generalized kahler manifolds commuting complex structures and split tangent bundles we study generalized kahler manifolds for which the corresponding complex structures commute and classify completely the compact four dimensional ones

symmetries in generalized kahler geometry we define the notion of a moment map and reduction in both generalized complex geometry and generalized kahler geometry as an application we give very simple explicit constructions of bi hermitian structures on mathbb c mathbb p n hirzebruch surfaces the blow up of mathbb c mathbb p n at arbitrarily many points and other toric varieties as well as complex grassmannians

unobstructed k deformations of generalized complex structures and bi hermitian structures abstract we introduce k deformations of generalized complex structures on a compact kahler manifold m x j with an effective anti canonical divisor and show that obstructions to k deformations of generalized complex structures on m always vanish applying unobstructed k deformations and the stability theorem for generalized kahler structures we construct deformations of bi hermitian structures in the form j j t h t on a compact kahler surface with a non zero holomorphic poisson structure then we prove that a compact kahler surface s admits a non trivial bi hermitian structure with the torsion condition and the same orientation if and only if s has a non zero holomorphic poisson structure we also obtain bi hermitian structures j j h on del pezzo surfaces degenerate del pezzo surfaces and some ruled surfaces for which the complex structure j is not equivalent to j under diffeomorphisms

generalized kahler potentials from supergravity we consider supersymmetric mathcal n 2 solutions with non vanishing ns three form building on worldsheet results we reduce the problem to a single generalized monge ampere equation on the generalized kahler potential k recently interpreted geometrically by lindstrom rocek von unge and zabzine one input in the procedure is a holomorphic function w that can be thought of as the effective superpotential for a d3 brane probe the procedure is hence likely to be useful for finding gravity duals to field theories with non vanishing abelian superpotential such as leigh strassler theories we indeed show that a purely ns precursor of the lunin maldacena dual to the deformed mathcal n 4 super yang mills falls in our class

modular forms and special cubic fourfolds abstract we study the degrees of special cubic divisors on moduli space of cubic fourfolds with at worst ade singularities in this paper we show that the generating series of the degrees of such divisors is a level three modular form

the complexity of finding minimum length generator sequences abstract the computational complexity of the following problem is investigated given a permutation group specified as a set of generators and a single target permutation which is a member of the group what is the shortest expression for the target permutation in terms of the generators the general problem is demonstrated to be psppace complete and indeed is shown to remain so even when the generator set is restricted to contain only two permutations the restriction on generator set cardinality is the best possible as the problem becomes soluble in polynomial time if the generator set cantains only one permutation an interesting feature of this problem is that it does not fall under the headings of two person games or formal languages which cover the great majority of known pspace complete problems some restricted versions of the problem in which the generator set is fixed rather than being part of the problem instance are also investigated and shown to be computationally tractable one result of this kind is that determining the most compact expression of a permutation in terms of cyclicly adjacent transpositions can be achieved in polynomial time thus from an initial arrangement of distinct objects on a circle one can quickly compute the smallest number of interchanges of adjacent objects required to realise any other arrangement surprisingly this problem appears substantially more difficult to solve than the related one for which a solution has been known for some time in which the objects are arranged on a line segment

reduction of courant algebroids and generalized complex structures we present a theory of reduction for courant algebroids as well as dirac structures generalized complex and generalized kahler structures which interpolates between holomorphic reduction of complex manifolds and symplectic reduction the enhanced symmetry group of a courant algebroid leads us to define extended actions and a generalized notion of moment map key examples of generalized kahler reduced spaces include new explicit bi hermitian metrics on cp2

instantons poisson structures and generalized kahler geometry using the idea of a generalized kahler structure we construct bihermitian metrics on cp2 and cp1 cp1 and show that any such structure on a compact 4 manifold m defines one on the moduli space of anti self dual connections on a fixed principal bundle over m we highlight the role of holomorphic poisson structures in all these constructions

geometric quantization and multiplicities of group representations the heisenberg uncertainty principle says that it is impossible to determine simultaneously the position and momentum of a quantum mechanical particle this can be rephrased as follows the smallest subsets of classical phase space in which the presence of a quantum mechanical particle can be detected are its lagrangian submanifolds for this reason it makes sense to regard the lagrangian submanifolds of phase space as being its true points see weinstein 17 now let g be a compact lie group and g x x x a hamiltonian action of g on x see w for definitions it is well known that the fixed points of this action form a symplectic submanifold of x see for instance guillemin and sternberg 5 however what can one say about the fixed points of g we will show that they are also the points of a symplectic manifold xc this manifold is the marsden weinstein reduction of x with respect to the zero orbit in g and will be described in sect 2 it was introduced in a completely different context fl om ours by marsden and weinstein 12 problems in classical mechanics can often be reduced to the study of hamiltonian systems on symplectic manifolds and problems in quantum mechanics to the study of linear operators on hilbert space this fact has inspired a number of efforts to quantize symplectic geometry by devising schemes for associating hilbert space to symplectic manifolds the no go theorems of groenwald and van hove impose some embarrassing limitations on all such schemes however it seems to be a useful idea heuristically to think of every symplectic manifold x a ic 1 as being symbiotically associated with a hilbert space xq a t in such a way that the classical observables on the first space correspond to quantum observables on the second space the heuristics further suggests that if g is a group of symmetries of x la i j it should also be a group of symmetries of xq a tum in this heuristic spirit we will state the main conjecture of this paper

the equivariant cohomology theory of twisted generalized complex manifolds it has been shown recently by kapustin and tomasiello that the mathematical notion of hamiltonian actions on twisted generalized kahler manifolds is in perfect agreement with the physical notion of general 2 2 gauged sigma models with three form fluxes in this article we study the twisted equivariant cohomology theory of hamiltonian actions on h twisted generalized complex manifolds if the manifold satisfies the overline partial partial lemma we establish the equivariant formality theorem if in addition the manifold satisfies the generalized kahler condition we prove the kirwan injectivity in this setting we then consider the hamiltonian action of a torus on an h twisted generalized calabi yau manifold and extend to this case the duistermaat heckman theorem for the push forward measure

borcherds products on unitary groups in the present paper we provide a construction of the multiplicative borcherds lift for unitary groups mathrm u 1 m which takes weakly holomorphic elliptic modular forms as input functions and lifts them to automorphic forms having infinite product expansions and taking their zeros and poles along heegner divisors in order to transfer borcherds theory to unitary groups we construct a suitable embedding of mathrm u 1 m into mathrm o 2 2m we also derive a formula for the values taken by the borcherds products at cusps of the symmetric domain of the unitary group further as an application of the lifting we obtain a modularity result for a generating series with heegner divisors as coefficients along the lines of borcherds generalization of the gross zagier kohnen theorem

the np completeness of some edge partition problems we show that for each fixed n geqq 3 it is np complete to determine whether an arbitrary graph can be edge partitioned into subgraphs isomorphic to the complete graph k n the np completeness of a number of other edge partition problems follows immediately

on the cost of interchange rearrangement in strings consider the following optimization problem given two strings over the same alphabet transform one into another by a succession of interchanges of two elements in each interchange the two participating elements exchange positions an interchange is given a weight that depends on the distance in the string between the two exchanged elements the object is to minimize the total weight of the interchanges this problem is a generalization of a classical problem on permutations where every element appears once the generalization considers general strings with possibly repeating elements and a function assigning weights to the interchanges the generalization to general strings with unit weights was mentioned by cayley in the 19th century and its complexity has been an open question since we solve this open problem and consider various weight functions as well

group actions on dg manifolds and exact courant algebroids let g be a lie group acting by diffeomorphisms on a manifold m and consider the image of t 1 g and t 1 m of g and m respectively in the category of differential graded manifolds we show that the obstruction to lift the action of t 1 g on t 1 m to an action on a mathbb r n bundle over t 1 m is measured by the g equivariant cohomology of m we explicitly calculate the differential graded lie algebra of the symmetries of the mathbb r n bundle over t 1 m and we use this differential graded lie algebra to understand which actions are hamiltonian we show how split exact courant algebroids could be obtained as the derived leibniz algebra of the symmetries of mathbb r 2 bundles over t 1 m and we use this construction to propose that the infinitesimal symmetries of a split exact courant algebroid should be encoded in the differential graded lie algebra of symmetries of a mathbb r 2 bundle over t 1 m with this setup at hand we propose a definition for an action of a lie group on an exact courant algebroid and we propose conditions for the action to be hamiltonian

on the equidistribution of some hodge loci we prove the equidistribution of the hodge locus for certain non isotrivial polarized variations of hodge structure of weight 2 with h 2 0 1 over complex quasi projective curves given some norm condition we also give an asymptotic on the growth of the hodge locus in particular this implies the equidistribution of elliptic fibrations in quasi polarized non isotrivial families of k3 surfaces

the rationality of vector valued modular forms associated with the weil representation in a recent paper duke math j 97 219 233 borcherds asks whether or not the spaces of vector valued modular forms associated to the weil representation have bases of modular forms whose fourier expansions have only integer coefficients we give an affirmative answer to borcherds question this strengthens and simplifies borcherds main theorem which is a generalization of a theorem of gross kohnen and zagier math ann 278 497 562

heegner divisors l functions and harmonic weak maass forms recent works mostly related to ramanujan s mock theta functions make use of the fact that harmonic weak maass forms can be combinatorial generating functions generalizing works of waldspurger kohnen and zagier we prove that such forms also serve as generating functions for central values and derivatives of quadratic twists of weight 2 modular l functions to obtain these results we construct differentials of the third kind with twisted heegner divisor by suitably generalizing the borcherds lift to harmonic weak maass forms the connection with periods fourier coefficients derivatives of l functions and points in the jacobian of modular curves is obtained by analyzing the properties of these differentials using works of scholl waldschmidt and gross and zagier

using swaps and deletes to make strings match given a source and a target string their swap delete edit distance is the minimum number of interchange consecutive symbols and delete operations applied to the source string to make it equal the target string we show that the swap delete edit distance of strings over alphabets of bounded size can be computed in polynomial time

manin triples for lie bialgebroids in his study of dirac structures a notion which includes both poisson structures and closed 2 forms t courant introduced a bracket on the direct sum of vector fields and 1 forms this bracket does not satisfy the jacobi identity except on certain subspaces in this paper we systematize the properties of this bracket in the definition of a courant algebroid this structure on a vector bundle e rightarrow m consists of an antisymmetric bracket on the sections of e whose jacobi anomaly has an explicit expression in terms of a bundle map e rightarrow tm and a field of symmetric bilinear forms on e when m is a point the definition reduces to that of a lie algebra carrying an invariant nondegenerate symmetric bilinear form for any lie bialgebroid a a over m a notion defined by mackenzie and xu there is a natural courant algebroid structure on a oplus a which is the drinfel d double of a lie bialgebra when m is a point conversely if a and a are complementary isotropic subbundles of a courant algebroid e closed under the bracket such a bundle with dimension half that of e is called a dirac structure there is a natural lie bialgebroid structure on a a whose double is isomorphic to e the theory of manin triples is thereby extended from lie algebras to lie algebroids our work gives a new approach to bihamiltonian structures and a new way of combining two poisson structures to obtain a third one we also take some tentative steps toward generalizing drinfel d s theory of poisson homogeneous spaces from groups to groupoids

algebraic formulas for the coefficients of mock theta functions and weyl vectors of borcherds products abstract we present some applications of the kudla millson and the millson theta lift the two lifts map weakly holomorphic modular functions to vector valued harmonic maass forms of weight 3 2 and 1 2 respectively we give finite algebraic formulas for the coefficients of ramanujan s mock theta functions f q and q in terms of traces of cm values of a weakly holomorphic modular function further we construct vector valued harmonic maass forms whose shadows are unary theta functions and whose holomorphic parts have rational coefficients this yields a rationality result for the coefficients of mock theta functions i e harmonic maass forms whose shadows lie in the space of unary theta functions moreover the harmonic maass forms we construct can be used to evaluate the petersson inner products of unary theta functions with harmonic maass forms giving formulas and rationality results for the weyl vectors of borcherds products

the weil representation and hecke operators for vector valued modular forms we define hecke operators for vector valued modular forms transforming with the weil representation associated to a discriminant form we describe the properties of the corresponding algebra of hecke operators and study the action on modular forms

fibonacci heaps and their uses in improved network optimization algorithms in this paper we develop a new data structure for implementing heaps priority queues our structure fibonacci heaps abbreviated f heaps extends the binomial queues proposed by vuillemin and studied further by brown f heaps support arbitrary deletion from an n item heap in o log n amortized time and all other standard heap operations in o 1 amortized time using f heaps we are able to obtain improved running times for several network optimization algorithms in particular we obtain the following worst case bounds where n is the number of vertices and m the number of edges in the problem graph o n log n m for the single source shortest path problem with nonnegative edge lengths improved from o m log m n 2 n o n 2 log n nm for the all pairs shortest path problem improved from o nm log m n 2 n o n 2 log n nm for the assignment problem weighted bipartite matching improved from o nm log m n 2 n o m m n for the minimum spanning tree problem improved from o m log log m n 2 n where m n min i u log i n m n note that m n log n if m n of these results the improved bound for minimum spanning trees is the most striking although all the results give asymptotic improvements for graphs of appropriate densities

blow ups and resolutions of strong kahler with torsion metrics abstract on a compact complex manifold we study the behaviour of strong kahler with torsion strong kt structures under small deformations of the complex structure and the problem of extension of a strong kt metric in this context we obtain the analogous result of miyaoka extension theorem studying the blow up of a strong kt manifold at a point or along a complex submanifold we prove that a complex orbifold endowed with a strong kt metric admits a strong kt resolution in this way we obtain new examples of compact simply connected strong kt manifolds

approximate string matching with stuck address bits a string s s m can be viewed as a set of pairs s i i s i s i 0 m 1 we follow the recent work on pattern matching with address errors and consider approximate pattern matching problems arising from the setting where errors are introduced to the location component i rather than the more traditional setting where errors are introduced to the content itself s i specifically we continue the work on string matching in the presence of address bit errors in this paper we consider the case where bits of i may be stuck either in a consistent or transient manner we formally define the corresponding approximate pattern matching problems and provide efficient algorithms for their resolution

cm values of regularized theta lifts and harmonic weak maas forms of weight 1 we study special values of regularized theta lifts at complex multiplication cm points in particular we show that cm values of borcherds products can be expressed in terms of finitely many fourier coefficients of certain harmonic weak maas forms of weight 1 as it turns out these coefficients are logarithms of algebraic integers whose prime ideal factorization is determined by special cycles on an arithmetic curve our results imply a conjecture of duke and li and give a new proof of the modularity of a certain arithmetic generating series of weight 1 studied by kudla rapoport and yang

a construction of antisymmetric modular forms for weil representations we give coefficient formulas for antisymmetric vector valued cusp forms with rational fourier coefficients for the weil representation associated to a finite quadratic module the forms we construct always span all cusp forms in weight at least three these formulas are useful for computing explicitly with theta lifts

regularized theta lifts for orthogonal groups over totally real fields we define a regularized theta lift from sl2 to orthogonal groups over totally real fields it takes harmonic whittaker forms to automorphic green functions and weakly holomorphic whittaker forms to meromorphic modular forms on orthogonal groups with zeros and poles supported on special divisors generalizing borcherds work on automorphic products to prove our results we use the spectral expansion of the lift and study its relationship with the cohomological theta lift of kudla and millson

transforming cabbage into turnip polynomial algorithm for sorting signed permutations by reversals genomes frequently evolve by reversals r i j that transform a gene order 1 i i 1 j 1 j n into 1 i j 1 i 1 j n reversal distance between permutations and sis the minimum number of reversals to transform into a analysis of genome rearrangements in molecular biology started in the late 1930 s when dobzhansky and sturtevant published a milestone paper presenting a rearrangement scenario with 17 inversions between the species of drosophilia analysis of genomes evolving by inversions leads to a combinatorial problem of sorting by reversals studied in detail recently we study sorting of signed permutations by reversals a problem that adequately models rearrangements in a small genomes like chloroplast or mitochondrial dna the previously suggested approximation algorithms for sorting signed permutations by reversals compute the reversal distance between permutations with an astonishing accuracy for both simulated and biological data we prove a duality theorem explaining this intriguing performance and show that there exists a hidden parameter that allows one to compute the reversal distance between signed permutations in polynomial time

the transposition median problem is np complete during recent years the genomes of more and more species have been sequenced providing data for phylogenetic reconstruction based on genome rearrangement measures where the most important distance measures are the reversal distance and the transposition distance the two main tasks in all phylogenetic reconstruction algorithms are to calculate pairwise distances and to solve the median of three problem while the reversal distance problem can be solved in linear time the reversal median problem has been proven to be np complete the status of the transposition distance problem is still open but it is conjectured to be more difficult than the reversal problem this in turn also suggests that the transposition median problem is np complete however this conjecture could not yet be proven we have now succeeded in giving a non trivial proof for the np completeness of the transposition median problem

generalized kahler geometry generalized kahler geometry is the natural analogue of kahler geometry in the context of generalized complex geometry just as we may require a complex structure to be compatible with a riemannian metric in a way which gives rise to a symplectic form we may require a generalized complex structure to be compatible with a metric so that it defines a second generalized complex structure we prove that generalized kahler geometry is equivalent to the bi hermitian geometry on the target of a 2 dimensional sigma model with 2 2 supersymmetry we also prove the existence of natural holomorphic courant algebroids for each of the underlying complex structures and that these split into a sum of transverse holomorphic dirac structures finally we explore the analogy between pre quantum line bundles and gerbes in the context of generalized kahler geometry

generalized moonshine ii borcherds products the goal of this paper is to construct infinite dimensional lie algebras by using infinite product identities and to use these lie algebras to reduce the generalized moonshine conjecture to a pair of hypotheses about group actions on vertex algebras and lie algebras we expect the lie algebras that we construct to manifest as algebras of physical states in an orbifold conformal field theory yet to be fully constructed with symmetries given by the monster simple group we introduce vector valued modular functions attached to families of modular functions of different levels and we prove infinite product identities for a distinguished class of automorphic functions on a product of two half planes we recast this result using the borcherds harvey moore singular theta lift and show that the vector valued functions attached to completely replicable modular functions with integer coefficients lift to automorphic functions with infinite product expansions at all cusps for each element of the monster simple group we construct an infinite dimensional lie algebra such that its denominator formula is an infinite product expansion of the automorphic function arising from that element s mckay thompson series these lie algebras have the unusual property that their simple roots and all root multiplicities are known we show that under certain hypotheses characters of groups acting on these lie algebras form functions on the upper half plane that are either constant or invariant under a genus zero congruence group