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

on the homology of the hilbert scheme of points in the plane geir ellingsrud 1 and stein arild stromme 2 i matematisk institutt universitetet i oslo blindern n oslo 3 norway 2 matematisk institutt universitetet i bergen n 5014 bergen norway although several authors have been interested in the hilbert scheme hilba ip z parametrizing finite subschemes of length d in the projective plane i1 i2 f 1 f2 br among others not much is known about the topological properties of this space the picard group has been calculated i f2 and the homology groups of hilb3 ip 2 have been computed hi in this paper we give a precise description of the additive structure of the homology of hilba ip2 applying the results of birula bialynicki b1 b2 on the cellular decompositions de fined by a torus action to the natural action of a maximal torus of sl 3 on hilbe ipz a rather easy consequence of the fact that this action has finitely many fixpoints is that the cycle maps between the chow groups and the homology groups are isomorphisms in particular there is no odd homology and the homology groups are all free the main objective of this work is to compute their ranks the betti numbers of hilba ipz as a byproduct of our method we get similar results on the homology of the punctual hilbert scheme and of the hilbert scheme of points in the affine plane it seems natural to generalize our results to any toric smooth surface however we give the results only for the rational ruled surfaces if with an indication of the necessary changes in the proofs for simplicity we work over the field of complex numbers but with an appropriate interpretation of the word homology our results remain valid over any base field

on decomposition matrices for graded algebras abstract we study the relations between the decomposition matrix of a module over a graded algebra and the decomposition matrix of its restriction to the grading subalgebra we show that under appropriate hypotheses if one of the decomposition matrices is stable unitriangular then so is the other a particular case is when one of them is square unitriangular then both are using these results we describe the decomposition matrices of the hecke algebras of the complex reflection groups g de e n with respect to the ones of the ariki koike algebras or the decomposition matrices of q schur algebras of type dn with respect to the ones of q schur algebras of type bn

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

asymptotically fast factorization of integers the paper describes a probabilistic algorithm for finding a factor of any large composite integer n the required input is the integer n together with an auxiliary sequence of random numbers it is proved that the expected number of operations which will be required is o exp 83qn n in in n l 2 for some constant f 0 asymptotically this algorithm is much faster than any previously analyzed algorithm for factoring integers earlier algorithms have all required o na operations where a 1 5

blocks of cyclotomic hecke algebras abstract this paper classifies the blocks of the cyclotomic hecke algebras of type g r 1 n over an arbitrary field rather than working with the hecke algebras directly we work instead with the cyclotomic schur algebras the advantage of these algebras is that the cyclotomic jantzen sum formula gives an easy combinatorial characterization of the blocks of the cyclotomic schur algebras we obtain an explicit description of the blocks by analyzing the combinatorics of jantzen equivalence we remark that a proof of the classification of the blocks of the cyclotomic hecke algebras was announced in 1999 unfortunately cox has discovered that this previous proof is incomplete

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

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

on the category o for rational cherednik algebras we study the category o of representations of the rational cherednik algebra aw attached to a complex reflection group w w e construct an exact functor called knizhnik zamolodchikov functor o hw mod where hw is the finite iwahori hecke algebra associated to w we prove that the knizhnik zamolodchikov functor induces an equiv alence between o otor the quotient of o by the subcategory of aw modules supported on the discriminant and the category of finite dimensional hw modules the standard aw modules go under this equivalence to certain modules arising in kazhdan lusztig theory of cells provided w is a weyl group and the hecke algebra hw has equal parameters we prove that the category o is equivalent to the module category over a finite dimensional algebra a generalized q schur algebra associated to w

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

reductions among number theoretic problems abstract many number theoretic problems such as integer factorization and the discrete logarithm problem have defied all attempts to classify their complexities thirteen such problems are considered none of which is known either to have a deterministic polynomial time solution or to be complete for any natural complexity class failing this the next best goal is to determine which among these are the easiest and which are the hardest problems toward this end this paper gives an overview of reductions among the problems two reductions are new a deterministic polynomial time reduction from squarefreeness to euler s function n and a probabilistic polynomial time reduction from order modulo a prime power to discrete logarithm modulo a prime power

some remarks on computing the square parts of integers abstract let n be a positive integer and suppose n p i a i is its prime factorization let n p i a i 1 so that n n is the largest squarefree factor of n we show that is deterministic polynomial time turing reducible to the euler function we also show that is reducible to the carmichael function we survey other recent work on computing the square part of an integer and give upper and lower bounds on the complexity of solving the problem

schur weyl reciprocity between quantum groups and hecke algebras of type g p p n abstract let u q gl m p be the quantized universal enveloping algebra of gl m p let be the automorphism of u q gl m p which is defined on generators by e i e i m f i f i m k j k j m for any i z p m z 0 m p 1 m and any j z p m z let h p p n be the hecke algebra of type g p p n with parameters q e where e is a primitive pth root of unity in this paper we establish a schur weyl reciprocity between h p p n and a twisted tensor product of u q gl m p and the group algebra for a cyclic group of order p by using the results in j hu j algebra 274 2004 446 490

characteristic varieties and vanishing cycles on presente une etude systematique des varietes caracteristiques des systemes holonomes a singularites regulieres ce sont des sous varietes lagrangiennes coniques dans le fibre cotangent t x

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

microlocalization of rational cherednik algebras we construct a microlocalization of the rational cherednik algebras h of type sn this is achieved by a quantization of the hilbert scheme hilb c2 of n points in c2 we then prove the equivalence of the category of h modules and that of modules over its microlocalization under certain conditions on the parameter

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

almost conservation law and global rough solutions to a nonlinear davey stewartson equation abstract we prove an almost conservation law to obtain global in time well posedness for the nonlinear davey stewartson equation in h s r 2 and s 4 7

the number of simple modules for the hecke algebras of type g r p n with an appendix by xiaoyi cui abstract we derive a parameterization of simple modules for the cyclotomic hecke algebras of type g r p n with p 1 and n 3 over fields of any characteristic coprime to p we give explicit formulas for the number of simple modules over these cyclotomic hecke algebras

on the decomposition numbers of the hecke algebra of type dn when n is even abstract let n 4 be an even integer let k be a field with char k 2 and q an invertible element in k such that i 1 n 1 1 q i 0 in this paper we study the decomposition numbers over k of the iwahori hecke algebra h q d n of type d n we obtain some equalities which relate its decomposition numbers with certain schur elements and the decomposition numbers of various iwahori hecke algebras of type a with the same parameter q when char k 0 this completely determine all of its decomposition numbers the main tools we used are the morita equivalence theorem established in j hu a morita equivalence theorem for hecke algebra h q d n when n is even manuscripta math 108 2002 409 430 and certain twining character formulae of weyl modules over a tensor product of two q schur algebras

divisors in residue classes constructively let r s n be integers satisfying 0 r 1 4 and let gcd r s 1 lenstra showed that the number of integer divisors of n equivalent to r mod s is upper bounded by o 1 4 2 we reexamine this problem showing how to explicitly construct all such divisors and incidentally improve this bound to o 1 4 3 2

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

hamiltonian reduction and nearby cycles for mirabolic d modules abstract we study holonomic d modules on sl n c c n called mirabolic modules analogous to lusztig s character sheaves we describe the supports of simple mirabolic modules we show that a mirabolic module is killed by the functor of hamiltonian reduction from the category of mirabolic modules to the category of representations of the trigonometric cherednik algebra if and only if the characteristic variety of the module is contained in the unstable locus we introduce an analogue of verdier s specialization functor for representations of cherednik algebras which agrees on category o with the restriction functor of bezrukavnikov and etingof in type a we also consider a verdier specialization functor on mirabolic d modules we show that hamiltonian reduction intertwines specialization functors on mirabolic d modules with the corresponding functors on representations of the cherednik algebra this allows us to apply known hodge theoretic purity results for nearby cycles in the setting considered by bezrukavnikov and etingof

q tensor space and q weyl modules we obtain the irreducible representations of the q schur algebra motivated by the fact that these representations give all the irreducible representations of gln q in the nondescribing characteristic the irreducible polynomial representations of the general linear groups in the describing characteristic are a special case of this construction the theory of polynomial representations of general linear groups is equivalent to the representation theory of schur algebras see green s book 5 and the bibliography therein in 4 we defined q analogues of schur algebras when q 1 these are the usual schur algebras and when q is a prime power representations of q schur algebras give a substantial part of the representation theory of finite general linear groups in the nondescribing characteristic case including all irreducible representations of these groups and important information about decomposition numbers it is natural to ask what features of the classical schur algebras have qanalogues in this paper we define q analogues of tensor space of weyl modules and of weight spaces thereby generalizing the main reslllts which appear in green s book 5 for example we classify the irreducible modules for qschur algebras we determine bases for q weyl modules compatible with weight spaces and we give results on composition multiplicities of the irreducible modules in q weyl modules the proofs are largely self contained so by specializing q to 1 we recover the corresponding results in 5 this paper therefore is relevant to the representation theory of symmetric groups and to the representation theory of general linear groups in the describing and in the nondescribing characteristics 1 the q schur algebra let r be a natural number let r be an integral domain and let q be a unit in r we denote the symmetric group on r letters by er the hecke algebra z is the r free r algebra with basis twlw e er where the multiplication is determined by the following rule if a i i 1 is a basic transposition in received by the editors august 15 1989 1980 mathematics subject classification 198 5 revision primary 1 6a64 1 6a6 5 secondary 20c30 this research was supported in part by nsf grant no dms 8802290 the authors gratefully acknowledge support received from nato grant no 0222 87 r 1991 american mathematical society 0002 9947 91 1 00 25 per page

endpoint strichartz estimates we prove an abstract strichartz estimate which implies previously unknown endpoint strichartz estimates for the wave equation in dimension n 4 and the schrodinger equation in dimension n 3 three other applications are discussed local existence for a nonlinear wave equation and strichartz type estimates for more general dispersive equations and for the kinetic transport equation

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

dunkl operators for complex reflection groups dunkl operators for complex reflection groups are defined in this paper these commuting operators give rise to a parameterized family of deformations of the polynomial de rham complex this leads to the study of the polynomial ring as a module over the rational cherednik algebra and a natural contravariant form on this module in the case of the imprimitive complex reflection groups g m p n the set of singular parameters in the parameterized family of these structures is described explicitly using the theory of non symmetric jack polynomials

the jantzen sum formula for cyclotomic schur algebras the cyclotomic q schur algebra was introduced by dipper james and mathas in order to provide a new tool for studying the ariki koike algebra we here prove an analogue of jantzen s sum formula for the cyclotomic q schur algebra among the applications is a criterion for certain specht modules of the ariki koike algebras to be irreducible

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

factoring integers with elliptic curves this paper is devoted to the description and analysis of a new algorithm to factor positive integers it depends on the use of elliptic curves the new method is obtained from pollard s p 1 method proc cambridge philos soc 76 1974 521 528 by replacing the multiplicative group by the group of points on a random elliptic curve it is conjectured that the algorithm determines a non trivial divisor of a composite number n in expected time at most k p log n 2 where p is the least prime dividing n and k is a function for which log k x 2 o 1 log x log log x for x x o in the worst case when n is the product of two primes of the same order of magnitude this is

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

cellular structures on hecke algebras of type b abstract the aim of this paper is to gather and try to unify several approaches for the modular representation theory of hecke algebras of type b we attempt to explain the connections between geck s cellular structures coming from kazhdan lusztig theory with unequal parameters and ariki s theorem on the canonical basis of the fock spaces

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

explicit bounds for primality testing and related problems many number theoretic algorithms rely on a result of ankeny which states that if the extended riemann hypothesis erh is true any nontrivial multiplicative subgroup of the integers modulo m omits a number that is o log2 m this has been generalized by lagarias montgomery and odlyzko to give a similar bound for the least prime ideal that does not split completely in an abelian extension of number fields this paper gives a different proof of this theorem in which explicit constants are supplied the bounds imply that if the erh holds a composite number m has a witness for its compositeness in the sense of miller or solovay strassen that is at most 2 log2m

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

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

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

sums of divisors perfect numbers and factoring let n be a positive integer and let sigma n denote the sum of the divisors of n e g sigma 6 1 2 3 6 12 we show computing sigma n is equivalent to factoring n in the following sense there is a random polynomial time algorithm that given sigma n produces the prime factorization of n and sigma n can be computed in polynomial time given the factorization of n we show that the same result holds for sigma k n the sum of the kth powers of divisors of nwe give three new examples of problems that are in gill s complexity class bpp perfect numbers multiply perfect numbers and amicable pairs these are the first natural sets in bpp that are not obviously in rp

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

rational cherednik algebras and hilbert schemes ii representations and sheaves let h c be the rational cherednik algebra of type a n 1 with spherical subalgebra u c e h c e then u c is filtered by order of differential operators with associated graded ring gr u c mathbb c mathfrac h mathfrac h w where is the n th symmetric group using the z algebra construction from gs it is also possible to associate to a filtered h c or u c module hat m on the hilbert scheme hilb n using this technique we study the representation theory of u c and h c and we relate it to hilb n and to the resolution of singularities hilb n mathfrac h mathfrac h w for example we prove the following if c 1 n so that l c triv is the unique one dimensional simple h c module then hat e l c triv mathcal o z n where z n 1 0 is the punctual hilbert scheme if c 1 n k for k in mathbb n then under a canonical filtration on the finite dimensional module l c triv gr e l c triv has a natural bigraded structure that coincides with that on h 0 z n mathscr l k where mathscr l mathcal o hilb n 1 this confirms conjectures of berest etingof and ginzburg beg2 conjectures 7 2 7 3 under mild restrictions on c the characteristic cycle of hat e c equals sum k z where k are kostka numbers and the z are known irreducible components of 1 mathfrak h w

using partial smoothness of p 1 for factoring polynomials modulo p let an arbitrarily small positive constant less than 1 and a poly nomial f with integer coefficients be fixed we prove unconditionally that j modulo p can be completely factored in deterministic polynomial time if p 1 has a lnp o 1 smooth divisor exceeding p we also address the issue of factoring f modulo p over finite extensions of the prime field f p and show that p 1 can be replaced by p k 1 k n for explicit classes of primes p

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

symplectic reflection algebras calogero moser space and deformed harish chandra homomorphism to any finite group sp v of automorphisms of a symplectic vector space v we associate a new multi parameter deformation h of the algebra v smash product of with the polynomial algebra on v the parameter runs over points of r where r number of conjugacy classes of symplectic reflections in the algebra h called a symplectic reflection algebra is related to the coordinate ring of a poisson deformation of the quotient singularity v this leads to a symplectic analogue of mckay correspondence which is most complete in case of wreath products if is the weyl group of a root system in a vector space and v then the algebras h are certain rational degenerations of the double affine hecke algebra introduced earlier by cherednik

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

cyclotomic double affine hecke algebras and affine parabolic category o using the orbifold kz connection we construct a functor from an affine parabolic category o of type a to the category o of a cyclotomic rational double affine hecke algebra h then we give several results concerning this functor

characteristic cycles for the loop grassmannian and nilpotent orbits x which is a linear combination of closures of conormal bundles to submanifolds of x intuitively the microlocal multiplicities c me asurethesingularity of at in settings related to representation theory a group g acts on x is g equivariant and the microlocal multiplicities play a significant but only partially understood role in representation theory see ro sv abv and kasa e g in this paper we compute microlocal multiplicities for certain cases of interest in representation theory let g be a connected reductive group with loop group lg and let p bethesubgroup of lg consisting of loops with positive fourier coefficients then p orbits on theloop grassmannian lg p correspond to the irreducible representations l of thedual re ductivegroup g for dominant weights and of a torus of

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

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

a deterministic version of pollard s p 1 algorithm in this article we present applications of smooth numbers to the unconditional derandomization of some well known integer factoring algorithms we begin with pollard s p 1 algorithm which finds in random polynomial time the prime divisors p of an integer n such that p 1 is smooth we show that these prime factors can be recovered in deterministic polynomial time we further generalize this result to give a partial derandomization of the k th cyclotomic method of factoring k 2 devised by bach and shallit we also investigate reductions of factoring to computing euler s totient function we point out some explicit sets of integers n that are completely factorable in deterministic polynomial time given n these sets consist roughly speaking of products of primes p satisfying with the exception of at most two certain conditions somewhat weaker than the smoothness of p 1 finally we prove that o ln n oracle queries for values of are sufficient to completely factor any integer n in less than exp 1 o 1 ln n 1 3 ln ln n 2 3 deterministic time

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

a note on the cauchy problem of the generalized davey stewartson equations in this note we establish some local and global existence results for the cauchy problem of a class of nonlinear dispersive equations which generalize the nonlinear schrodinger equations and the davey stewartson equations these results improve some previously obtained results by some other authors when they are restricted to certain special equations

a bilinear estimate with applications to the kdv equation u x 0 u0 x where u0 h r our principal aim here is to lower the best index s for which one has local well posedness in h r i e existence uniqueness persistence and continuous dependence on the data for a finite time interval whose size depends on u0 hs equation in 1 1 was derived by korteweg and de vries 21 as a model for long wave propagating in a channel a large amount of work has been devoted to the existence problem for the ivp 1 1 for instance see 9 10 the inverse scattering method applies to this problem and under appropriate decay assumptions on the data several existence results have been established see 5 6 14 28 33 another approach inherited from hyperbolic problems relies on the energy estimates and in particular shows that 1 1 is locally well posed in h r for s 3 2 see 2 3 12 29 30 31 using these results and conservation laws global in time well posedness in h r s 2 was established see 3 12 30 also global in time weak solutions in the energy space h r were constructed in 34 in 13 and 22 a local smoothing effect for solutions of 1 1 was discovered this combined with the conservation laws was used in 13 and 22 to construct global in time weak solutions with data in h r and even in l r in 16 we introduced oscillatory integral techniques to establish local well posedness of 1 1 in h r s 3 4 and hence global in time well posedness in h r s 1 in 16 we showed how to obtain the above mentioned result by picard iteration in an appropriate function space in 4 j bourgain introduced new function spaces adapted to the linear operator t 3 x for which there are good bilinear estimates for the nonlinear term x u 2 using these spaces bourgain was able to establish local well posedness of 1 1 in h r l r and hence by a conservation