stacks similar to the stack of perverse sheaves we introduce on a topological space x a class of stacks of abelian categories we call stacks of type p this class of stacks includes the stack of perverse sheaves of any perversity constructible with respect to a fixed stratification and is singled out by fairly innocuous axioms we show that some basic structure theory for perverse sheaves holds for a general stack of type p such a stack is locally equivalent to a macpherson vilonen construction and under certain connectedness conditions its category of global objects is equivalent to the category of modules over a finite dimensional algebra to prove these results we develop a rudimentary tilting formalism for stacks of type p another sense in which these stacks are similar to stacks of perverse sheaves

character sheaves v this paper is part of a series s 131 devoted to the study of a class g of irreducible perverse sheaves called character sheaves on a connected reductive algebraic group g the numbering of chapters sections and references will continue that of s 131 this paper is a step towards the classification of character sheaves on g one of the main results is the following one under certain assumptions there is a natural surjective map with finite fibers from g to the set of all pairs 9 c up to conjugacy by the weyl group where 9 is a tame local system on the maximal torus and c is a two sided cell in the stabilizer w of 9 in the weyl group the assumptions made on g are

on category o over triangular generalized weyl algebras abstract we analyze the bgg category o over a large class of generalized weyl algebras henceforth termed gwas given such a triangular gwa for which category o decomposes into a direct sum of subcategories we study in detail the homological properties of blocks with finitely many simples as consequences we show that the endomorphism algebra of a projective generator of such a block is quasi hereditary finite dimensional and graded koszul we also classify all tilting modules in the block as well as all submodules of all projective and tilting modules finally we present a novel connection between blocks of triangular gwas and young tableaux which provides a combinatorial interpretation of morphisms and extensions between objects of the block

perverse sheaves of categories and some applications abstract we study perverse sheaves of categories their connections to classical algebraic geometry we show how perverse sheaves of categories encode naturally derived categories of coherent sheaves on p 1 bundles semiorthogonal decompositions and relate them to a recent proof of segal that all autoequivalences of triangulated categories are spherical twists furthermore we show that perverse sheaves of categories can be used to represent certain degenerate calabi yau varieties

on a class of holonomic d modules on mn c related to the action of gln c gln c abstract we give an algebraic description of regular holonomic d modules whose characteristic variety is the union of conormal bundles to the orbits of the action of the general linear group on square matrices

singular chain intersection homology for traditional and super perversities we introduce a singular chain intersection homology theory which generalizes that of king and which agrees with the deligne sheaf intersection homology of goresky and macpherson on any topological stratified pseudomanifold compact or not with constant or local coecients and with traditional perversities or superperversities those satisfying p 2 0 for the case p 2 1 these latter perversitie were introduced by cappell and shaneson and play a key role in their superduality theorem for embeddings we further describe the sheafification of this singular chain complex and its adaptability to broader classes of stratified spaces 2000 mathematics subject classification primary 55n33 secondary 32s60 57n80

the invariant holonomic system on a semisimple lie algebra on etudie la theorie geometrique des representations du groupe de weyl du point de vue des systemes holonomes sur des algebres de lie

witt spaces a geometric cycle theory for ko homology at odd primes introduction this paper studies a class of stratified piecewiselinear pseudomanifolds which we call witt spaces characterized by natural local intersection homology conditions 11 we compute the cobordism groups by introducing an invariant taking values in the witt group of symmetric bilinear forms over the rationals w q these pseudomanifolds solve a problem posed by d sullivan 25 to construct a class of p l cycles with signature which represent the connected version of ko homology at odd primes ko 0d z 1 2 the result was one of the first applications of the intersection homology theory of goresky and macpherson 11 specifically the rational intersection homology groups of witt spaces satisfy a poincare duality theorem to each witt space x we associate a p l invariant w x with values in w q this invariant generalizes the signature of manifolds and satisfies cobordism invariance additivity and a product formula adapting classical surgery spherical modification to this setting we prove that w x determines the cobordism class of x and obtain an explicit description of the witt cobordism groups the only nontrivial groups occur in dimensions 4k and for k 0 they are w q sullivan 25 27 has shown that a class of p l pseudomanifolds equipped with such an invariant possesses canonical orientations in ko 0 z 1 2 the orientations induce a natural transformation it of homology theories from witt space bordism to ko 0 z 1 2 the structure of w q is known 14 and we conclude that it is an equivalence at odd primes in this paper we study the cobordism theory of a class of stratified p l pseudomanifolds by means of the recent intersection homology theory of goresky and macpherson 11 goresky and macpherson were interested in finding a class of spaces with cobordism invariant signature for the purpose of extending the hirzebruch l class to the setting of manifolds with

assignment of tools to machines in a flexible manufacturing system abstract subgradient optimization is employed to solve the problem of assigning tools to machines in a flexible manufacturing system fms machines in the fms have a limited number of slots in the tool mafazine and tools may require multiple slots tools are grouped based upon a pairwise similarity coefficient that indicates the frequency of successive operations that require both tools typical solution strategies have included graph theoretic heuristics lagrangian relaxation is utilized in the algorithm developed in this article the problem is formulated as a linear integer program after dualizing two sets of constraints two integer subproblems are formed in which the first is further decomposed into several knapsack subproblems the second subproblem can be solved by a linear network code a subgradient algorithm is developed to solve the dual problem the algorithm is compared to a graph theoretic heuristic that utilizes cluster analysis for most test problems the subgradient algorithm is superior but is sensitive to convergence parameters

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

perverse sheaves koszul ic modules and the quiver for the category mathcal o for a stratified topological space we introduce the category of ic modules which are linear algebra devices with the relations described by the equation d 2 0 we prove that the category of mixed ic modules is equivalent to the category of mixed perverse sheaves for flag varieties as an application we describe an algorithm calculating the quiver underlying the bgg category mathcal o for arbitrary simple lie algebra thus answering a question which goes back to i m gelfand

koszul duality for toric varieties we show that certain categories of perverse sheaves on affine toric varieties x and x v defined by dual cones are koszul dual in the sense of beilinson ginzburg and soergel 1996 the functor expressing this duality is constructed explicitly by using a combinatorial model for mixed sheaves on toric varieties

on the uniqueness of bivariant chern class and bivariant riemann roch transformations the existence of bivariant chern classes was conjectured by w fulton and r macpherson and proved by j p brasselet for cellular morphisms of analytic varieties however its uniqueness has been unsolved since then in this paper we show that restricted to morphisms whose target varieties are possibly singular but rational homology manifolds such as orbifolds the bivariant chern classes with rational coefficients are uniquely determined we also discuss some related results and problems

solving machine loading problem of a flexible manufacturing system with constraint based genetic algorithm abstract machine loading problem of a flexible manufacturing system is known for its complexity this problem encompasses various types of flexibility aspects pertaining to part selection and operation assignments along with constraints ranging from simple algebraic to potentially very complex conditional constraints from the literature it has been seen that simple genetic algorithm based heuristics for this problem lead to constraint violations and large number of generations this paper extends the simple genetic algorithm and proposes a new methodology constraint based genetic algorithm cbga to handle a complex variety of variables and constraints in a typical fms loading problem to achieve this aim three new genetic operators constraint based initialization crossover and mutation are introduced the methodology developed here helps avoid getting trapped at local minima the application of the algorithm is tested on standard data sets and its superiority is demonstrated the solution approach is illustrated by a simple example and the robustness of the algorithm is tested on five well known functions

the decomposition theorem perverse sheaves and the topology of algebraic maps we give a motivated introduction to the theory of perverse sheaves culminating in the decomposition theorem of beilinson bernstein deligne and gabber a goal of this survey is to show how the theory develops naturally from classical constructions used in the study of topological properties of algebraic varieties while most proofs are omitted we discuss several approaches to the decomposition theorem indicate some important applications and examples

d modules on representations of capelli type abstract let g v be an irreducible multiplicity free finite dimensional representation of a connected reductive complex group g as classified by v g kac 17 and g its derived subgroup denote by g the lie algebra of g and u g its universal enveloping algebra assume that there exists a polynomial f generating the algebra of g invariant polynomials on v c v g c f and such that f c v g such representations are said to be of capelli type if the algebra of g invariant differential operators is the image of the center of u g under the differential of the g action they fall into eight cases given by r howe and t umeda 14 five infinite families and three exceptional examples we prove that the category of regular holonomic d v modules invariant under the action of g is equivalent to the category of graded modules of finite type over a suitable algebra a except for few special cases indeed the levasseur s conjecture 28 conjecture 5 17 p 508 fails in these cases because of the disconnectedness of the stabilizers of some smaller orbits

on the reductive borel serre compactification l p cohomology of arithmetic groups for large p the l 2 cohomology of an arithmetic quotient of an hermitian symmetric space i e of a locally symmetric variety is known to have the topological interpretation as the intersection homology of its baily borel satake compactification in this article we observe that even without the hermitian hypothesis the l p cohomology of an arithmetic quotient for p finite and sufficiently large is isomorphic to the ordinary cohomology of its reductive borel serre compactification we use this to generalize a theorem of mumford concerning homogeneous vector bundles their invariant chern forms and the canonical extensions of the bundles here though we are referring to canonical extensions to the reductive borel serre compactification of any arithmetic quotient to achieve that we give a systematic discussion of vector bundles and chern classes on stratified spaces

homological theory of recollements of abelian categories abstract we investigate several homological aspects of recollements of abelian categories in particular we study how various homological invariants and dimensions of the categories involved in a recollement situation are related and when recollements of abelian categories induce recollements at the level of the bounded derived categories finally we give applications to global finitistic and representation dimension of rings and artin algebras and to rouquier s dimension of triangulated categories

on categories of equivariant d modules abstract let x be a variety with an action by an algebraic group g in this paper we discuss various properties of g equivariant d modules on x such as the decompositions of their global sections as representations of g when g is reductive and descriptions of the categories that they form when g acts on x with finitely many orbits the category of equivariant d modules is equivalent to the category of finite dimensional representations of a finite quiver with relations we describe explicitly these categories for irreducible g modules x that are spherical varieties and show that in such cases the quivers are almost always representation finite i e with finitely many indecomposable representations

gorenstein projective modules and symmetric recollements we introduce compatible bimodules if m is a compatible a b bimodule then the gorenstein projective modules over algebra am0b are explicitly described and if is gorenstein then this description implies that m is compatible as an application if m is compatible then there is a left recollement of the stable category gp and if is gorenstein and ma is projective then there is a symmetric recollement of the singularity category dsgb

perverse sheaves over real hyperplane arrangements let h be an arrangement of real hyperplanes in r the complexification of h defines a natural stratification of c we denote by perv c h the category of perverse sheaves on c smooth with respect to this stratification we give a description of perv c h as the category of representations of an explicit quiver with relations whose vertices correspond to real faces of h of all dimensions the relations are of monomial nature they identify some pairs of paths in the quiver they can be formulated in terms of the oriented matroid associated to h

bimodule monomorphism categories and rss equivalences via cotilting modules abstract the monomorphism category m a m b induced by a bimodule m b a is the subcategory of mod consisting of x y such that m b y x is a monic a map where a m 0 b and a b are artin algebras in general m a m b is not the monomorphism category induced by quivers it could describe the gorenstein projective modules this monomorphism category is a resolving subcategory of mod if and only if m b is projective in this case it has enough injective objects and auslander reiten sequences and can be also described as the left perpendicular category of a unique basic cotilting module if m satisfies the condition ip see subsection 1 6 then the stable category of m a m b admits a recollement of additive categories which is in fact a recollement of singularity categories if m a m b is a frobenius category ringel schmidmeier simson equivalence between m a m b and its dual is introduced if m is an exchangeable bimodule then an rss equivalence is given by a bimodule which is a two sided cotilting module with a special property and the nakayama functor n gives an rss equivalence if and only if both a and b are frobenius algebras

moduli of regular holonomic d modules with normal crossing singularities this paper solves the global moduli problem for regular holonomic dmodules with normal crossing singularities on a nonsingular complex projective variety this is done by introducing a level structure which gives rise to pre d modules and then introducing a notion of semi stability and applying geometric invariant theory to construct a coarse moduli scheme for semistable pre d modules a moduli is constructed also for the corresponding perverse sheaves and the riemann hilbert correspondence is represented by an analytic morphism between these moduli spaces

perverse sheaves and finite dimensional algebras we study finite dimensional algebras which arise from categories of perverse sheaves on complex algebraic varieties

intersection homology ii in 19 20 we introduced topological invariants ih x called intersection homology groups for the study of singular spaces x these groups depend on the choice of a perversity p a perversity is a function from 2 3 to the non negative integers such that both c and c 2 c are positive and increasing functions of c 2 1 the group ihr is defined for spaces x called pseudomanifolds a pseudomanifold of dimension n is a space that admits a stratification x x n x n z x n 3 x 1 x 0

equivariant d modules on 2 2 2 hypermatrices abstract let v c 2 c 2 c 2 be the space of 2 2 2 hypermatrices endowed with the natural group action of gl gl 2 c gl 2 c gl 2 c the category of gl equivariant coherent left d modules on v is equivalent to the category of representations of a quiver with relations in this article we give a construction of each simple object and study their gl equivariant structure using this information we go on to explicitly describe the corresponding quiver with relations as an application we compute all iterations of local cohomology with support in the orbit closures of v

the defect recollement the macpherson vilonen construction and pp formulas abstract for any abelian category a auslander constructed a localisation w fp a op ab a called the defect which is the left adjoint to the yoneda embedding y a fp a op ab if a has enough projectives then this localisation is part of a recollement called the defect recollement we show that this recollement is an instance of the macpherson vilonen construction if and only if a is hereditary we also discuss several subcategories of fp a op ab which arise as canonical features of the defect recollement and characterise them by properties of their projective presentations and their orthogonality with other subcategories we apply some parts of the defect recollement to the model theory of modules let r be a ring and let be a pp pair when r is an artin algebra we show that there is a smallest pp formula such that which agrees with on injectives and that there is a largest pp formula such that and r r when r is left coherent we show that there is a largest pp formula such that which agrees with on injectives and that the pp pair is isomorphic to a pp formula if and only if and that there is a smallest pp formula such that and r r we also show that for any pp pair w d r d r where d is the elementary duality of pp formulas we also give expressions for w in terms of free realisation of and

microdifferential systems and the codimension three conjecture in this paper we give a proof of a fundamental conjecture the codimension three conjecture for microdierential holonomic systems the conjecture states that any regular holonomic module extends uniquely beyond an analytic subset that is at least of codimension three in its support our result can also be interpreted from a topological point of view as a statement about microlocal perverse sheaves however our proof is entirely in the context of microdierential holonomic systems

harmonic maps between surfaces and teichmuller spaces we study harmonic maps between two distinct compact riemann surfaces of the same genus our aim is to examine the role of the curvature in the relationship between energies of harmonic maps and conformal metrics on the target surface we consider the energy of the energy minimizing map between the surfaces as a function of the metrics within the conformal class of the target and solve the variational problem of minimizing this functional when we restrict the upper curvature bound and normalize the area of the metrics in doing so we show that the upper curvature bound on the normalized target metric is an obstruction in obtaining a degree 1 conformal harmonic map between distinct riemann surfaces

geometric langlands duality and representations of algebraic groups over commutative rings as such it can be viewed as a first step in the geometric langlands program the connected complex reductive groups have a combinatorial classification by their root data in the root datum the roots and the co roots appear in a symmetric manner and so the connected reductive algebraic groups come in pairs if g is a reductive group we write g for its companion and call it the dual group g the notion of the dual group itself does not appear in satake s paper but was introduced by langlands together with its various elaborations in li l2 and is a cornerstone of the langlands program it also appeared later in physics mo gno in this paper we discuss the basic relationship between g and g we begin with a reductive g and consider the affine grassmannian qx the grassmannian for the loop group of g for technical reasons we work with formal algebraic loops the affine grassmannian is an infinite dimen sional complex space we consider a certain category of sheaves the spherical perverse sheaves on r these sheaves can be multiplied using a convolution product and this leads to a rather explicit construction of a hopf algebra by what has come to be known as tannakian formalism the resulting hopf algebra turns out to be the ring of functions on g in this interpretation the spherical perverse sheaves on the affine grassman nian correspond to finite dimensional complex representations of g thus instead of defining g in terms of the classification of reductive groups we pro vide a canonical construction of g starting from g we can carry out our construction over the integers the spherical perverse sheaves are then those with integral coefficients but the grassmannian remains a complex algebraic object

intersection homology with field coefficients k witt spaces and k witt bordism we construct geometric examples of pseudomanifolds that satisfy the witt condition for intersection homology poincare duality with respect to certain fields but not others we then compute the bordism theory of k witt spaces for an arbitrary field k extending results of siegel for k 2008 wiley periodicals inc

gorenstein defect categories of triangular matrix algebras abstract we apply the technique of recollement to study the gorenstein defect categories of triangular matrix algebras first we construct a left recollement of gorenstein defect categories for a triangular matrix algebra under some conditions using it we give a categorical interpretation of the gorenstein properties of the triangular matrix algebras obtained by x w chen b l xiong and p zhang respectively second under some additional conditions a recollement of gorenstein defect categories for a triangular matrix algebra is constructed as an application for a special kind of triangular matrix algebras which are called simple gluing algebras we describe their singularity categories and gorenstein defect categories

geometry and braiding of stokes data fission and wild character varieties a family of new algebraic poisson varieties will be constructed generalising the complex character varieties of riemann surfaces then the well known poisson mapping class group actions on the character varieties will be generalised

recollements of abelian categories and special types of comma categories abstract in this paper we first study conditions under which a recollement relative to abelian categories induces a new recollement relative to abelian categories and comma categories then we apply our results to deduce results about recollements relative to categories of modules furthermore if a ring r is morita equivalent to s we obtain that the upper triangular matrix ring r m 0 t is morita equivalent to s n 0 t for an arbitrary ring t r t bimodule m and s t bimodule n depending on m

additivity and non additivity for perverse signatures a well known property of the signature of closed oriented 4n dimensional manifolds is novikov additivity which states that if a manifold is split into two manifolds with boundary along an oriented smooth hypersurface then the signature of the original manifold equals the sum of the signatures of the resulting manifolds with boundary wall showed that this property is not true of signatures on manifolds with boundary and that the dierence from additivity could be described as a certain maslov triple index perverse signatures are signatures dened for any stratied pseudomanifold using the intersection homology groups of goresky and macpherson in the case of witt spaces the middle perverse signature is the same as the witt signature this paper proves a generalization to perverse signatures of wall s non additivity theorem for signatures of manifolds with boundary under certain topological conditions on the dividing hypersurface novikov additivity for perverse signatures may be deduced as a corollary in particular siegel s version of novikov additivity for witt signatures is a special case of this corollary

solving capacitated clustering problems abstract this paper presents two effective algorithms for clustering n entities into p mutually exclusive and exhaustive groups where the size of each group is restricted as its objective the clustering model minimizes the sum of distance between each entity and a designated group median empirical results using both a primal heuristic and a hybrid heuristic subgradient method for problems having n 100 i e 10 100 binary variables show that the algorithms locate close to optimal solutions without resorting to tree enumeration the capacitated clustering model is applied to the problem of sales force territorial design

on the weil petersson metric on teichmuller space teichmuller space for a compact oriented surface m without boundary is described as the quotient ji b0 where s is the space of almost complex structures on m compatible with a given orientation and 8 0 are those c diffeomorphisms homotopic to the identity there is a natural 3 0 invariant l2 riemannian structure on si which induces a riemannian structure on si 3 n infinitesimally this is the bilinear pairing suggested by andre weil the weil petersson riemannian struc ture the structure is shown to be kahler with respect to a naturally induced complex structure on s0

whitney stratified chains and cochains this paper contains the technical constructions necessary for a geometric cycle definition of cohomology and homology in the context of whitney stratifications cup and cap products are interpreted as the transverse intersection of geometric cocycles and cycles