number theory and finite group theory representation stability is also used by church 2012 19 to give broad generalizations and new proofs of classical homological stability theorems for configuration spaces on oriented manifolds

spectral triples and associated connes de rham complex for the quantum su 2 and the quantum sphere in this article we construct spectral triples for the c algebra of continuous functions on the quantum su 2 group and the quantum sphere there have been various approaches towards building a calculus on quantum spaces but there seem to be very few instances of computations outlined in chapter 6 5 we give detailed computations of the associated connes de rham complex and the space of l2 forms