spectra of digraphs in this primarily expository paper we survey classical and some more recent results on the spectra of digraphs equivalently the spectra of 0 1 matrices with emphasis on the spectral radius

mixed braid group actions from deformations of surface singularities we consider a set of toric calabi yau varieties which arise as deformations of the small resolutions of type a surface singularities by careful analysis of the heuristics of b brane transport in the associated gauged linear sigma models we predict the existence of a mixed braid group action on the derived category of each variety and then prove that this action does indeed exist this generalizes the braid group action found by seidel and thomas for the undeformed resolutions we also show that the actions for different deformations are related in a way that is predicted by the physical heuristics

generalized calabi yau manifolds and the chiral de rham complex abstract we show that the chiral de rham complex of a generalized calabi yau manifold carries n 2 supersymmetry we discuss the corresponding topological twist for this n 2 algebra we interpret this as an algebroid version of the super sugawara or kac todorov construction

projection methods isotone projection cones and the complementarity problem one studies some iterative methods for solving complementarity problems explicit or implicit in hilbert spaces the iterative methods presented are based on the fixed point theory on the coincidence equations on convex cones and on some properties of projection operators onto a special cone named the isotone projection cone the ordering defined by the cone is essential for the complementarity theory the idea is to use systematically the ordering in the iterative methods for solving complementarity problems in this sense one shows that the isotone projection cone is a good instrument

non commutative crepant resolution of minimal nilpotent orbit closures of type a and mukai flops abstract in this article we construct a non commutative crepant resolution nccr of a minimal nilpotent orbit closure b 1 of type a and study relations between an nccr and crepant resolutions y and y of b 1 more precisely we show that the nccr is isomorphic to the path algebra of the double beilinson quiver with certain relations and we reconstruct the crepant resolutions y and y of b 1 as moduli spaces of representations of the quiver we also study the kawamata namikawa s derived equivalence between crepant resolutions y and y of b 1 in terms of an nccr we also show that the p twist on the derived category of y corresponds to a certain operation of the nccr which we call multi mutation and that a multi mutation is a composition of iyama wemyss s mutations

on the geometry of a siegel modular threefold introduction 317 1 compactifications 318 2 humbert surfaces 324 3 the chern numbers of s n 331 4 symmetric quartic threefolds 334 5 the modular threefold of level 2 335 6 kummer surfaces 338 7 prym surfaces 340 8 humbert surfaces again 345

the orbifold chow ring of toric deligne mumford stacks the orbifold chow ring of a deligne mumford stack defined by abramovich graber and vistoli 2 is the algebraic version of the orbifold cohomology ring in troduced by w chen and ruan 7 8 by design this ring incorporates numerical invariants such as the orbifold euler characteristic and the orbifold hodge num bers of the underlying variety the product structure is induced by the degree zero part of the quantum product in particular it involves gromov witten invariants inspired by string theory and results in batyrev 3 and yasuda 28 one expects that in nice situations the orbifold chow ring coincides with the chow ring of a resolution of singularities fantechi and g ttsche 14 and uribe 25 verify this conjecture when the orbifold is symn 5 where 5 is a smooth projective surface with ks 0 and the resolution is hilbn the initial motivation for this project was to compare the orbifold chow ring of a simplicial toric variety with the chow ring of a cr pant resolution to achieve this goal we first develop the theory of toric deligne mumford stacks modeled on simplicial toric varieties a toric deligne mumford stack corresponds to a combinatorial object called a stacky fan as a first approximation this object is a simplicial fan with a distinguished lattice point on each ray in the fan more precisely a stacky fan s is a triple consisting of a finitely generated abelian group n a simplicial fan e in q z n with n rays and a map zn n where the image of the standard basis in zn generates the rays in e a rational simplicial fan e produces a canonical stacky fan s n e where n is the distinguished lattice and is the map defined by the minimal lattice points on the rays hence there is a natural toric deligne mumford stack associated to every simplicial toric variety a stacky fan encodes a group action on a quasi affine variety and the toric deligne mumford stack is the quotient if e corresponds to a smooth toric variety x e and s is the canonical stacky fan associated to e then we simply have x z we show that many of the basic concepts such as open and closed toric substacks line bundles and maps between toric deligne mumford stacks correspond to combinatorial notions we expect that many more results about toric varieties lift to the realm of stacks and hope that toric deligne mumford stacks will serve as a useful testing ground for general theories

applications of homological mirror symmetry to hypergeometric systems duality conjectures abstract homological mirror symmetry for crepant resolutions of gorenstein toric singularities leads to a pair of conjectures on certain hypergeometric systems of pdes we explain these conjectures and verify them in some cases

stability conditions and crepant small resolutions in this paper we describe the spaces of stability conditions on the triangulated categories associated to three dimensional crepant small resolutions the resulting spaces have chamber structures such that each chamber corresponds to a birational model together with a special fourier mukai transform we observe that these spaces are covering spaces over certain open subsets of finite dimensional vector spaces and determine their deck transformations

mellin barnes integrals as fourier mukai transforms we study the generalized hypergeometric system introduced by gelfand kapranov and zelevinsky and its relationship with the toric deligne mumford dm stacks recently studied by borisov chen and smith we construct series solutions with values in a combinatorial version of the chen ruan orbifold cohomology and in the k theory of the associated dm stacks in the spirit of the homological mirror symmetry conjecture of kontsevich we show that the k theory action of the fourier mukai functors associated to basic toric birational maps of dm stacks are mirrored by analytic continuation transformations of mellin barnes type

an upper bound for the energy of radial digraphs abstract let d be a digraph with n vertices a arcs c 2 closed walks of length 2 and spectral radius d recently ayyaswamy balachandran and gutman 2011 1 proved that when d a c 2 2 n 1 it is possible to construct an upper bound for the energy of digraphs which improves the mcclelland inequality for the energy of strongly connected digraphs given in rada 2009 16 it is our interest in this paper to show that for general digraphs the inequality d a c 2 2 n does not hold however we introduce the class of radial digraphs that satisfy the spectral radius condition above and for this class of digraphs we improve the bound for the energy given in ayyaswamy et al 2011 1

flops flips and perverse point sheaves on threefold stacks 1 2 non gorenstein threefolds as stacks in this paper we are concerned with generalizing the results of br00 ch02 to some q gorenstein threefolds using algebraic stacks as commented in ch02 section 1 8 it should be pointed out that kawamata obtained some very general results ka01 ka02 also using algebraic stacks concentrating on equivalences or embeddings of derived categories in birational transformations to avoid excessive overlap with kawamata s work we emphasize the moduli construction of birational transformations

on the rigidity theorems of witten in this paper we prove the rigidity theorems predicted by witten in 1986 about the index of certain elliptic operators on manifolds with an s1 action w witten s insight was the culmination of an interesting interchange of ideas between him and hopkins landweber ochanine and stong for the detailed history we refer the reader to la the present account is essentially a reinterpretation of the second author s taubes t original proof of the theorem the senior author s contribution was solely to notice that the rather densely written arguments of the original manuscript could be formulated in terms of the well known fixed point formulae of equivariant index theory and equivariant cohomology in this context the final proof then appears in direct lineage of the atiyah hirzebruch theorem concerning the vanishing of the a genus of spin manifolds admitting a circle action a h and of the even older idea of lusztig concerning circle actions in the complex case there remains however the beautiful physics inspired novelty of connecting these techniques with elliptic function theory

on almost regular tournament matrices abstract spectral and determinantal properties of a special class m n of 2n 2n almost regular tournament matrices are studied in particular the maximum perron value of the matrices in this class is determined and shown to be achieved by the brualdi li matrix which has been conjectured to have the largest perron value among all tournament matrices of even order we also establish some determinantal inequalities for matrices in m n and describe the structure of their associated walk spaces

complex surfaces with equivalent derived categories we examine the extent to which a smooth minimal complex projective surface x is determined by its derived category of coherent sheaves dx to do this we find for each such surface x the set of surfaces y for which there exists a fourier mukai transform d y to d x

projective elliptic genera and elliptic pseudodifferential genera abstract in this paper we construct for the first time the projective elliptic genera for a compact oriented manifold equipped with a projective complex vector bundle such projective elliptic genera are rational q series that have topological definition and also have analytic interpretation via the fractional index theorem in 25 without requiring spin condition we prove the modularity properties of these projective elliptic genera as an application we construct elliptic pseudodifferential genera for any elliptic pseudodifferential operator this suggests the existence of putative s 1 equivariant elliptic pseudodifferential operators on loop space whose equivariant indices are elliptic pseudodifferential genera

superconformal structures on generalized calabi yau metric manifolds we construct an embedding of two commuting copies of the n 2 superconformal vertex algebra in the space of global sections of the twisted chiral anti chiral de rham complex of a generalized calabi yau metric manifold including the case when there is a non trivial h flux and non vanishing dilaton the 4 corresponding brst charges are well defined on any generalized kahler manifold this allows one to consider the half twisted model defining thus the chiral de rham complex of a generalized kahler manifold the classical limit of this result allows one to recover the celebrated generalized kahler identities as the degree zero part of an infinite dimensional lie superalgebra attached to any generalized kahler manifold as a byproduct of our study we investigate the properties of generalized calabi yau metric manifolds in the lie algebroid setting

d equivalence and k equivalence let x and y be smooth projective varieties over they are called d equivalent if their derived categories of bounded complexes of coherent sheaves are equivalent as triangulated categories and k equivalent if they are birationally equivalent and the pull backs of their canonical divisors to a common resolution coincide we expect that the two equivalences coincide for birationally equivalent varieties we shall provide a partial answer to the above problem in this paper

properties of the brualdi li tournament matrix the brualdi li tournament matrix is conjectured to have the largest spectral radius among all tournament matrices of even order in this paper two forms of the characteristic polynomial of the brualdi li tournament matrix are found using the first form it is shown that the roots of the characteristic polynomial are simple and that the brualdi li tournament matrix is diagonalizable using the second form an expression is found for the coefficients of the powers of the variable in the characteristic polynomial these coefficients give information about the cycle structure of the cycles of length 1 5 of the directed graph associated with the brualdi li tournament matrix

projection onto simplicial cones by picard s method abstract by using moreau s decomposition theorem for projecting onto cones the problem of projecting onto a simplicial cone is reduced to finding the unique solution of a nonsmooth system of equations it is shown that picard s method applied to the system of equations associated with the problem of projecting onto a simplicial cone generates a sequence that converges linearly to the solution of the system numerical experiments are presented making the comparison between picard s and semi smooth newton s methods to solve the nonsmooth system associated with the problem of projecting a point onto a simplicial cone

elliptic genera of singular varieties the notions of orbifold elliptic genus and elliptic genus of singular varieties are introduced and the relation between them is studied the elliptic genus of singular varieties is given in terms of a resolution of singularities and extends the elliptic genus of calabi yau hypersurfaces in fano gorenstein toric varieties introduced earlier the orbifold elliptic genus is given in terms of the fixed point sets of the action we show that the generating function for the orbifold elliptic genus ellorb x 6n p for symmetric groups 6n acting on n fold products coincides with the one proposed by r dijkgraaf g moore e verlinde and h verlinde the two notions of elliptic genera are conjectured to coincide

coherent sheaves on configuration schemes abstract we introduce and study configuration schemes which are obtained by glueing usual schemes along closed embeddings the category of coherent sheaves on a configuration scheme is investigated smooth configuration schemes provide alternative resolutions of singularities of usual singular schemes we consider in detail the case when the singular scheme is a union of hyperplanes in a projective space

elliptic genera of symmetric products and second quantized strings in this note we prove an identity that equates the elliptic genus partition function of a supersymmetric sigma model on the n fold symmetric product m n sn of a manifold m to the partition function of a second quantized string theory on the space m s 1 the generating function of these elliptic genera is shown to be almost an automorphic form for o 3 2 z in the context of d brane dynamics this result gives a precise computation of the free energy of a gas of d strings inside a higher dimensional brane

spectral radii of tournament matrices whose graphs are related by an arc reversal abstract given an irreducible tournament matrix t and a pair of distinct indices i and j let t i j be the matrix obtained from t by transposing its principal submatrix on rows and columns i and j we establish one condition on rows i and j of t under which the spectral radius of t i j is no smaller than that of t and another condition on the i th and j th entries of the left and right perron vectors of t under which the spectral radius of t i j must be strictly smaller than that of t these conditions are used to compare the spectral radii of a class of toeplitz tournament matrices and the resulting comparison sheds light on some conjectures of brualdi and li further if t yields equality in a certain lower bound on the spectral radius of a tournament matrix then for any i and j we provide simple necessary and sufficient conditions for the spectral radius of t i j to be larger than that of t to be smaller than that of t and to be equal to that of t

perron vector ordering for a subclass of tournament matrices abstract we consider the class of n n tournament matrices having scor vector 1 1 2 3 n 4 n 3 n 2 n 2 1 for each such matrix we give formulas for the entries in its perron vector in terms of the corresponding perron value this leads to a discussion of the ordering of the entries in the perron vector and so yields some insight into the kendall wei method for ranking players in a round robin competition finally we characterize all matrices from this class such that the ordering of the entries in the perron vector coincides with the ordering of the entries in the score vector

lagrangian approach to sheaves of vertex algebras we explain how sheaves of vertex algebras are related to mathematical structures encoded by a class of lagrangians the exposition is focused on two examples the wzw model and the 1 1 supersymmetric model we conclude by showing how to construct a family of vertex algebras with base the barannikov kontsevich moduli space thus furnishing the b model moduli for witten s half twisted model

elliptic genus of phases of n 2 theories we discuss an algebro geometric description of witten s phases of n 2 theories and propose a definition of their elliptic genus provided some conditions on singularities of the phases are met for landau ginzburg phase one recovers elliptic genus of lg models proposed in physics literature in early 1990s for certain transitions between phases we derive invariance of elliptic genus from an equivariant form of mckay correspondence for elliptic genus as special cases one obtains landau giznburg calabi yau correspondence for elliptic genus of weighted homogeneous potentials as well as certain hybrid cy correspondences

chiral de rham complex we define natural sheaves of vertex algebras over smooth manifolds which may be regarded as semi infinite de rham complexes of certain d modules over the loop spaces for calabi yau manifolds they admit n 2 supersymmetry connection with wakimoto modules is discussed

group inverses of m matrices associated with nonnegative matrices having few eigenvalues abstract we continue here earlier investigations of the structure of group generalized inverses a of singular and irreducible m matrices a ri m where m is an n n nonnegative and irreducible matrix whose spectral radius is r and where m is assumed to belong to one of several special classes of nonnegative matrices we now focus on cases where m has only a few distinct eigenvalues we are particularly interested in instances of such matrices m which cause all or almost all of the off diagonal entries of the corresponding a to be nonpositive we then apply our results to the study of the sign pattern of the off diagonal entries of the group generalized inverse of a ri m where m comes from one of the following families of nonnegative matrices 1 the magic square of order n 4k generated by matlab 2 the doubly regular tournament matrices the latter type is an example of a special type of a nonnegative matrix m for which the group inverse of the associated m matrix satisfies a a t for some 0

on the better behaved version of the gkz hypergeometric system we consider a version of the generalized hypergeometric system introduced by gelfand kapranov and zelevinsky gkz suited for the case when the underlying lattice is replaced by a finitely generated abelian group in contrast to the usual gkz hypergeometric system the rank of the better behaved gkz hypergeometric system is always the expected one we give largely self contained proofs of many properties of this system the discussion is intimately related to the study of the variations of hodge structures of hypersurfaces in algebraic tori

isotone retraction cones in hilbert spaces abstract a mapping is called isotone if it is monotone increasing with respect to the order induced by a pointed closed convex cone finding the pointed closed convex generating cones for which the projection mapping onto the cone is isotone is a difficult problem which was analyzed in isac and nemeth 1986 1990 1992 1 2 3 4 5 such cones are called isotone projection cones in particular it was shown that any isotone projection cone is latticial isac 1990 2 this problem is extended by replacing the projection mapping with continuous retractions onto the cone by introducing the notion of sharp mappings it is shown that a pointed closed convex generating cone is latticial if and only if there is a continuous retraction onto the cone whose complement is sharp several particular cases are considered and examples are given

compact moduli of plane curves we construct a compactication md of the moduli space of plane curves of degreed we regard a plane curvec p 2 as a surface divisor pair p 2 c and dene md as a moduli space of pairs x d where x is a degeneration of the plane we show that if d is not divisible by 3 the stackmd is smooth and the degenerate surfaces x can be described explicitly msc2000 14h10 14j10 14e30

elliptic genera of toric varieties and applications to mirror symmetry the paper contains a proof that elliptic genus of a calabi yau manifold is a jacobi form finds in which dimensions the elliptic genus is determined by the hodge numbers and shows that elliptic genera of a calabi yau hypersurface in a toric variety and its mirror coincide up to sign the proof of the mirror property is based on the extension of elliptic genus to calabi yau hypersurfaces in toric varieties with gorenstein singularities

characters of topological n 2 vertex algebras are jacobi forms on the moduli space of elliptic supercurves abstract we show that trace functions on modules of topological n 2 super vertex algebras give rise to conformal blocks on elliptic supercurves we show that they satisfy a system of linear partial differential equations with respect to the modular parameters of the supercurves under some finiteness condition on the vertex algebra these differential equations can be interpreted as a connection on the vector bundle of conformal blocks we show that this connection is equivariant with respect to a natural action of the jacobi modular group on the modular parameters and the trace functions in the appendix we prove the convergence of the trace functions

small contractions of symplectic 4 folds we classify small contractions of holomorphically symplectic 4 folds ams msc 14e30 14j35

equivalences of derived factorization categories of gauged landau ginzburg models abstract for a given fourier mukai equivalence of bounded derived categories of coherent sheaves on smooth quasi projective varieties we construct fourier mukai equivalences of derived factorization categories of gauged landau ginzburg lg models as an application we obtain some equivalences of derived factorization categories of k equivalent gauged lg models this result is an equivariant version of the result of baranovsky and pecharich and it also gives a partial answer to segal s conjecture as another application we prove that if the kernel of the fourier mukai equivalence is linearizable with respect to a reductive affine algebraic group action then the derived categories of equivariant coherent sheaves on the varieties are equivalent this result is shown by ploog for finite groups case

a chiral borel weil bott theorem abstract we compute the cohomology of modules over the algebra of twisted chiral differential operators over the flag manifold this is applied to 1 finding the character of g integrable irreducible highest weight modules over the affine lie algebra at the critical level and 2 computing a certain elliptic genus of the flag manifold the main tool is a result that interprets the drinfeld sokolov reduction as a derived functor

the resolution property for schemes and stacks we prove the equivalence of two fundamental properties of algebraic stacks being a quotient stack in a strong sense and the resolution property which says that every coherent sheaf is a quotient of some vector bundle moreover we prove these properties in the important special case of orbifolds whose associated algebraic space is a scheme mathematics subject classification primary 14a20 secondary 14l30

equivariant autoequivalences for finite group actions the familiar fourier mukai technique can be extended to an equivariant setting where a finite group g acts on a smooth projective variety x in this paper we compare the group of invariant autoequivalences aut db x g with the group of autoequivalences of dg x we apply this method in three cases hilbert schemes on k3 surfaces kummer surfaces and canonical quotients

algebraic multiplicity of the eigenvalues of a tournament matrix let f denote the set of irreducible n x n tournament matrices here arc our main results 1 for all n 3 every matrix in k has at least three distinct eigenvalues such a matrix has exactly three distinct eigenvalues if and only if it is a hadamard tournament matrix 2 for all n 2 3 there is a matrix in y having n distinct eigenvalues 3 if cr denotes the maximum algebraic multiplicity of 0 as an eigenvalue of the matrices in z then ln 21 2 6 4 if r is the minimum perron value i e spectral radius of all matrices in r then 2 8

flops and equivalences of derived categories for threefolds with only terminal gorenstein singularities the main purpose of this paper is to show that bridgeland s moduli space of perverse point sheaves for certain flopping contractions gives the flops and the fourier mukai transform given by the birational correspondence of the flop is an equivalence between bounded derived categories

hodge elliptic genera and how they govern k3 theories the complex hodge elliptic genus and its conformal field theoretic counterpart were recently introduced by kachru and tripathy refining the traditional complex elliptic genus we construct a different so called chiral hodge elliptic genus which is expected to agree with the generic conformal field theoretic hodge elliptic genus in contrast to the complex hodge elliptic genus as originally defined for k3 surfaces x the chiral hodge elliptic genus is shown to be independent of all moduli moreover employing kapustin s results on infinite volume limits it is shown that it agrees with the generic conformal field theoretic hodge elliptic genus of k3 theories while the complex hodge elliptic genus does not this new invariant governs part of the field content of k3 theories supporting the idea that all their spaces of states have a common subspace which underlies the generic conformal field theoretic hodge elliptic genus and thereby the complex elliptic genus mathematically this space is modelled by the sheaf cohomology of the chiral de rham complex of x it decomposes into irreducible representations of the n 4 superconformal algebra such that the multiplicity spaces of all massive representations have precisely the dimensions required in order to furnish the representation of the mathieu group m 24 that is predicted by mathieu moonshine this is interpreted as evidence in favour of the ideas of symmetry surfing which have been proposed by taormina and the author along with the claim that the sheaf cohomology of the chiral de rham complex is a natural home for mathieu moonshine these investigations also imply that the generic chiral algebra of k3 theories is precisely the n 4 superconformal algebra at central charge c 6 if the usual predictions on infinite volume limits from string theory hold true

the algebra of continuous piecewise polynomials abstract for a triangulated d dimensional region r d we consider the algebra c 0 of all continuous piecewise polynomial functions on we find generators for c 0 as an r algebra and use these to give an isomorphism between c 0 and a quotient of the face ring a of we then study the structure of c 0 as a module over r r y 1 y d the polynomial ring in d indeterminates giving generators for c 0 as an r module these form a free basis when is a shellable complex in general we show that c 0 is a free r module whenever is a disk

toric varieties and modular forms let n r be a lattice and let deg n be a piecewise linear function that is linear on the cones of a complete rational polyhedral fan under certain conditions on deg the data n deg determines a function f that is a holomorphic modular form of weight r for the congruence subgroup 1 l moreover by considering all possible pairs n deg we obtain a natural subring l of modular forms with respect to 1 l we construct an explicit set of generators for l and show that l is stable under the action of the hecke operators finally we relate l to the hirzebruch elliptic genera that are modular with respect to 1 l

tilting generators via ample line bundles abstract it is known that a tilting generator on an algebraic variety x gives a derived equivalence between x and a certain non commutative algebra in this paper we present a method to construct a tilting generator from an ample line bundle and construct it in several examples

reconstruction and finiteness results for fourier mukai partners abstract we develop some methods for studying the fourier mukai partners of an algebraic variety as applications we prove that abelian varieties have finitely many fourier mukai partners and that they are uniquely determined by their derived category of coherent d modules we also generalize a famous theorem due to bondal and orlov

berglund hubsch mirror symmetry via vertex algebras we give a vertex algebra proof of the berglund hubsch duality of nondegenerate invertible potentials we suggest a way to unify it with the batyrev borisov duality of reflexive gorenstein cones

knot homology via derived categories of coherent sheaves i the mathfrak sl 2 case using derived categories of equivariant coherent sheaves we construct a knot homology theory which categorifies the quantum mathfrak sl m knot polynomial our knot homology naturally satisfies the categorified moy relations and is conjecturally isomorphic to khovanov rozansky homology our construction is motivated by the geometric satake correspondence and is related to manolescu s by homological mirror symmetry

normal matrices and their principal submatrices of co order one abstract let a be a normal matrix v be any of its indices a v be the matrix obtained from a by deleting the v th row and column and be an eigenvalue of a v in our paper we construct the eigenspace of a associated with from that of a v in particular it is shown that if there is a unique jordan block of size strictly greater than one in the part of the jordan form of a v corresponding to then the geometric multiplicity of decreases by one under the transition from a v to a in other words the typical change of the spectral properties holds for the results obtained are applied to circulant matrices moreover in appendix to our paper we consider almost regular tournament matrices as principal submatrices of co order one of regular tournament matrices in particular it is observed that the brualdi li tournament matrix b 2 n of order 2n is permutationally similar to a principal submatrix of co order one of the circulant matrix of order 2n 1 with the first row 0 1 1 n 0 0 n as a consequence of this fact the weak brualdi li conjecture is formulated for principal submatrices of co order one of the adjacency matrices of cayley tournaments

duality and monodromy reducibility of a hypergeometric systems we study hypergeometric systems ha in the sense of gelfand kapranov and zelevinsky under two aspects the structure of their holonomically dual system and reducibility of their rank module we prove in the first part that rank jumping parameters always correspond to reducible systems we show further that the property of being reducible is invariant modulo the lattice and obtain as a special instance a theorem of alicia dickenstein and timur sadykov on reducibility of mellin systems in the second part we study a conjecture of nobuki takayama which states that the holonomic dual of ha is of the form ha for suitable we prove the conjecture for all matrices a and generic parameter exhibit an example that shows that in general the conjecture cannot hold and present a refined version of the conjecture questions on both duality and reducibility have been quite difficult to answer with classical methods this paper may be seen as an example of the usefulness and scope of applications of the homological tools for a hypergeometric systems developed in matusevich et al j amer math soc 18 919 941 2005

braid group actions on derived categories of coherent sheaves this paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety x the motivation for this is m kontsevich s homological mirror conjecture together with the occurrence of certain braid group actions in symplectic geometry one of the main results is that when dim x geq 2 our braid group actions are always faithful we describe conjectural mirror symmetries between smoothings and resolutions of singularities which lead us to find examples of braid group actions arising from crepant resolutions of various singularities relations with the mckay correspondence and with exceptional sheaves on fano manifolds are given moreover the case of an elliptic curve is worked out in some detail

knot homology via derived categories of coherent sheaves ii mathfrak sl _ m case using derived categories of equivariant coherent sheaves we construct a knot homology theory which categorifies the quantum mathfrak sl m knot polynomial our knot homology naturally satisfies the categorified moy relations and is conjecturally isomorphic to khovanov rozansky homology our construction is motivated by the geometric satake correspondence and is related to manolescu s by homological mirror symmetry

characteristic classes of singular toric varieties in this paper we compute the motivic chern classes and homology hirzebruch characteristic classes of possibly singular toric varieties which in the context of complete toric varieties fit nicely with a generalized hirzebruch riemann roch theorem as important special cases we obtain new or recover well known formulae for the baum fulton macpherson todd or macpherson s chern classes of toric varieties as well as for the thom milnor l classes of simplicial projective toric varieties we present two different perspectives for the computation of these characteristic classes of toric varieties first we take advantage of the torus orbit decomposition and the motivic properties of the motivic chern and respectively homology hirzebruch classes to express the latter in terms of dualizing sheaves and respectively the dual todd classes of closures of orbits this method even applies to torus invariant subspaces of a given toric variety the obtained formula is then applied to weighted lattice point counting in lattice polytopes and their subcomplexes yielding generalized pick type formulae second in the case of simplicial toric varieties we compute our characteristic classes by using the lefschetz riemann roch theorem of edidin graham in the context of the geometric quotient description of such varieties in this setting we define mock hirzebruch classes of simplicial toric varieties which specialize to the mock chern mock todd and mock l classes of such varieties and investigate the difference between the actual homology hirzebruch class and the mock hirzebruch class we show that this difference is localized on the singular locus and we obtain a formula for it in which the contribution of each singular cone is identified explicitly finally the two methods of computing characteristic classes are combined for proving several characteristic class formulae originally obtained by cappell and shaneson in the early 1990s 2015 wiley periodicals inc

on the minimum perron value for an irreducible tournament matrix abstract perron values of tournament matrices have been of interest to a number of authors recently in 1983 brualdi and li made a conjecture as to which matrix minimizes the perron value over the class of irreducible n n tournament matrices we prove that conjecture using both matrix theoretic and graph theoretic techniques

vertex algebras and mirror symmetry mirror symmetry for calabi yau hypersurfaces in toric varieties is by now well established however previous approaches to it did not uncover the underlying reason for mirror varieties to be mirror we are able to calculate explicitly vertex algebras that correspond to holomorphic parts of a and b models of calabi yau hypersurfaces and complete intersections in toric varieties we establish the relation between these vertex algebras for mirror calabi yau manifolds this should eventually allow us to rewrite the whole story of toric mirror symmetry in the language of sheaves of vertex algebras our approach is purely algebraic and involves simple techniques from toric geometry and homological algebra as well as some basic results of the theory of vertex algebras ideas of this paper may also be useful in other problems related to maps from curves to algebraic varieties

chiral rings of vertex algebras of mirror symmetry we calculate chiral rings of the n 2 vertex algebras constructed from the combinatorial data of toric mirror symmetry and show that they coincide with the description of stringy cohomology conjectured previously in a joint work with a mavlyutov this constitutes an important reality check of the vertex algebra approach to mirror symmetry

fourier mukai transformations on k3 surfaces with ρ 1 and atkin lehner involutions abstract we show that there is a surjection from the fourier mukai transformations on projective k3 surfaces with the picard number x 1 to so called the group of atkin lehner involutions this was expected in hosono lian oguiso and yau s paper

a semi smooth newton method for a special piecewise linear system with application to positively constrained convex quadratic programming in this paper a special piecewise linear system is studied it is shown that under a mild assumption the semi smooth newton method applied to this system is well defined and the method generates a sequence that converges linearly to a solution besides we also show that the generated sequence is bounded for any starting point and a formula for any accumulation point of this sequence is presented as an application we study the convex quadratic programming problem under positive constraints the numerical results suggest that the semi smooth newton method achieves accurate solutions to large scale problems in few iterations

chiral differential operators on supermanifolds the first part of this paper provides a new description of chiral differential operators cdos in terms of global geometric quantities the main result is a recipe to define all sheaves of cdos on a smooth cs manifold its ingredients consist of an affine connection and an even 3 form that trivializes p1 with fixed two suitable 3 forms define isomorphic sheaves of cdos if and only if their difference is exact moreover conformal structures are in one to one correspondence with even 1 forms that trivialize c1 applying our work in the first part we then construct what may be called chiral dolbeault complexes of a complex manifold m and analyze conditions under which these differential vertex superalgebras admit compatible conformal structures or extra gradings by fermion numbers when m is compact their cohomology computes in various cases the witten genus the two variable elliptic genus and a spinc version of the witten genus this part contains some new results as well as provides a geometric formulation of certain known facts from the study of holomorphic cdos and models

eigenvalues of oriented graph matrices abstract we give bounds on the real and imaginary parts of the eigenvalues of an oriented graph matrix prove that each of the irreducible oriented graph matrices of order n 3 has at least three distinct eigenvalues and determine such irreducible matrices with three distinct eigenvalues we also consider spectral properties of the bipartite oriented graph matrices and the multiequipartite oriented graph matrices

tournament matrices with extremal spectral properties abstract for a tournament matrix m of order n we define its walk space wm to be span m j 1 j 0 n 1 where 1 is the all ones vector we show that the dimension of wm equals the number of eigenvalues of m whose real parts are greater than 1 2 we then focus on tournament matrices whose walk space has particularly simple structure and characterize them in terms of their spectra specifically we characterize those tournament matrices such that mj1 is an eigenvector of m for some j 0 we also characterize the tournament matrices m such that jn 2m is a skew hadamard matrix throughout we illustrate our results with examples

autoequivalences of derived categories on the minimal resolutions of an singularities on surfaces in this article we study the group of autoequivalences of derived categories of coherent sheaves on the minimal resolution of an singularities on surfaces our main result is to find generators of this group

derived equivalences for cotangent bundles of grassmannians via categorical sl2 actions we construct an equivalence of categories from a strong categorical sl 2 action following the work of chuang rouquier as an application we give an explicit natural equivalence between the derived categories of coherent sheaves on cotangent bundles to complementary grassmannians

mckay correspondence for elliptic genera we establish a correspondence between orbifold and singular elliptic genera of a global quotient while the former is defined in terms of the fixed point set of the action the latter is defined in terms of the resolution of singularities as a byproduct the second quantization formula of dijkgraaf moore verlinde and verlinde is extended to arbitrary kawamata log terminal pairs

pick s inequality and tournaments abstract let t be a tournament on n nodes and let a be its adjacency matrix a brauer and i gentry observed that an inequality due to g pick implies that im 1 2 cot 2n for all eigenvalues of a we say that t is a pick or p tournament if equality holds for some we determine when equality holds in pick s inequality for arbitrary real matrices and use this to show that the p tournaments can all be constructed from the transitive tournament m by reversing the arcs between the sets of certain node partitions or cuts u the cuts are specified by 1 n vectors u such that utw 0 where w 1 n 1 t e i n this links the cuts to cyclotomic polynomials there is at least one p tournament on n nodes if and only if n 2k k 1 up to isomorphism there is precisely one p tournament on n nodes if and only if n p or n 2p for some odd prime p for odd n up toisomorphism the only regular p tournament on n nodes has matrix zn circ 0 1 1 0 0 a composition rule is used on the matrices zp to form all p tournaments on n 2kpl nodes l 1

iterative methods for nonlinear complementarity problems on isotone projection cones in this paper we present a recursion related to a nonlinear complementarity problem defined by a closed convex cone in a hilbert space and a continuous mapping defined on the cone if the recursion is convergent then its limit is a solution of the nonlinear complementarity problem in the case of isotone projection cones sufficient conditions are given for the mapping so that the recursion to be convergent

elliptic classes of schubert varieties we introduce new notions in elliptic schubert calculus the twisted borisov libgober classes of schubert varieties in general homogeneous spaces g p while these classes do not depend on any choice they depend on a set of new variables for the definition of our classes we calculate multiplicities of some divisors in schubert varieties which were only known for full flag varieties before our approach leads to a simple recursions for the elliptic classes comparing this recursion with r matrix recursions of the so called elliptic weight functions of rimanyi tarasov varchenko we prove that weight functions represent elliptic classes of schubert varieties

on the elliptic genus of generalised kummer varieties borisov and libgober 2 recently proved a conjecture of dijkgraaf moore verlinde and verlinde see 6 on the elliptic genus of a hilbert scheme of points on a surface we show how their result can be used together with our work on complex genera of generalised kummer varieties 17 to deduce the following formula conjectured by kawai and yoshioka 15 on the elliptic genus of a generalised kummer variety a n of dimension 2 n 1 here is the weak jacobi form of weight 1 and index and v n is the hecke operator sending jacobi forms of index r to jacobi forms of index nr see 7