a statistical mechanical description of quantum entanglement we present a description of finite dimensional quantum entanglement based on a study of the space of all convex decompositions of a given density matrix on this space we construct a system of real polynomial equations describing separable states we further study this system using methods of statistical mechanics as an example we finally apply our techniques to werner states of two qubits and obtain a sufficient criterion for separability

tilting and refined donaldson thomas invariants abstract we study tilting for the heart a of the canonical t structure of the finite dimensional derived category of the ginzburg algebra for a quiver with potential q w we give conditions on that the stable objects for a central charge on a define a sequence of simple tilts from a to a 1 on that conditions the refined donaldson thomas invariant associated to q w is independent of the chosen central charge

a priori probabilities of separable quantum states zyczkowski horodecki sanpera and lewenstein zhsl recently proposed a natural measure on the n dimensional quantum systems but expressed surprise when it led them to conclude that for n 2 2 disentangled separable systems are more probable 0 632 0 002 in nature than entangled ones we contend however that zhsl s rejected intuition has in fact a sound theoretical basis and that the a priori probability of disentangled 2 2 systems should more properly be viewed as considerably less than 0 5 we arrive at this conclusion in two quite distinct ways the first based on classical and the second quantum considerations both approaches however replace in whole or part the zhsl product measure by ones based on the volume elements of monotone metrics which in the classical case amounts to adopting the jeffreys prior of bayesian theory only the quantum theoretic analysis which yields the smallest probabilities of disentanglement uses the minimum number of parameters possible that is n2 1 as opposed to n2 n 1 although this over parametrization as recently indicated by byrd should be avoidable however despite substantial computation we are not able to obtain precise estimates of these probabilities and the need for additional possibly supercomputer analyses is indicated particularly so for higher dimensional quantum systems such as the 2 3 ones which we also study here

moduli stacks and invariants of semistable objects on k3 surfaces abstract for a k3 surface x and its bounded derived category of coherent sheaves d x we have the notion of stability conditions on d x in the sense of t bridgeland in this paper we show that the moduli stack of semistable objects in d x with a fixed numerical class and a phase is represented by an artin stack of finite type over c then following d joyce s work we introduce the invariants counting semistable objects in d x and show that the invariants are independent of a choice of a stability condition

extremal spacings between eigenphases of random unitary matrices and their tensor products extremal spacings between eigenphases of random unitary matrices of size n pertaining to circular ensembles are investigated explicit probability distributions for the minimal spacing for various ensembles are derived for n 4 we study ensembles of tensor product of k random unitary matrices of size n which describe independent evolution of a composite quantum system consisting of k subsystems in the asymptotic case as the total dimension n nk becomes large the nearest neighbor distribution p s becomes poissonian but statistics of extreme spacings p smin and p smax reveal certain deviations from the poissonian behavior

cusps of the kahler moduli space and stability conditions on k3 surfaces ma int j math 20 6 727 750 2009 established a bijection between fourier mukai partners of a k3 surface and cusps of the kahler moduli space the kahler moduli space can be described as a quotient of bridgeland s stability manifold we study the relation between stability conditions near to a cusp and the associated fourier mukai partner y in the following ways 1 we compare the heart of to the heart of coherent sheaves on y 2 we construct y as moduli space of stable objects an appendix is devoted to the group of auto equivalences of mathcal d b x which respect the component stab dagger x of the stability manifold

semiclassical level spacings when regular and chaotic orbits coexist the authors calculate semiclassical limiting level spacing distributions p s for systems whose classical energy surface is divided into a number of separate region in which motion is regular or chaotic in the calculation it is assumed that the spectrum is the superposition of statistically independent sequences of levels from each of the classical phase space regions sequences from regular regions having poisson distributions and those from irregular regions having wigner distributions the formulae for p s depend on the sum of the liouville measures of all the classical regular regions and on the separate liouville measures of the significant chaotic regions

marginal distribution of the s matrix elements for dyson s measure and some applications in recent attempts to construct a statistical theory of nuclear reactions by doing statistics directly on the s matrix elements dyson s measure which remains invariant under an automorphism that maps the space of unitary and symmetric matrices into itself is of fundamental importance the authors study some of the marginal distributions of the individual s matrix elements or of groups of them that arise from dyson s measure to understand the problem better a similar discussion is first carried out for haar s measure of unitary matrices which do not have the restriction of symmetry and some of the effects of this restriction are thus exhibited some applications of these mathematical results to reaction theory are discussed

birational geometry of singular moduli spaces of o grady type abstract following bayer and macri we study the birational geometry of singular moduli spaces m of sheaves on a k3 surface x which admit symplectic resolutions more precisely we use the bayer macri map from the space of bridgeland stability conditions stab x to the cone of movable divisors on m to relate wall crossing in stab x to birational transformations of m we give a complete classification of walls in stab x and show that every minimal birational model of m in the sense of the log minimal model program appears as a moduli space of bridgeland semistable objects on x an essential ingredient of our proof is an isometry between the orthogonal complement of a mukai vector inside the algebraic mukai lattice of x and the neron severi lattice of m which generalises results of yoshioka as well as perego and rapagnetta moreover this allows us to conclude that the symplectic resolution of m is deformation equivalent to the 10 dimensional irreducible holomorphic symplectic manifold found by o grady

global entangling properties of the coupled kicked tops we study global entangling properties of the system of coupled kicked tops testing various hypotheses and predictions concerning entanglement in quantum chaotic systems in order to analyze the averaged initial entanglement production rate and the averaged asymptotic entanglement various ensembles of initial product states are evolved two different ensembles with natural probability distribution are considered product states of independent spin coherent states and product states of random states it appears that the choice of either of these ensembles results in significantly different averaged entanglement behavior we investigate also a relation between the averaged asymptotic entanglement and the mean entanglement of eigenvectors of the evolution operator lower bound on the averaged asymptotic entanglement is derived expressed in terms of the eigenvector entanglement

truncations of random unitary matrices we analyse properties of non hermitian matrices of size m constructed as square submatrices of unitary orthogonal random matrices of size n m distributed according to the haar measure in this way we define ensembles of random matrices and study the statistical properties of the spectrum located inside the unit circle in the limit of large matrices this ensemble is characterized by the ratio m n for the truncated cue we analytically derive the joint density of eigenvalues and all correlation functions in the strongly non unitary case universal ginibre behaviour is found for n m fixed and n to the universal resonance width distribution with n m open channels is recovered

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

averages on the unitary group and applications to the problem of disordered conductors in a recent approach to the study of disordered conductors a maximum entropy criterion is used to perform statistical calculations directly on the transfer matrix for the full physical system quantities of great importance in that approach are averages of products of the u and u matrix elements u being a unitary matrix evaluated with the invariant measure of the unitary group the authors evaluate those averages that contain up to four u and four u matrix elements since they are the ones needed in the calculation of the average and covariance of transmission and reflection coefficients the evaluation involves no direct integration at all but makes use solely of the invariance of the measure the fact that the matrix elements ua alpha are commuting c numbers and the unitarity of u the physical consequences for the average and covariance of transmission and reflection coefficients are briefly discussed

the space of stability conditions on the local projective plane we study the space of stability conditions on the total space of the canonical bundle over the projective plane we explicitly describe a chamber of geometric stability conditions and show that its translates via autoequivalences cover a whole connected component we prove that this connected component is simply connected we determine the group of autoequivalences preserving this connected component which turns out to be closely related to 1 3 finally we show that there is a submanifold isomorphic to the univer sal covering of a moduli space of elliptic curves with 1 3 level struc ture the morphism is 1 3 equivariant and is given by solutions of picard fuchs equations this result is motivated by the notion of stability and by mirror symmetry

inclusion relations among separability criteria we revisit the application of different separability criteria by recourse to an exhaustive monte carlo exploration involving the pertinent state space of pure and mixed states the corresponding chain of implications of different criteria is in such a way numerically elucidated we also quantify for a bipartite system of arbitrary dimension the proportion of states that can be distilled according to a definite criterion our work can be regarded as a complement to the recent review paper by terhal b 2002 theor comput sci 287 313 some questions posed there receive an answer here

new features of quantum discord uncovered by q entropies the notion of quantum discord introduced by ollivier and zurek phys rev lett 88 2001 017901 see also henderson and vedral j phys a 34 2001 6899 has attracted increasing attention in recent years as an entropic quantifier of non classical features pertaining to the correlations exhibited by bipartite quantum systems here we generalize the notion so as to encompass power law q entropies that reduce to the standard shannon entropy in the limit q 1 and study the concomitant consequences the ensuing new discord like measures we advance describe aspects of non classicality that are different from those associated with the standard quantum discord a particular manifestation of this difference concerns a feature related to order let d1 stand for the standard shannon based discord measure and dq for the q 1 one if two quantum states a b are such that d1 a d1 b this order relation does not remain invariant under a change from d1 to dq

induced measures in the space of mixed quantum states we analyse several product measures in the space of mixed quantum states in particular we study measures induced by the operation of partial tracing the natural rotationally invariant measure on the set of all pure states of a n k composite system induces a unique measure in the space of n n mixed states or in the space of k k mixed states if the reduction takes place with respect to the first subsystem for k n the induced measure is equal to the hilbert schmidt measure which is shown to coincide with the measure induced by singular values of non hermitian random gaussian matrices pertaining to the ginibre ensemble we compute several averages with respect to this measure and show that the mean entanglement of n n pure states behaves as lnn 1 2

mmp via wall crossing for moduli spaces of stable sheaves on an enriques surface abstract we use wall crossing in the bridgeland stability manifold to systematically study the birational geometry of the moduli space m v of semistable objects of class v for a generic stability condition on an arbitrary enriques surface x in particular we show that for any other generic stability condition the two moduli spaces m v and m v are birational as a consequence we show that for primitive v of odd rank m v is birational to a hilbert scheme of points similarly in even rank we show that m v is birational to a moduli space of torsion sheaves supported on a hyperelliptic curve when l v 1 as an added bonus of our work we prove that the donaldson mukai map v v pic m v is an isomorphism for these classes finally we use our classification to fully describe the geometry of the only two examples of moduli of stable sheaves on x that are uniruled and thus not k trivial

interpolating ensembles of random unitary matrices we investigate transitions between ensembles of unitary random matrices modeling changes of statistical properties of quantum chaotic systems which are periodically time dependent the transitions simulating change of integrability are modeled with the help of numerically generated one parameter families of ensembles interpolating between an ensemble of diagonal unitary matrices and a circular unitary or circular orthogonal ensemble in an analogous manner we described transitions between circular orthogonal and unitary ensembles corresponding to the time reversal symmetry breaking perturbation of a chaotic system in all cases we present results concerning statistics of the quasi energy levels and eigenvectors textcopyright 1996 the american physical society

stability conditions on threefolds with nef tangent bundles abstract in this paper we prove the bogomolov gieseker type inequality conjecture for threefolds with nef tangent bundles as a corollary there exist bridgeland stability conditions on these threefolds

invariant norm quantifying nonlinear content of hamiltonian systems given a hamiltonian system one can represent it using a symplectic map this symplectic map is specified by a set of homogeneous polynomials which are uniquely determined by the hamiltonian in this paper we construct an invariant norm in the space of homogeneous polynomials of a given degree this norm is a function of parameters characterizing the original hamiltonian system such a norm has several potential applications

a note on factoring unitary matrices abstract the note concerns three factorizations of unitary matrices a recent factorization allowing to obtain any unitary matrix of dimension 2n from four unitary matrices of dimension n the cosine sine decomposition and a new factorization decomposing a given unitary matrix into a product of two orthogonal matrices and a diagonal unitary matrix

on weyl groups and artin groups associated to orbifold projective lines abstract we associate a generalized root system in the sense of kyoji saito to an orbifold projective line via the derived category of finite dimensional representations of a certain bound quiver algebra we generalize results by saito and takebayashi and yamada for elliptic weyl groups and elliptic artin groups to the weyl groups and the fundamental groups of the regular orbit spaces associated to the generalized root systems moreover we study the relation between this fundamental group and a certain subgroup of the autoequivalence group of a triangulated subcategory of the derived category of 2 calabi yau completion of the bound quiver algebra

smoothness and poisson structures of bridgeland moduli spaces on poisson surfaces let x be a projective smooth holomorphic poisson surface in other words whose anti canonical bundle is effective we show that moduli spaces of certain bridgeland stable objects on x are smooth moreover we construct poisson structures on these moduli spaces

orthogonally spherical objects and spherical fibrations abstract we introduce a relative version of the spherical objects of seidel and thomas 32 define an object e in the derived category d z x to be spherical over z if the corresponding functor from d z to d x gives rise to autoequivalences of d z and d x in a certain natural way most known examples come from subschemes of x fibred over z this categorifies to the notion of an object of d z x orthogonal over z we prove that such an object is spherical over z if and only if it possesses certain cohomological properties similar to those in the original definition of a spherical object we then interpret this geometrically in the case when our objects are actual flat fibrations in x over z

stability conditions on k3 surfaces this article contains a description of one connected component of the space of stability conditions on the bounded derived category of coherent sheaves on a complex algebraic k3 surface

random unistochastic matrices an ensemble of random unistochastic orthostochastic matrices is defined by taking squared moduli of elements of random unitary orthogonal matrices distributed according to the haar measure on u n or o n an ensemble of symmetric unistochastic matrices is obtained with use of unitary symmetric matrices pertaining to the circular orthogonal ensemble we study the distribution of complex eigenvalues of bistochastic unistochastic and orthostochastic matrices in the complex plane we compute averages entropy traces over the ensembles of unistochastic matrices and present inequalities concerning the entropies of products of bistochastic matrices

peculiarities of quantum discord s geometric measure some properties of the quantum discord based on the geometric measure advanced by dakic et al 2010 phys rev lett 105 190502 are discussed here by recourse to a systematic survey of the two qubit state space with emphasis on werner and mem states we explore the dependence of quantum discord on the degree of mixedness of the bipartite states and also its connection with non locality as measured by the maximum violation of a bell inequality within the chsh scenario some considerations regarding the xy model are also made

derived automorphism groups of k3 surfaces of picard rank 1 we give a complete description of the group of exact autoequivalences of the bounded derived category of coherent sheaves on a k3 surface of picard rank 1 we do this by proving that a distinguished connected component of the space of stability conditions is preserved by all autoequivalences and is contractible

random bures mixed states and the distribution of their purity ensembles of random density matrices determined by various probability measures are analysed a simple and efficient algorithm to generate at random density matrices distributed according to the bures measure is proposed this procedure may serve as an initial step in performing the bayesian approach to quantum state estimation based on the bures prior we study the distribution of purity of random mixed states the moments of the distribution of purity are determined for quantum states generated with respect to the bures measure this calculation serves as an exemplary application of the deform and study approach in the theory of integrable systems leading to one of painleves transcendents

numerical algorithms for use in quantum information quantum information theory is the new field of physics and electrical engineering that arose from the application of fundamental physics concepts in communications and computing in this paper aiming to calculate some properties of quantum communication systems and the quantum entanglement measure for c2 c2 systems based on relative entropy two numerical algorithms are presented the first one is based on the codification of the possible solution in a binary string and in the application in that string of an assembly algorithm such as one used in dna construction the second one is the construction of a genetic algorithm where a string of density matrices and quantum gates in the reproduction stage are used both algorithms are used in situations where the best solution needs to be found numerical simulations are presented and the advantages and disadvantages of the algorithms are discussed

derivation of amplitude equations by the renormalization group method a proper formulation in the perturbative renormalization group method is presented to deduce amplitude equations the formulation makes it possible not only avoiding a serious difficulty in the previous reduction to amplitude equations by eliminating all of the secular terms but also consistent derivation of higher order correction to amplitude equations

tilt stability vanishing theorems and bogomolov gieseker type inequalities abstract we investigate the tilt stability of stable sheaves on projective varieties with respect to certain tilt stability conditions depends on two parameters constructed by bridgeland 11 see also 1 6 5 for a stable sheaf we give effective bounds of these parameters such that the stable sheaf is tilt stable these allow us to prove new vanishing theorems for stable sheaves and an effective serre vanishing theorem for torsion free sheaves using these results we also prove bogomolov gieseker type inequalities for the third chern character of a stable sheaf on p 3

derived equivalences of k3 surfaces and orientation every fourier mukai equivalence between the derived categories of two k3 surfaces induces a hodge isometry of their cohomologies viewed as hodge structures of weight two endowed with the mukai pairing we prove that this hodge isometry preserves the natural orientation of the four positive directions this leads to a complete description of the action of the group of all autoequivalences on cohomology very much like the classical torelli theorem for k3 surfaces determining all hodge isometries that are induced by automorphisms

composed ensembles of random unitary matrices composed ensembles of random unitary matrices are defined via products of matrices each pertaining to a given canonical circular ensemble of dyson we investigate statistical properties of spectra of some composed ensembles and demonstrate their physical relevance we also discuss the methods of generating random matrices distributed according to invariant haar measure on the orthogonal and unitary group

stability and fourier mukai transforms on elliptic fibrations abstract we systematically develop bridgeland s 7 and bridgeland maciocia s 10 techniques for studying elliptic fibrations and identify criteria that ensure 2 term complexes are mapped to torsion free sheaves under a fourier mukai transform as an application we construct an open immersion from a moduli of stable complexes to a moduli of gieseker stable sheaves on elliptic threefolds as another application we give various 1 1 correspondences between fibrewise semistable torsion free sheaves and codimension 1 sheaves on weierstrass surfaces

hall normalization constants for the bures volumes of the n state quantum systems we report the results of certain integrations of quantum theoretic interest relying in this regard upon recently developed parametrizations of boya et al 1998 preprint quant ph 9810084 of the n n density matrices in terms of squared components of the unit n 1 sphere and the n n unitary matrices firstly we express the normalized volume elements of the bures minimal monotone metric for n 2 and 3 thereby obtaining bures prior probability distributions over the two and three state systems then as a first step in extending these results to n 3 we determine that the hall normalization constant cn for the marginal bures prior probablity distribution over the n 1 dimensional simplex of the n eigenvalues of the n n density matrices is for n 4 equal to 71 680 2 since we also find that c3 35 it follows that c4 is simply equal to 211c3 c2 itself is known to equal 2 the constant c5 is also found it too is associated with a remarkably simple decomposition involving the product of the eight consecutive prime numbers from 3 to 23 we also preliminarily investigate several cases n 5 with the use of quasi monte carlo integration we hope that the various analyses reported will prove useful in deriving a general formula which evidence suggests will involve the bernoulli numbers for the hall normalization constant for arbitrary n this would have diverse applications including quantum inference and universal quantum coding

spectral gap of doubly stochastic matrices generated from equidistributed unitary matrices to a unitary matrix u we associate a doubly stochastic matrix m by taking the squared modulus of each element of u to study the connection between onset of quantum chaos on graphs and ergodicity of the underlying markov chain specified by m we study the limiting distribution of the spectral gap of m when u is taken from the circular unitary ensemble and the dimension n of u is taken to infinity we prove that the limiting distribution is degenerate the gap tends to its maximal value 1 the shape of the gap distribution for finite n is also discussed

completions branched covers artin groups and singularity theory we study the curvature of metric spaces and branched covers of riemannian manifolds with applications in topology and algebraic geometry here curvature bounds are expressed in terms of the cat inequality we prove a general cat extension theorem giving sufficient conditions on and near the boundary of a locally cat metric space for the completion to be cat we use this to prove that a branched cover of a complete riemannian manifold is locally cat if and only if all tangent spaces are cat 0 and the base has sectional curvature bounded above by we also show that the branched cover is a geodesic space using our curvature bound and a local asphericity assumption we give a sufficient condition for the branched cover to be globally cat and the complement of the branch locus to be contractible we conjecture that the universal branched cover of cn over the mirrors of a finite coxeter group is cat 0 conditionally on this conjecture we use our machinery to prove the arnoldpham thom conjecture on k 1 spaces for artin groups also conditionally we prove the asphericity of moduli spaces of amply lattice polarized k3 surfaces and of the discriminant complements of all the unimodal hypersurface singularities in arnold s hierar

conditional q entropies and quantum separability a numerical exploration we revisit the relationship between quantum separability and the sign of the relative q entropies of composite quantum systems the q entropies depend on the density matrix eigenvalues pi through the quantity q ipqi renyi s and tsallis measures constitute particular instances of these entropies we perform a systematic numerical survey of the space of mixed states of two qubit systems in order to determine as a function of the degree of mixture and for different values of the entropic parameter q the volume in state space occupied by those states characterized by positive values of the relative entropy similar calculations are performed for qubit qutrit systems and for composite systems described by hilbert spaces of larger dimensionality we pay particular attention to the limit case q our numerical results indicate that as the dimensionalities of both subsystems increase typical mixed states of composite quantum systems tend as far as their relative q entropies are concerned to behave in a classical way

generating random density matrices we study various methods to generate ensembles of random density matrices of a fixed size n obtained by partial trace of pure states on composite systems structured ensembles of random pure states invariant with respect to local unitary transformations are introduced to analyze statistical properties of quantum entanglement in bi partite systems we analyze the distribution of schmidt coefficients of random pure states such a distribution is derived in the case of a superposition of k random maximally entangled states for another ensemble obtained by performing selective measurements in a maximally entangled basis on a multi partite system we show that this distribution is given by the fuss catalan law and find the average entanglement entropy a more general class of structured ensembles proposed containing also the case of bures forms an extension of the standard ensemble of structureless random pure states described asymptotically as n by the marchenko pastur distribution

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

random unitary matrices methods of constructing random matrices typical of circular unitary and circular orthogonal ensembles are presented we generate numerically random unitary matrices and show that the statistical properties of their spectra level spacing distribution number variance and eigenvectors entropy participation ratio eigenvector statistics confer to the predictions of the random matrix theory for both cue and coe

the space of stability conditions on abelian threefolds and on some calabi yau threefolds we describe a connected component of the space of stability conditions on abelian threefolds and on calabi yau threefolds obtained as the crepant resolution of a finite quotient of an abelian threefold our proof includes the following essential steps 1 we simultaneously strengthen a conjecture by the first two authors and toda and prove that it follows from a more natural and seemingly weaker statement this conjecture is a bogomolov gieseker type inequality involving the third chern character of tilt stable two term complexes on smooth projective threefolds we extend it from complexes of tilt slope zero to arbitrary tilt slope 2 we show that this stronger conjecture implies the so called support property of bridgeland stability conditions and the existence of an explicit open subset of the space of stability conditions 3 we prove our conjecture for abelian threefolds thereby reproving and generalizing a result by maciocia and piyaratne important in our approach is a more systematic understanding on the behaviour of quadratic inequalities for semistable objects under wall crossing closely related to the support property

bridgeland stability conditions on the acyclic triangular quiver abstract using results in our previous paper non semistable exceptional objects in hereditary categories we focus here on studying the topology of the space of bridgeland stability conditions on d b rep k q where q and k is an algebraically closed field in particular we prove that this space is contractible in the previous paper it was shown that it is connected

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

birkhoff s polytope and unistochastic matrices n 3 and n 4 the set of bistochastic or doubly stochastic n n matrices is a convex set called birkhoff s polytope which we describe in some detail our problem is to characterize the set of unistochastic matrices as a subset of birkhoff s polytope for n 3 we present fairly complete results for n 4 partial results are obtained an interesting difference between the two cases is that there is a ball of unistochastic matrices around the van der waerden matrix for n 3 while this is not the case for n 4

moduli spaces of stable sheaves on abelian surfaces in this paper we consider basic problems on moduli spaces of stable sheaves on abelian surfaces our main assumption is the primitivity of the associated mukai vector we determine the deformation types albanese maps bogomolov factors and their weight 2 hodge structures we also discuss the deformation types of moduli spaces of stable sheaves on k3 surfaces

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

the minimal model program for the hilbert scheme of points on p2 and bridgeland stability abstract in this paper we study the birational geometry of the hilbert scheme p 2 n of n points on p 2 we discuss the stable base locus decomposition of the effective cone and the corresponding birational models we give modular interpretations to the models in terms of moduli spaces of bridgeland semi stable objects we construct these moduli spaces as moduli spaces of quiver representations using g i t and thus show that they are projective there is a precise correspondence between wall crossings in the bridgeland stability manifold and wall crossings between mori cones for n 9 we explicitly determine the walls in both interpretations and describe the corresponding flips and divisorial contractions

the invariant measure for scattering matrices with block symmetries we find the invariant measure for two new types of s matrices relevant for chaotic scattering from a cavity in a waveguide the s matrices considered can be written as a matrix of blocks each of rank n in which the two diagonal blocks are identical and the two off diagonal blocks are identical the s matrices are unitary in addition they may be symmetric because of time reversal symmetry the invariant measure with and without the condition of symmetry is given explicitly in terms of the invariant measures for the well known circular unitary and orthogonal ensembles some implications are drawn for the resulting statistical distribution of the transmission coefficient through a chaotic cavity

bridgeland s stabilities on abelian surfaces in this paper we shall study the structure of walls for bridgeland s stability conditions on abelian surfaces in particular we shall study the structure of walls for the moduli spaces of rank 1 complexes on an abelian surface with the picard number 1

quantization of a class of piecewise affine transformations on the torus we present a unified framework for the quantization of a family of discrete dynamical systems of varying degrees of chaoticity the systems to be quantized are piecewise affine maps on the two torus viewed as phase space and include the automorphisms translations and skew translations we then treat some discontinuous transformations such as the baker map and the sawtooth like maps our approach extends some ideas from geometric quantization and it is both conceptually and calculationally simple

stability conditions via spherical objects an object in the bounded derived category db x of coherent sheaves on a complex projective k3 surface x is spherical if it is rigid and simple although spherical objects form only a discrete set in the moduli stack of complexes they determine much of the structure of x and db x here we show that a stability condition on db x is determined by the stability of spherical objects

the monge metric on the sphere and geometry of quantum states topological and geometrical properties of the set of mixed quantum states in the n dimensional hilbert space are analysed assuming that the corresponding classical dynamics takes place on the sphere we use the vector su 2 coherent states and the generalized husimi distributions to define the monge distance between two arbitrary density matrices the monge metric has a simple semiclassical interpretation and induces a non trivial geometry among all pure states the distance from the maximally mixed state proportional to the identity matrix admits the largest value for the coherent states while the delocalized chaotic states are close to this contrasts the geometry induced by the standard trace hilbert schmidt or bures metrics for which the distance from is the same for all pure states we discuss possible physical consequences including unitary time evolution and the process of decoherence we introduce also a simplified monge metric defined in the space of pure quantum states and more suitable for numerical computation

a categorical invariant for cubic threefolds abstract we prove a categorical version of the torelli theorem for cubic threefolds more precisely we show that the non trivial part of a semi orthogonal decomposition of the derived category of a cubic threefold characterizes its isomorphism class

classical 1d maps quantum graphs and ensembles of unitary matrices we study a certain class of classical one dimensional piecewise linear maps for these systems we introduce an infinite family of markov partitions in equal cells the symbolic dynamics generated by these systems is described by bi stochastic doubly stochastic matrices we analyse the structure of graphs generated from the corresponding symbolic dynamics we demonstrate that the spectra of quantized graphs corresponding to the regular classical systems have locally poissonian statistics while quantized graphs derived from classically chaotic systems display statistical properties characteristic of the circular unitary ensemble even though the corresponding unitary matrices are sparse