betti numbers of holomorphic symplectic quotients via arithmetic fourier transform a fourier transform technique is introduced for counting the number of solutions of holomorphic moment map equations over a finite field this technique in turn gives information on betti numbers of holomorphic symplectic quotients as a consequence simple unified proofs are obtained for formulas of poincare polynomials of toric hyperkahler varieties recovering results of bielawski dancer and hausel sturmfels poincare polynomials of hilbert schemes of points and twisted atiyah drinfeld hitchin manin adhm spaces of instantons on 2 recovering results of nakajima yoshioka and poincare polynomials of all nakajima quiver varieties as an application a proof of a conjecture of kac on the number of absolutely indecomposable representations of a quiver is announced

counting representations of quivers over finite fields abstract by counting the numbers of isomorphism classes of representations indecomposable or absolutely indecomposable of quivers over finite fields with fixed dimension vectors we obtain a multi variable formal identity if the quiver has no edge loops this identity turns out to be a q analogue of the kac denominator identity modulus a conjecture of kac

a computational criterion for the kac conjecture abstract we give a criterion for the kac conjecture asserting that the free term of the polynomial counting the absolutely indecomposable representations of a quiver over a finite field of given dimension coincides with the corresponding root multiplicity of the associated kac moody algebra our criterion suits very well for computer tests

normality of marsden weinstein reductions for representations of quivers we prove that the marsden weinstein reductions for the moment map associated to representations of a quiver are normal varieties we give an application to conjugacy classes of matrices

smooth models of quiver moduli for any moduli space of stable representations of quivers certain smooth varieties compactifying projective space fibrations over the moduli space are constructed the boundary of this compactification is analyzed explicit formulas for the betti numbers of the smooth models are derived in the case of moduli of simple representations explicit cell decompositions of the smooth models are constructed

superpotential algebras and manifolds we study a special class of calabi yau algebras in the sense of ginzburg those arising as the fundamental group algebras of acyclic manifolds motivated partly by the usefulness of superpotential descriptions in motivic donaldson thomas theory we investigate the question of whether these algebras admit superpotential presentations we establish that the fundamental group algebras of a wide class of acyclic manifolds including all hyperbolic manifolds do not admit such descriptions disproving a conjecture of ginzburg regarding them we also describe a class of manifolds that do admit such descriptions and discuss a little their motivic donaldson thomas theory finally some links with topological field theory are described

arithmetic harmonic analysis on character and quiver varieties ii we study connections between the topology of generic character varieties of fundamental groups of punctured riemann surfaces macdonald polynomials quiver representations hilbert schemes on c x x c x modular forms and multiplicities in tensor products of irreducible characters of finite general linear groups c 2012 elsevier inc all rights reserved

motivic donaldson thomas invariants of the conifold and the refined topological vertex abstract we compute the motivic donaldson thomas theory of the resolved conifold in all chambers of the space of stability conditions of the corresponding quiver the answer is a product formula whose terms depend on the position of the stability vector generalizing known results for the corresponding numerical invariants our formulae imply in particular a motivic form of the dt pt correspondence for the resolved conifold the answer for the motivic pt series is in full agreement with the prediction of the refined topological vertex formalism

superconvergence analysis of a two grid finite element method for ginzburg landau equation abstract in this paper a two grid method tgm is presented for the complex ginzburg landau equation in which the original nonlinear system is analyzed on the coarse grid based on the newton iteration method for the backward euler fully discrete scheme and then a simple linearized problem developed through taylor s expansion on the fine grid is solved by use of the character of the element and some special techniques the superclose estimation in the h1 norm is deduced for the tgm scheme furthermore the global superconvergent result is derived through interpolation postprocessing skill numerical results illustrate that the proposed tgm is very effective and its computing cost is much less than that of the traditional galerkin finite element method fem without loss of accuracy

counting absolutely cuspidals for quivers for an arbitrary quiver q i omega and dimension vector mathbf d in mathbb n i we define the dimension of absolutely cuspidal functions on the moduli stacks of representations of dimension mathbf d of a quiver q over a finite field mathbb f q and prove that it is a polynomial in q which we conjecture to be positive and integral we obtain a closed formula for these dimensions of spaces of cuspidals for totally negative quivers

counting using hall algebras i quivers abstract we survey some results on counting the rational points of moduli spaces of quiver representations we then make generalizations to grassmannians and flags of quiver representations these results have nice applications to the cluster algebra along the way we use the full hopf structure of the hall algebra of a quiver

finite dimensional behaviour for the derivative ginzburg landau equation in two spatial dimensions abstract in this paper we study the long time behaviour of the solutions for the generalized derivative ginzburg landau equation in two spatial dimensions we show the existence of the global attractor for this equation and establish the estimates of the upper bounds of hausdorff and fractal dimensions for the global attractor

the rational cohomology of overline mathcal m _4 we present two approaches to the study of the cohomology of moduli spaces of curves together they allow us to compute the rational cohomology of the moduli space m over bar 4 of stable comple

hall littlewood functions and the a2 rogers ramanujan identities we prove an identity for hall littlewood symmetric functions labelled by the lie algebra a 2 through specialization this yields a simple proof of the a 2 rogers ramanujan identities of andrews schilling and the author c 2005 elsevier inc all rights reserved

fourier spectral approximation to long time behaviour of the derivative three dimensional ginzburg landau equation in this paper we consider a derivative ginzburg landau equation with periodic initial value condition in three dimensional space a fully discrete galerkin fourier spectral approximation scheme is constructed and then the dynamical behaviour of the discrete system is analysed firstly the existence of global attractors an of the discrete system are proved by a priori estimate of the discrete solution next the convergence of approximate attractors is proved by error estimates of the discrete solution furthermore the long time convergence as n and 0 simultaneously as well as the numerical long time stability of the discrete scheme are obtained

quivers from higgs bundles over p1 and quiver gauge theory in this paper we associate a higgs bundle over p1 with a higgs quiver we need to consider the higgs representation of higgs quivers where the difference of higgs representation from the usual linear representation is putting an op1 module at each vertex in the quiver and then changing the higgs field on the underlying vector bundle is equivalent to changing the higgs representations we can also treat the space of higgs fields as a space of the representations of the so called higgs structure quiver with the action of a reductive group we produce some new quivers from the higgs structure quiver and its representations such as double quivers and bogomol nyi prasad sommerfield quivers related to quiver gauge theory and study some topology of moduli space of corresponding stable representations encoded in the structure of a higgs bundle we also introduce some algebraic structures on some vector spaces constructed from the higgs structure quiver as the generalizations of classical hall algebra and cohomological hall algebra via bipartite quivers the complexities of these algebraic structures are reflected in the nonassociativity of multiplications arising from preservation of some z2 r graded structure

erratum to transitivity of generic semigroups of area preserving surface diffeomorphisms theorem 12 in 2 claims that for a moser generic diffeomorphism f and an invariant residual domain u there are no periodic points in the boundary of u this is claimed to be a direct consequence of 3 however mather s theorem is weaker than that specifically it says that the rotation number of the prime ends compactification on any end of u is irrational on the sphere mather s result implies what is claimed in theorem 12 and the proof which can be found in 1 uses the existence of homoclinic intersections for periodic points of f guaranteed by a theorem of pixton this can also be done in t2 using 4 however for higher genus the claim remains open we will prove a weaker version of theorem 12 of 2 to do this we define for each f a cr residual set g f and we prove that the statement of theorem 12 of 2 holds if we additionally require that the residual domain u be g invariant for some g g f this is enough for our proofs the modifications needed in the statements and proofs of 2 are as follows

the structure of the tautological ring in genus one we prove getzler s claims about the cohomology of the moduli space of stable curves of genus one that is that the even cohomology ring is spanned by the strata classes and that all relations between these classes follow from the witten dijkgraaf verlinde verlinde wdvv relation and getzler s relation in particular the even cohomology ring is isomorphic to the tautological ring

counting quiver representations over finite fields via graph enumeration let be a quiver on n vertices v1 v2 vn with gij edges between vi and vj and let nn hua gave a formula for a q the number of isomorphism classes of absolutely indecomposable representations of over the finite field fq with dimension vector kac showed that a q is a polynomial in q with integer coefficients using hua s formula we show that for each integer s 0 the sth derivative of a q with respect to q when evaluated at q 1 is a polynomial in the variables gij and we compute the highest degree terms in this polynomial our formulas for these coefficients depend on the enumeration of certain families of connected graphs

on the initial value problem for the generalized two dimensional ginzburg landau equation abstract the ginzburg landau type complex partial differential equations are simplified mathematical models for various pattern formation systems in mechanics physics and chemistry most work so far concentrates on ginzburg landau type equations with one spatial dimension 1d in this paper the authors study a complex generalized ginzburg landau equation with two spatial dimensions 2d sufficient conditions for the existence and uniqueness of global solutions for the initial value problem of the generalized 2d ginzburg landau equation are obtained this rigorously establishes the foundation for further investigation of this type of model

noncommutative mirror symmetry for punctured surfaces in 2013 abouzaid auroux efimov katzarkov and orlov showed that the wrapped fukaya categories of punctured spheres and finite unbranched covers of punctured spheres are derived equivalent to the categories of singularities of a superpotential on certain crepant resolutions of toric 3 dimensional singularities we generalize this result to other punctured riemann surfaces and reformulate it in terms of certain noncommutative algebras coming from dimer models in particular given any consistent dimer model we can look at a subcategory of noncommutative matrix factorizations and show that this category is mathtt a infty isomorphic to a subcategory of the wrapped fukaya category of a punctured riemann surface the connection between the dimer model and the punctured riemann surface then has a nice interpretation in terms of a duality on dimer models

torus orbits on homogeneous varieties and kac polynomials of quivers in this paper we prove that the counting polynomials of certain torus orbits in products of partial flag varieties coincides with the kac polynomials of supernova quivers which arise in the study of the moduli spaces of certain irregular meromorphic connections on trivial bundles over the projective line we also prove that these polynomials can be expressed as a specialization of tutte polynomials of certain graphs providing a new way to compute them we also discuss connections with the gel fand macpherson correspondence

a new approach to kac s theorem on representations of valued quivers by using the ringel hall algebra approach we first present a new proof of kac s theorem for species over a finite field the method used here is quite different from earlier techniques in sv sevenhant and van den bergh have discovered an interesting relation between a conjecture of kac on representations of quivers and the structure of the ringel hall algebras in some special cases we second show that this relation still holds for species

indecomposable vector bundles and stable higgs bundles over smooth projective curves we prove that the number of geometrically indecomposable vector bundles of xed rank r and degree d over a smooth projective curve x dened over a nite eld is given by a polynomial depending only on r d and the genus g of x in the weil numbers of x we provide a closed formula expressed in terms of generating series for this polynomial we also show that the same polynomial computes the number of points of the moduli space of stable higgs bundles of rank r and degree d over x this entails a closed formula for the poincar e polynomial of the moduli spaces of stable higgs bundles over a compact riemann surface and hence also for the poincar e polynomials of the twisted character varieties for the groups gl r

the frobenius morphism in invariant theory abstract let r be the homogeneous coordinate ring of the grassmannian g gr 2 n defined over an algebraically closed field of characteristic p 0 in this paper we give a completely characteristic free description of the decomposition of r considered as a graded r p module into indecomposables frobenius summands as a corollary we obtain a similar decomposition for the frobenius pushforward of the structure sheaf of g and we obtain in particular that this pushforward is almost never a tilting bundle on the other hand we show that r provides a noncommutative resolution for r p when p n 2 generalizing a result known to be true for toric varieties in both the invariant theory and the geometric setting we observe that if the characteristic is not too small the frobenius summands do not depend on the characteristic in a suitable sense in the geometric setting this is an explicit version of a general result by bezrukavnikov and mirkovic on frobenius decompositions for partial flag varieties we are hopeful that it is an instance of a more general p uniformity principle

induction theorems on the stable rationality of the center of the ring of generic matrices following procesi and formanek the center of the division ring of rn x n generic matrices over the complex numbers c is stably equivalent to the fixed field under the action of sn of the function field of the group algebra of a zsn lattice denoted by gn we study the question of the stable rationality of the center cn over the complex numbers when rn is a prime in this module theoretic setting let n be the normalizer of an rn sylow subgroup of sn let m be a zsn lattice we show that under certain conditions on m inductionrestriction from n to sn does not affect the stable type of the corresponding field in particular c gn and c zg0zn gn are stably isomorphic and the isomorphism preserves the sn action we further reduce the problem to the study of the localization of gn at the prime n all other primes behave well we also present new simple proofs for the stable rationality of cn over c in the cases r 5 and rn 7

kac s conjecture from nakajima quiver varieties we prove a generating function formula for the betti numbers of nakajima quiver varieties we prove that it is a q deformation of the weyl kac character formula in particular this implies that the constant term of the polynomial counting the number of absolutely indecomposable representations of a quiver equals the multiplicity of a certain weight in the corresponding kac moody algebra which was conjectured by kac in 1982

the semi classical approximation for modular operads we study the contribution of one loop graphs the semi classical expansion problem in the setting of modular operads as an application we calculate the betti numbers of the deligne mumford knudsen moduli spaces of stable curves of genus 1 with n marked smooth points

rhombic embeddings of planar quad graphs given a finite or infinite planar graph all of whose faces have degree 4 we study embeddings in the plane in which all edges have length 1 that is in which every face is a rhombus we give a necessary and sucient condition for the existence of such an embedding as well as a description of the set of all such embeddings r etant donne un graphe planaire fini ou infini dont toutes les faces sont de degre 4 onetudie ses plongements dans le plan dont toutes les aretes sont de longueur 1 c estdire dont toutes les faces sont des losanges on donne une condition necessaire et susante pour l existence d un tel plongement et on decrit l ensemble de ces plongements

mixed hodge polynomials of character varieties we calculate the e polynomials of certain twisted gl n character varieties mathcal m n of riemann surfaces by counting points over finite fields using the character table of the finite group of lie type text gl n mathbb f q and a theorem proved in the appendix by n katz we deduce from this calculation several geometric results for example the value of the topological euler characteristic of the associated pgl n character variety the calculation also leads to several conjectures about the cohomology of mathcal m n an explicit conjecture for its mixed hodge polynomial a conjectured curious hard lefschetz theorem and a conjecture relating the pure part to absolutely indecomposable representations of a certain quiver we prove these conjectures for n 2

wall crossing free fermions and crystal melting we describe wall crossing for local toric calabi yau manifolds without compact four cycles in terms of free fermions vertex operators and crystal melting firstly to each such manifold we associate two states in the free fermion hilbert space the overlap of these states reproduces the bps partition function corresponding to the non commutative donaldson thomas invariants given by the modulus square of the topological string partition function secondly we introduce the wall crossing operators which represent crossing the walls of marginal stability associated to changes of the b field through each two cycle in the manifold bps partition functions in non trivial chambers are given by the expectation values of these operators thirdly we discuss crystal interpretation of such correlators for this whole class of manifolds we describe evolution of these crystals upon a change of the moduli and find crystal interpretation of the flop transition and the dt pt transition the crystals which we find generalize and unify various other calabi yau crystal models which appeared in literature in recent years

the construction of ale spaces as hyper kahler quotients on decrit la construction d une famille particuliere de 4 varietes hyper kahler les espaces asymptotiquement localement euclidiens ale on decrit une 4 variete de riemann avec juste une extremite qui a l infini ressemble au quotient r 4 de l espace euclidien r 4 par un groupe fini d identifications

the gorenstein conjecture fails for the tautological ring of mathcal overline m _ 2 n we prove that for n equal to at least one of the integers 8 12 16 20 the tautological ring r bullet overline mathcal m 2 n is not gorenstein in fact our n equals the smallest integer such that there is a non tautological cohomology class of even degree on overline mathcal m 2 n by work of graber and pandharipande such a class exists on overline mathcal m 2 20 and we present some evidence indicating that n is in fact 20

stability for an abelian category abstract the main goal of the article is to give the definition of algebraic stability that would permit us to consider stability not only for algebraic vector bundles or torsion free coherent sheaves but for the abelian category of coherent sheaves or for whatever abelian category we present an axiomatic description of the algebraic stability on an abelian category and prove some general results then the stability for coherent sheaves on a projective variety is constructed which generalizes gieseker stability stabilities for graded modules and for quiver representations are also discussed the constructions could be used for other abelian categories as well

stable rationality of certain pgln quotients stable rationality of the field of matrix invariants m n m n pgl n is proved forn 5 andn 7 in combination with existing results this shows that v pgl n is stably rational wheneverv is an almost free representation ofpgl n andn divides 420 22 3 5 7

consistency conditions for brane tilings abstract given a brane tiling on a torus we provide a new way to prove and generalise the recent results of szendr i mozgovoy and reineke regarding the donaldson thomas theory of the moduli space of framed cyclic representations of the associated algebra using only a natural cancellation type consistency condition we show that the algebras are 3 calabi yau and calculate donaldson thomas type invariants of the moduli spaces two new ingredients to our proofs are a grading of the algebra by the path category of the associated quiver modulo relations and a way of assigning winding numbers to pairs of paths in the lift of the brane tiling to the universal cover these ideas allow us to generalise the above results to all consistent brane tilings on k 1 surfaces we also prove a converse no consistent brane tiling on a sphere gives rise to a 3 calabi yau algebra

asymptotics for the generalized two dimensional ginzburg landau equation abstract in this paper the authors have studied a generalized ginzburg landau equation in two spatial dimensions 2d they have shown that this equation under periodic boundary conditions has the maximal attractor with finite hausdorff dimension this rigorously establishes the foundation for further investigation of this type of model

global existence of solutions to the derivative 2d ginzburg landau equation abstract in this paper we study a complex derivative ginzburg landau equation with two spatial variables 2d we obtain sufficient conditions for the existence and uniqueness of global solutions for the initial boundary value problem of the derivative 2d ginzburg landau equation and improve the known results

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

cohomology of moduli spaces of curves of genus three via point counts in this article we consider the moduli space of smooth n pointed non hyperelliptic curves of genus 3 in the pursuit of cohomological information about this space we make s n equivariant counts of

the equivariant euler characteristic of moduli spaces of curves we derive a formula for the sn equivariant euler characteristic of the moduli space mg n of genus g curves with n marked points

on the noncommutative donaldson thomas invariants arising from brane tilings abstract given a brane tiling that is a bipartite graph on a torus we can associate with it a quiver potential and a quiver potential algebra under certain consistency conditions on a brane tiling we prove a formula for the donaldson thomas type invariants of the moduli space of framed cyclic modules over the corresponding quiver potential algebra we relate this formula with the counting of perfect matchings of the periodic plane tiling corresponding to the brane tiling we prove that the same consistency conditions imply that the quiver potential algebra is a 3 calabi yau algebra we also formulate a rationality conjecture for the generating functions of the donaldson thomas type invariants

hyperkahler metrics and supersymmetry we describe two constructions of hyperkahler manifolds one based on a legendre transform and one on a sympletic quotient these constructions arose in the context of supersymmetric nonlinear models but can be described entirely geometrically in this general setting we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience

cellular resolutions of noncommutative toric algebras from superpotentials abstract this paper constructs cellular resolutions for classes of noncommutative algebras analogous to those introduced by bayer and sturmfels 1998 2 in the commutative case to achieve this we generalise the dimer model construction of noncommutative crepant resolutions of three dimensional toric algebras by associating a superpotential and a notion of consistency to toric algebras of arbitrary dimension for abelian skew group algebras and algebraically consistent dimer model algebras we introduce a cell complex in a real torus whose cells describe uniformly all maps in the minimal projective bimodule resolution of a we illustrate the general construction of for an example in dimension four arising from a tilting bundle on a smooth toric fano threefold to highlight the importance of the incidence function on

the harder narasimhan system in quantum groups and cohomology of quiver moduli methods of harder and narasimhan from the theory of moduli of vector bundles are applied to moduli of quiver representations using the hall algebra approach to quantum groups an analog of the harder narasimhan recursion is constructed inside the quantized enveloping algebra of a kac moody algebra this leads to a canonical orthogonal system the hn system in this algebra using a resolution of the recursion an explicit formula for the hn system is given as an application explicit formulas for betti numbers of the cohomology of quiver moduli are derived generalizing several results on the cohomology of quotients in linear algebra type situations

morita equivalences and azumaya loci from higgsing dimer algebras abstract let a a be a cyclic contraction of dimer algebras with a non cancellative and a cancellative a is then prime noetherian and a finitely generated module over its center in contrast a is often not prime nonnoetherian and an infinitely generated module over its center we present certain morita equivalences that relate the representation theory of a with that of a we then characterize the azumaya locus of a in terms of the azumaya locus of a and give an explicit classification of the simple a modules parameterized by the azumaya locus furthermore we show that if the smooth and azumaya loci of a coincide then the smooth and azumaya loci of a coincide this provides the first known class of algebras that are nonnoetherian and infinitely generated modules over their centers with the property that their smooth and azumaya loci coincide

on the number of absolutely indecomposable representations of a quiver limburgs univ ctr dept wni b 3590 diepenbeek belgium sevenhant b limburgs univ ctr dept wni univ campus bldg d b 3590 diepenbeek belgium

positivity for kac polynomials and dt invariants of quivers we give a cohomological interpretation of both the kac polynomial and the rened donaldson thomas invariants of quivers this interpretation yields a proof of a conjecture of kac from 1982 and gives a new perspective on recent work of kontsevich soibelman this is achieved by computing via an arithmetic fourier transform the dimensions of the isotypical components of the cohomology of associated nakajima quiver varieties under the action of a weyl group the generating function of the corresponding poincar e polynomials is an extension of hua s formula for kac polynomials of quivers involving hall littlewood symmetric functions the resulting formulae contain a wide range of information on the geometry of the quiver varieties

the euler characteristic of the moduli space of curves let fg 1 g 1 be the mapping class group consisting of all isotopy classes of base point and orientation preserving homeomorphisms of a closed oriented surface f of genus g let 1 be its euler characteristic in the sense of wall that is z f i fgl f l z e f where f is any torsion free subgroup of finite index in f 1 and e is a contractible space on which f acts freely and properly discontinuously an example of such a space is the teichmiiller space 1 and g f can be interpreted as the orbifold euler characteristic of f i join the moduli space of curves of genus g with base point the purpose of this paper is to prove the following formula for f i

cohomological hall algebras vertex algebras and instantons we define an action of the double of cohomological hall algebra of kontsevich and soibelman on the cohomology of the moduli space of spiked instantons of nekrasov we identify this action with the one of the affine yangian of mathfrak gl 1 based on that we derive the vertex algebra at the corner mathcal w r 1 r 2 r 3 of gaiotto and rapcak we conjecture that our approach works for a big class of calabi yau categories including those associated with toric calabi yau 3 folds

noncommutative projective curves and quantum loop algebras we show that the hall algebra of the category of coherent sheaves on a weighted projective line over a finite field provides a realization of the quantized enveloping algebra of a certain nilpotent subalgebra of the affinization of the corresponding kac moody algebra in particular this yields a geometric realization of the quantized enveloping algebra of elliptic or 2 toroidal algebras of types d 4 1 1 e 1 1 6 e 1 1 7 and e 8 1 1 in terms of coherent sheaves on weighted projective lines of genus one or equivalently in terms of equivariant coherent sheaves on elliptic curves

transitivity of generic semigroups of area preserving surface diffeomorphisms we prove that the action of the semigroup generated by a cr generic pair of area preserving diffeomorphisms of a compact orientable surface is transitive

absolutely indecomposable representations and kac moody lie algebras a conjecture of kac states that the polynomial counting the number of absolutely indecomposable representations of a quiver over a finite field with given dimension vector has positive coefficients and furthermore that its constant term is equal to the multiplicity of the corresponding root in the associated kac moody lie algebra in this paper we prove these conjectures for indivisible dimension vectors