codimensions of algebras and growth functions abstract let a be an algebra over a field f of characteristic zero and let c n a n 1 2 be its sequence of codimensions we prove that if c n a is exponentially bounded its exponential growth can be any real number 1 this is achieved by constructing for any real number 1 an f algebra a such that lim n c n a n exists and equals the methods are based on the representation theory of the symmetric group and on properties of infinite sturmian and periodic words

ringel duality for perverse sheaves on hypertoric varieties abstract motivated by the polynomial representation theory of the general linear group and the theory of symplectic singularities we study a category of perverse sheaves with coefficients in a field k on any affine unimodular hypertoric variety m our main result is that this is a highest weight category whose ringel dual is the corresponding category for the gale dual hypertoric variety m on the way to proving our main result we confirm a conjecture of finkelberg kubrak in the case of hypertoric varieties we also show that our category is equivalent to representations of a combinatorially defined algebra recently introduced in a related paper

on the equations and classification of toric quiver varieties toric quiver varieties moduli spaces of quiver representations are studied given a quiver and a weight there is an associated quasi projective toric variety together with a canonical embedding into projective space it is shown that for a quiver with no oriented cycles the homogeneous ideal of this embedded projective variety is generated by elements of degree at most 3 in each fixed dimension d up to isomorphism there are only finitely many d dimensional toric quiver varieties a procedure for their classification is outlined

corrections to fast multilevel computation of low rank representation of mathcal h matrix blocks several asymptotic complexity expressions in 1 were overestimated and require correction all the errors resulted from the same incorrect assumption at the end of section ii the computational complexity of calculating the singular value decomposition svd of an m times n matrix mathbf a was presented as mathcal o m 2 n n 3 the cost of computing the full svd mathbf a mathbf u sigma mathbf v dagger using the r svd algorithm 2 assuming m ge n otherwise m and n are interchanged the algorithm in 1 however uses the svd for computing sigma mathbf v dagger and only the first n columns of mathbf u at an mathcal o mn 2 n 3 cost only 2 indeed the standard svd implementations ensure that this computation costs mathcal o mn min m n 3 accordingly 1 should have used svd costs of mathcal o m 2 n mathcal o mn 2 rather than mathcal o n 3 mathcal o m 2 n when m using these impacts several complexity expressions in 1 and its conclusion as detailed next

gromov invariants for holomorphic maps from riemann surfaces to grassmannians two compactifications of the space of holomorphic maps of fixed degree from a compact riemann surface to a grassmannian are studied it is shown that the uhlenbeck compactification has the structure of a projective variety and is dominated by the algebraic compactification coming from the grothendieck quot scheme the latter may be embedded into the moduli space of solutions to a generalized version of the vortex equations studied by bradlow this gives an effective way of computing certain intersection numbers known as gromov invariants on the space of holomorphic maps into grassmannians we carry out these computations in the case where the riemann surface has genus one department of mathematics university of utah salt lake city utah 84112 e mail address bertram math utah edu department of mathematics brown university providence rhode island 02912 e mail address daskal gauss math brown edu department of mathematics university of california irvine california 92717 e mail address raw math uci edu license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

quasialgebra structure of the octonions abstract we show that the octonions are a twisting of the group algebra of z 2 z 2 z 2 in the quasitensor category of representations of a quasi hopf algebra associated to a group 3 cocycle in particular we show that they are quasialgebras associative up to a 3 cocycle isomorphism we show that one may make general constructions for quasialgebras exactly along the lines of the associative theory including quasilinear algebra representation theory and an automorphism quasi hopf algebra we study the algebraic properties of quasialgebras of the type which includes the octonions further examples include the higher 2n onion cayley algebras and examples associated to hadamard matrices

on differential operators on complete symmetric varieties of type a1 and a2 abstract in this paper we consider the algebra of global differential operators d x on wonderful compactifications x of symmetric spaces g h of type a 1 and a 2 we will first construct a global differential operator on these varieties that does not come from the infinitesimal action of the lie algebra of g we will then focus on type a 2 where we will show that d x is an algebra of finite type and that for any invertible sheaf l on x h 0 x l is either 0 or a simple left d x l module finally we will show with the help of local cohomology that this still holds for higher cohomology groups h i x l

a randomized blocked algorithm for efficiently computing rank revealing factorizations of matrices this manuscript describes a technique for computing partial rank revealing factorizations such as a partial qr factorization or a partial singular value decomposition the method takes as input a tolerance varepsilon and an m times n matrix boldsymbol mathsf a and returns an approximate low rank factorization of boldsymbol mathsf a that is accurate to within precision varepsilon in the frobenius norm or some other easily computed norm the rank k of the computed factorization which is an output of the algorithm is in all examples we examined very close to the theoretically optimal varepsilon rank the proposed method is inspired by the gram schmidt algorithm and has the same o mnk asymptotic flop count however the method relies on randomized sampling to avoid column pivoting which allows it to be blocked and hence accelerates practical computations by reducing communication numerical experiments demonstrate that the accuracy of the scheme is for every matrix that was

a dynamic programming method for building free algebras abstract we are interested in deciding if a given nonassociative polynomial h is an identity for a variety of nonassociative algebras we present an algorithm for constructing a certain homomorphic image of a free nonassociative algebra which is sufficient to answer the question the algorithm resembles dynamic programming in that the algebra is built by constructing a sequence of subspaces the basis of each subspace is determined by the basis of previous subspaces the number of arithmetic operations required to construct the algebra is bounded by a polynomial in the degree of h and the dimension of the resulting algebra

groupe de picard des varietes de modules de fibres semi stables sur les courbes algebriques soient x une courbe algebrique projective lisse de genre g 2 sur r d des entiers avec r 2 on note u r d la variete de modules des fibres algebriques semi stables sur x de rang r et de degre d us r d l ouvert de u r d correspondant aux fibres stables on sait que u r d est une variete algebrique projective irreductible et normale si r et d ne sont pas premiers entre eux u r d n est pas lisse sauf dans une exception le cas ou g 2 r 2 et d est pair on supposera dans cet article qu on n est pas dans ce cas on a alors codimu r d u r d us r d 2 et u r d us r d est le lieu des points singuliers de u r d si l est un fibre en droites de degre d sur x on note u r l la sous variete fermee de u r d correspondant aux fibres vectoriels de determinant isomorphe a l on etudie dans cet article les groupes de picard pic u r d et pic u r l on montre d abord que les varietes u r d et u r l sont localement factorielles on calcule ensuite pic u r d et pic u r l qui sont decrits au moyen des diviseurs thetas generalises

git stability of weighted pointed curves in the late 1970s mumford established chow stability of smooth unpointed genus g curves embedded by complete linear systems of degree d 2g 1 and at about the same time gieseker established asymptotic hilbert stability that is stability of m th hilbert points for some large values of m under the same hypotheses both of them then use an indirect argument to show that nodal deligne mumford stable curves are git stable the case of marked points lay untouched until 2006 when elizabeth baldwin proved that pointed deligne mumford stable curves are asymptotically hilbert stable actually she proved this for stable maps which includes stable curves as a special case her argument is a delicate induction on g and the number of marked points n elliptic tails are glued to the marked points one by one ultimately relating stability of an n pointed genus g curve to gieseker s result for genus g n unpointed curves there are three ways one might wish to improve upon baldwin s results first one might wish to construct moduli spaces of weighted pointed curves or maps it appears that baldwin s proof can accommodate some but not all sets of weights second one might wish to study hilbert stability for small values of m since baldwin s proof uses gieseker s proof as the base case it is not easy to see how it could be modified to yield an approach for small m finally the minimal model program for moduli spaces of curves has generated interest in git for 2 3 or 4 canonical linear systems due to its use of elliptic tails baldwin s proof cannot be used to study these as elliptic tails are known to be git unstable in these cases in this paper i give a direct proof that smooth curves with distinct weighted marked points are asymptotically hilbert stable with respect to a wide range of parameter spaces and linearizations some of these yield the coarse moduli space of deligne mumford stable pointed curves m g n and hassett s moduli spaces of weighted pointed curves m g a while other linearizations may give other quotients which are birational to these and which may admit interpretations as moduli spaces the full construction of the moduli spaces is not contained in this paper only the proof that smooth curves with distinct weighted marked points are stable which is the key new result needed for the construction for this i follow gieseker s approach to reduce to the git problem to a combinatorial problem though the solution is very different

lie superalgebras graded by p n and q n abstract in this article we study lie superalgebras graded by the root systems p n and q n

flexible lie admissible algebras abstract this paper investigates finite dimensional flexible lie admissible algebras a over fields of characteristic 0 under these hypotheses the vector space a with the lie product x y xy yx is a lie algebra denoted by a the main result of this work gives a characterization of those flexible lie admissible algebras for which the solvable radical of a is a direct summand of a included in this class of algebras are all flexible lie admissible a for which a is a reductive lie algebra our technique is to view a as a module for a certain semisimple lie algebra of derivations of a and to see what restrictions the module structure imposes on the multiplication of a a subsequent investigation will show that this module approach can also be used to determine the flexible lie admissible algebras a for which the radical of a is abelian

higher genus wall crossing in landau ginzburg theory abstract for a fermat quasi homogeneous polynomial we study the associated weighted fan jarvis ruan witten theory with narrow insertions we prove a wall crossing formula in all genera via localization on a master space which is constructed by introducing an additional tangent vector to the moduli problem this is a landau ginzburg theory analogue of the higher genus quasi map wall crossing formula proved by ciocan fontanine and kim it generalizes the genus 0 result by ross ruan and the genus 1 result by guo ross

quivers cones and polytopes abstract let q be a quiver without oriented cycles we consider the polytope of flows in q with input these polytopes are closely related to the combinatorial structure of the quiver in particular to its spanning subtrees furthermore we consider a system of cones which turns out to be a fan and can be seen as a base for the family of all flow polytopes for the various inputs finally we present several examples

compact kahler quotients of algebraic varieties and geometric invariant theory abstract given an action of a complex reductive lie group g on a normal variety x we show that every analytically zariski open subset of x admitting an analytic hilbert quotient with projective quotient space is given as the set of semistable points with respect to some g linearised weil divisor on x applying this result to hamiltonian actions on algebraic varieties we prove that semistability with respect to a momentum map is equivalent to git semistability in the sense of mumford and hausen it follows that the number of compact momentum map quotients of a given algebraic hamiltonian g variety is finite as further corollary we derive a projectivity criterion for varieties with compact kahler quotient

fast construction of hierarchical matrix representation from matrix vector multiplication we develop a hierarchical matrix construction algorithm using matrix vector multiplications based on the randomized singular value decomposition of low rank matrices the algorithm uses o logn applications of the matrix on structured random test vectors and o nlogn extra computational cost where n is the dimension of the unknown matrix numerical examples on constructing green s functions for elliptic operators in two dimensions show efficiency and accuracy of the proposed algorithm

derived categories of bhk mirrors abstract we prove a derived analogue to the results of borisov clarke kelly and shoemaker on the birationality of berglund hubsch krawitz mirrors heavily bootstrapping off work of seidel and sheridan we obtain homological mirror symmetry for berglund hubsch krawitz mirror pencils to hypersurfaces in projective space

mirror symmetry and toric degenerations of partial flag manifolds in this paper we propose and discuss a mirror construction for complete intersections in partial flag manifolds f n1 nl n this construction includes our previous mirror construction for complete intersection in grassmannians and the mirror construction of givental for complete flag manifolds the key idea of our construction is a degeneration of f n1 nl n to a certain gorenstein toric fano variety p n1 nl n which has been investigated by gonciulea and lakshmibai we describe a natural small crepant desingularization of p n1 nl n and prove a generalized version of a conjecture of gonciulea and lakshmibai on the singular locus of p n1 nl n mathematisches institut eberhard karls universitat tubingen d 72076 tubingen germany email address batyrev bastau mathematik uni tuebingen de department of mathematics oklahoma state university stillwater ok 74078 usa email address ciocan math okstate edu department of mathematics university of california davis davis ca 95616 usa email address bumsig math ucdavis edu fb 17 mathematik johannes gutenberg universitat mainz d 55099 mainz germany email address straten mathematik uni mainz de

volumes and ehrhart polynomials of flow polytopes the lidskii formula for the type a n root system expresses the volume and ehrhart polynomial of the flow polytope of the complete graph with nonnegative integer netflows in terms of kostant partition functions for every integer polytope the volume is the leading coefficient of the ehrhart polynomial the beauty of the lidskii formula is the revelation that for these polytopes their ehrhart polynomial function can be deduced from their volume function baldoni and vergne generalized lidskii s result for flow polytopes of arbitrary graphs g while their formulas are combinatorial in nature their proofs are based on residue computations in this paper we construct canonical polytopal subdivisions of flow polytopes which we use to prove the baldoni vergne lidskii formulas in contrast with the original computational proof of these formulas our proof reveal their geometry and combinatorics we conclude by exhibiting enumerative properties of the lidskii formulas via our canonical polytopal subdivisions

on the variation in the cohomology of the symplectic form of the reduced phase space is called the momentum mapping of the hamiltonian t action given 1 1 the condition 1 2 just means that t acts along the fibers of j for the basic definitions and properties of non commutative hamiltonian group actions see am the results of this paper can easily be extended to hamiltonian actions of arbitrary compact connected lie groups by applying our results to the action of its maximal toms and using the equivariance of the momentum mapping for some more details see the remarks at the end of sect 2 we will assume throughout this paper that the momentum map is proper that is j i u is compact for each compact subset u of t now let r be a regular value of j that is t j tmm t is surjective for all meye j 1 4 then ye is a smooth submanifold of m compact because

spinors lagrangians and rank 2 higgs bundles this article considers the dirac operator on a riemann surface coupled to a symplectic holomorphic vector bundle w each spinor in the null space which is a holomorphic section of w k1 2 generates through the moment map a higgs field and varying w one obtains a holomorphic lagrangian subvariety in the moduli space of higgs bundles applying this to the irreducible symplectic representations of sl 2 c we obtain lagrangian submanifolds of the rank 2 higgs bundle moduli space which link up with m period points on the prym variety of the spectral curve as well as brill noether loci on the moduli space of semistable bundles the case of genus 2 is investigated in some detail

projectivity and birational geometry of bridgeland moduli spaces we construct a family of nef divisor classes on every moduli space of stable complexes in the sense of bridgeland this divisor class varies naturally with the bridgeland stability condition for a generic stability condition on a k3 surface we prove that this class is ample thereby generalizing a result of minamide yanagida and yoshioka our result also gives a systematic explanation of the relation between wall crossing for bridgeland stability and the minimal model program for the moduli space we give three applications of our method for classical moduli spaces of sheaves on a k3 surface 1 we obtain a region in the ample cone in the moduli space of gieseker stable sheaves only depending on the lattice of the k3 2 we determine the nef cone of the hilbert scheme of n points on a k3 surface of picard rank one when n is large compared to the genus 3 we verify the hassett tschinkel huybrechts sawon conjecture on the existence of a birational lagrangian fibration for the hilbert scheme in a new family of cases

all the git quotients at once let g be an algebraic torus acting on a smooth variety v we study the relationship between the various git quotients of v and the symplectic quotient of the cotangent bundle of v

mmp for moduli of sheaves on k3s via wall crossing nef and movable cones lagrangian fibrations we use wall crossing with respect to bridgeland stability conditions to systematically study the birational geometry of a moduli space m of stable sheaves on a k3 surface x a we describe the nef cone the movable cone and the effective cone of m in terms of the mukai lattice of x b we establish a long standing conjecture that predicts the existence of a birational lagrangian fibration on m whenever m admits an integral divisor class d of square zero with respect to the beauville bogomolov form these results are proved using a natural map from the space of bridgeland stability conditions mathop mathrm stab nolimits x to the cone mathop mathrm mov nolimits x of movable divisors on m this map relates wall crossing in mathop mathrm stab nolimits x to birational transformations of m in particular every minimal model of m appears as a moduli space of bridgeland stable objects on x

a combinatorial model for computing volumes of flow polytopes we introduce new families of combinatorial objects whose enumeration computes volumes of flow polytopes these objects provide an interpretation based on parking functions of baldoni and vergne s generalization of a volume formula originally due to lidskii we recover known flow polytope volume formulas and prove new volume formulas for flow polytopes a highlight of our model is an elegant formula for the flow polytope of a graph we call the caracol graph as by products of our work we uncover a new triangle of numbers that interpolates between catalan numbers and the number of parking functions we prove the log concavity of rows of this triangle along with other sequences derived from volume computations and we introduce a new ehrhart like polynomial for flow polytope volume and conjecture product formulas for the polytopes we consider

toroidal varieties and the weak factorization theorem we develop the theory of stratified toroidal varieties which gives together with the theory of birational cobordisms 73 a proof of the weak factorization conjecture for birational maps in characteristic zero a birational map between complete nonsingular varieties over an algebraically closed field k of characteristic zero is a composite of blow ups and blow downs with smooth centers

projective toric varieties as fine moduli spaces of quiver representations this paper proves that every projective toric variety is the fine moduli space for stable representations of an appropriate bound quiver to accomplish this we study the quiver q with relations r corresponding to the finite dimensional algebra mathop rm end nolimits textstyle bigoplus nolimits i 0 r l i where cal l scr o x l 1 ldots l r is a list of line bundles on a projective toric variety x the quiver q defines a smooth projective toric variety called the multilinear series cal l and a map x longrightarrow cal l we provide necessary and sufficient conditions for the induced map to be a closed embedding as a consequence we obtain a new geometric quotient construction of projective toric varieties under slightly stronger hypotheses on cal l the closed embedding identifies x with the fine moduli space of stable representations for the bound quiver q r

fast multilevel computation of low rank representation of mathcal h matrix blocks a physics based algorithm for accelerating the computation of method of moments matrix blocks low rank approximation is presented the algorithm relies on efficient sampling of phase and amplitude compensated interactions using nonuniform grids rank revealing analysis is applied in a multilevel fashion to matrices of reduced column and row dimensions that describe subdomains interactions with these coarse grids rather than to the original matrix blocks as a result significant savings are achieved especially for the inherently more compressible dynamic quasi planar and quasi static cases the algorithm s reduced storage and computation time requirements are estimated analytically and verified numerically for representative examples

mirror symmetry langlands duality and the hitchin system among the major mathematical approaches to mirror symmetry are those of batyrev borisov and strominger yau zaslow syz the first is explicit and amenable to computation but is not clearly related to the physical motivation the second is the opposite furthermore it is far from obvious that mirror partners in one sense will also be mirror partners in the other this paper concerns a class of examples that can be shown to satisfy the requirements of syz but whose hodge numbers are also equal this provides significant evidence in support of syz moreover the examples are of great interest in their own right they are spaces of flat slr connections on a smooth curve the mirror is the corresponding space for the langlands dual group pglr these examples therefore throw a bridge from mirror symmetry to the duality theory of lie groups and more broadly to the geometric langlands program