derived category automorphisms from mirror symmetry inspired by the homological mirror symmetry conjecture of kontsevich 30 we construct new classes of automorphisms of the bounded derived category of coherent sheaves on a smooth quasi projective variety msc 2000 18e30 14j32

kernel pca for novelty detection kernel principal component analysis kernel pca is a non linear extension of pca this study introduces and investigates the use of kernel pca for novelty detection training data are mapped into an infinite dimensional feature space in this space kernel pca extracts the principal components of the data distribution the squared distance to the corresponding principal subspace is the measure for novelty this new method demonstrated a competitive performance on two dimensional synthetic distributions and on two real world data sets handwritten digits and breast cancer cytology

multiple kernel learning for dimensionality reduction in solving complex visual learning tasks adopting multiple descriptors to more precisely characterize the data has been a feasible way for improving performance the resulting data representations are typically high dimensional and assume diverse forms hence finding a way of transforming them into a unified space of lower dimension generally facilitates the underlying tasks such as object recognition or clustering to this end the proposed approach termed mkl dr generalizes the framework of multiple kernel learning for dimensionality reduction and distinguishes itself with the following three main contributions first our method provides the convenience of using diverse image descriptors to describe useful characteristics of various aspects about the underlying data second it extends a broad set of existing dimensionality reduction techniques to consider multiple kernel learning and consequently improves their effectiveness third by focusing on the techniques pertaining to dimensionality reduction the formulation introduces a new class of applications with the multiple kernel learning framework to address not only the supervised learning problems but also the unsupervised and semi supervised ones

on classification of n 2 supersymmetric theories we find a relation between the spectrum of solitons of massiven 2 quantum field theories ind 2 and the scaling dimensions of chiral fields at the conformal point the condition that the scaling dimensions be real imposes restrictions on the soliton numbers and leads to a classification program for symmetricn 2 conformal theories and their massive deformations in terms of a suitable generalization of dynkin diagrams which coincides with the a d e dynkin diagrams for minimal models the landau ginzburg theories are a proper subset of this classification in the particular case of lg theories we relate the soliton numbers with intersection of vanishing cycles of the corresponding singularity the relation between soliton numbers and the scaling dimensions in this particular case is a well known application of picard lefschetz theory

gtm the generative topographic mapping latent variable models represent the probability density of data in a space of several dimensions in terms of a smaller number of latent or hidden variables a familiar example is factor analysis which is based on a linear transformations between the latent space and the data space in this paper we introduce a form of non linear latent variable model called the generative topographic mapping for which the parameters of the model can be determined using the em algorithm gtm provides a principled alternative to the widely used self organizing map som of kohonen 1982 and overcomes most of the significant limitations of the som we demonstrate the performance of the gtm algorithm on a toy problem and on simulated data from flow diagnostics for a multi phase oil pipeline

the physical mirror equivalence for the local mathbb p 2 in this paper we consider the total space of the canonical bundle of mathbb p 2 and we use a proposal by hosono together with results of seidel and auroux katzarkov orlov to deduce the physical mirror equivalence between d b mathbb p 2 k mathbb p 2 and the derived fukaya category of its mirror which assigns the expected central charge to bps states by construction our equivalence is compatible with the mirror map relating the complex and the kahler moduli spaces and with the computation of gromov witten invariants

nonlinear topological component analysis application to age invariant face recognition we introduce a novel formalism that performs dimensionality reduction and captures topological features such as the shape of the observed data to conduct pattern classification this mission is achieved by 1 reducing the dimension of the observed variables through a kernelized radial basis function technique and expressing the latent variables probability distribution in terms of the observed variables 2 disclosing the data manifold as a 3 d polyhedron via the alpha shape constructor and extracting topological features and 3 classifying a data set using a mixture of multinomial distributions we have applied our methodology to the problem of age invariant face recognition experimental results obtained demonstrate the efficiency of the proposed methodology named nonlinear topological component analysis when compared with some state of the art approaches

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

quantum symmetries and exceptional collections we study the interplay between discrete quantum symmetries at certain points in the moduli space of calabi yau compactifications and the associated identities that the geometric realization of d brane monodromies must satisfy we show that in a wide class of examples both local and compact the monodromy identities in question always follow from a single mathematical statement one of the simplest examples is the mathbb z 5 symmetry at the gepner point of the quintic and the associated d brane monodromy identity

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

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

gross fibrations syz mirror symmetry and open gromov witten invariants for toric calabi yau orbifolds given a toric calabi yau orbifold x whose underlying toric variety is semi projective we construct and study a non toric lagrangian torus bration on x which we call the gross bration we apply the strominger yau zaslow recipe to the gross bration of a toric modication of x to construct its instanton corrected mirror where the instanton corrections come from genus 0 open orbifold gromov witten invariants which are virtual counts of holomorphic orbi disks inx bounded by bers of the gross bration we explicitly evaluate all these invariants by rst proving an open closed equality and then employing the toric mirror theorem for suitable toric parital compactications of x our calculations are then applied to 1 prove a conjecture of gross siebert on a relation between genus 0 open orbifold gromov witten invariants and mirror maps ofx this is called the open mirror theorem which leads to an enumerative meaning of mirror maps and 2 demonstrate how open orbifold gromov witten invariants for toric calabi yau orb ifolds change under toric crepant resolutions this is an open analogue of ruan s crepant resolution conjecture

conformation based hidden markov models application to human face identification hidden markov models hmms and their variants are capable to classify complex and structured objects however one of their major restrictions is their inability to cope with shape or conformation intrinsically hmm based techniques have difficulty predicting the n dimensional shape formed by the symbols of the visible observation vo sequence in order to fulfill this crucial need we propose a novel paradigm that we named conformation based hidden markov models cohmms this new formalism classifies vo sequences by embedding the nodes of an hmm state transition graph in a euclidean vector space this is accomplished by modeling the noise contained in the shape composed by the vo sequence we cover the one level as well as the multilevel cohmms five problems are assigned to a multilevel cohmm 1 sequence probability evaluation 2 statistical decoding 3 structural decoding 4 shape decoding and 5 learning we have applied the cohmms formalism to human face identification tested on different benchmarked face databases the results show that the multilevel cohmms outperform the embedded hmms as well as some standard hmm based models

lagrangian floer theory on compact toric manifolds i we introduced the notion of weakly unobstructed lagrangian submanifolds and constructed their potential function po purely in terms of a model data in fooo3 in this article we carry out explicit calculations involving po on toric manifolds and study the relationship between this class of lagrangian submanifolds with the earlier work of givental g1 which advocates that the quantum cohomology ring is isomorphic to the jacobian ring of a certain function called the landau ginzburg superpotential combining this study with the results from fooo3 we also apply the study to various examples to illustrate its implications to symplectic topology of lagrangian fibers of toric manifolds in particular we relate it to the hamiltonian displacement property of lagrangian fibers and to entov polterovich s symplectic quasi states

a construction of frobenius manifolds from stability conditions 2018 london mathematical society a finite quiver q without loops or 2 cycles defines a cy3 triangulated category d q and a finite heart a q d q we show that if q satisfies some strong conditions then the space of stability conditions a q supported on this heart admits a natural family of semisimple frobenius manifold structures constructed using the invariants counting semistable objects in d q in the case of an evaluating the family at a special point we recover a branch of the saito frobenius structure of the an singularity y2 xn 1 we give examples where applying the construction to each mutation of q and evaluating the families at a special point yields a different branch of the maximal analytic continuation of the same semisimple frobenius manifold in particular we check that this holds in the case of an n5

equivalences of derived catgories of sheaves on smooth stacks we extend orlov s representability theorem on the equivalence of derived categories of sheaves to the case of smooth stacks associated to normal projective varieties with only quotient singularities

d branes categories and n 1 supersymmetry we show that boundary conditions in topological open string theory on calabi yau cy manifolds are objects in the derived category of coherent sheaves as foreseen in the homological mirror symmetry proposal of kontsevich together with conformal field theory considerations this leads to a precise criterion determining the supersymmetry preserving branes at any point in cy moduli space completing the proposal of ii stability

algebraic deformations of toric varieties i general constructions abstract we construct and study noncommutative deformations of toric varieties by combining techniques from toric geometry isospectral deformations and noncommutative geometry in braided monoidal categories our approach utilizes the same fan structure of the variety but deforms the underlying embedded algebraic torus we develop a sheaf theory using techniques from noncommutative algebraic geometry the cases of projective varieties are studied in detail and several explicit examples are worked out including new noncommutative deformations of grassmann and flag varieties our constructions set up the basic ingredients for a thorough study of instantons on noncommutative toric varieties which will be the subject of the sequel to this paper

limit stable objects on calabi yau 3 folds the notion of limit stability is introduced in 31 to give a limiting degeneration of bridgeland stability in this article we give the survey of the results in 31 30 we introduce the limit stability counting invariants of limit stable objects and see that the wall crossing phenomena yields the rationality conjecture of pandharipandethomas generating functions

homological mirror symmetry for singularities of type d we prove homological mirror symmetry for lefschetz fibrations obtained as sebastiani thom sums of polynomials of types a or d the proof is based on the behavior of the fukaya category under sebastiani thom summation of a polynomial of type d

distance metric optimization driven convolutional neural network for age invariant face recognition propose a new distance metric optimization driven deep learning framework for age invariant face recognition learn feature representations and the similarity measure simultaneously in an end to end way train the joint network using a novel optimization method and carefully designed training strategies the experimental results demonstrate the effectiveness of our approach despite the great advances in face related works in recent years face recognition across age remains a challenging problem the traditional approaches to this problem usually include two basic steps feature extraction and the application of a distance metric sometimes common space projection is also involved on the one hand handling these steps separately ignores the interactions of these components and on the other hand the fixed distance threshold of measurement affects the models robustness in this paper we present a novel distance metric optimization driven learning approach that integrates these traditional steps via a deep convolutional neural network which learns feature representations and the decision function in an end to end way given the labelled training images we first generate a large number of pairs with a certain proportion of matched and unmatched pairs for matched pairs we try to select as many different age instances as possible for each person to learn the identification information that is not affected by age then taking these pairs as input we aim to enlarge the differences between the unmatched pairs while reducing the variations between the matched pairs and we update the model parameters by using the mini batch stochastic gradient descent sgd algorithm specifically the distance matrix is used as the top fully connected layer and the bottom layers representing the image features are integrated with it seamlessly thus the image features and the distance metric can be optimized simultaneously by backward propagation in particular we introduce several training strategies to reduce the computational cost and overcome insufficient memory capacity we evaluate our method on three tasks age invariant face identification on the morph database age invariant face retrieval on the cacd database and age invariant face verification on cacd vs database the experimental results demonstrate the effectiveness of our approach

graph embedding and extensions a general framework for dimensionality reduction a large family of algorithms supervised or unsupervised stemming from statistics or geometry theory has been designed to provide different solutions to the problem of dimensionality reduction despite the different motivations of these algorithms we present in this paper a general formulation known as graph embedding to unify them within a common framework in graph embedding each algorithm can be considered as the direct graph embedding or its linear kernel tensor extension of a specific intrinsic graph that describes certain desired statistical or geometric properties of a data set with constraints from scale normalization or a penalty graph that characterizes a statistical or geometric property that should be avoided furthermore the graph embedding framework can be used as a general platform for developing new dimensionality reduction algorithms by utilizing this framework as a tool we propose a new supervised dimensionality reduction algorithm called marginal fisher analysis in which the intrinsic graph characterizes the intraclass compactness and connects each data point with its neighboring points of the same class while the penalty graph connects the marginal points and characterizes the interclass separability we show that mfa effectively overcomes the limitations of the traditional linear discriminant analysis algorithm due to data distribution assumptions and available projection directions real face recognition experiments show the superiority of our proposed mfa in comparison to lda also for corresponding kernel and tensor extensions

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

multiple kernel learning algorithms in recent years several methods have been proposed to combine multiple kernels instead of using a single one these different kernels may correspond to using different notions of similarity or may be using information coming from multiple sources different representations or different feature subsets in trying to organize and highlight the similarities and differences between them we give a taxonomy of and review several multiple kernel learning algorithms we perform experiments on real data sets for better illustration and comparison of existing algorithms we see that though there may not be large differences in terms of accuracy there is difference between them in complexity as given by the number of stored support vectors the sparsity of the solution as given by the number of used kernels and training time complexity we see that overall using multiple kernels instead of a single one is useful and believe that combining kernels in a nonlinear or data dependent way seems more promising than linear combination in fusing information provided by simple linear kernels whereas linear methods are more reasonable when combining complex gaussian kernels

quantum conditional mutual information and approximate markov chains a state on a tripartite quantum system a otimes b otimes c forms a markov chain if it can be reconstructed from its marginal on a otimes b by a quantum operation from b to b otimes c we show that the quantum conditional mutual information i a c b of an arbitrary state is an upper bound on its distance to the closest reconstructed state it thus quantifies how well the markov chain property is approximated

noncommutative projective schemes an analogue of the concept of projective scheme is defined for noncommutative n graded algebras using the quotient category c of graded right a modules modulo its full subcategory of torsion modules we define proj a c a s where a is the object corresponding to the module aa and s is the autoequivalence defined by the shift of degrees the triples equivalent to proj a for a right noetherian graded algebra a are characterized in terms of a condition on extensions

legendrian fronts for affine varieties in this article we study weinstein structures endowed with a lefschetz fibration in terms of the legendrian front projection first we provide a systematic recipe for translating from a weinstein lefschetz bifibration to a legendrian handlebody then we present several new applications of this technique to symplectic topology this includes the detection of flexibility and rigidity for several families of weinstein manifolds and the existence of closed exact lagrangian submanifolds in particular we prove that the koras russell cubic is stein deformation equivalent to c3 and we verify the affine parts of the algebraic mirrors of two weinstein 4 folds

the mckay correspondence as an equivalence of derived categories the classical mckay correspondence relates representations of a finite subgroup g sl 2 c to the cohomology of the well known minimal resolution of the kleinian singularity c2 g gonzalez sprinberg and verdier 10 interpreted the mckay correspondence as an isomorphism on k theory observing that the representation ring of g is equal to the g equivariant k theory of c2 more precisely they identify a basis of the k theory of the resolution consisting of the classes of certain tautological sheaves associated to the irreducible representations of g

stability conditions on text cy _n categories associated to a_n quivers and period maps in this paper we study the space of stability conditions on a certain n calabi yau mathrm cy n category associated to an a n quiver recently bridgeland and smith constructed stability conditions on some mathrm cy 3 categories from meromorphic quadratic differentials with simple zeros generalizing their results to higher dimensional calabi yau categories we describe the space of stability conditions as the universal cover of the space of polynomials of degree n 1 with simple zeros in particular central charges of stability conditions on mathrm cy n categories are constructed as the periods of quadratic differentials with zeros of order n 2 which are associated to polynomials

stokes matrices for the quantum cohomologies of a class of orbifold projective lines we prove the dubrovin s conjecture for the stokes matrices for the quantum cohomology of orbifold projective lines the conjecture states that the stokes matrix of the first structure connection of the frobenius manifold constructed from the gromov witten theory coincides with the euler matrix of a full exceptional collection of the bounded derived category of the coherent sheaves our proof is based on the homological mirror symmetry primitive forms of affine cusp polynomials and the picard lefschetz theory

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

hessian eigenmaps locally linear embedding techniques for high dimensional data we describe a method for recovering the underlying parametrization of scattered data m i lying on a manifold m embedded in high dimensional euclidean space the method hessian based locally linear embedding derives from a conceptual framework of local isometry in which the manifold m viewed as a riemannian submanifold of the ambient euclidean space r n is locally isometric to an open connected subset of euclidean space r d because does not have to be convex this framework is able to handle a significantly wider class of situations than the original isomap algorithm the theoretical framework revolves around a quadratic form h f m h f m f dm defined on functions f m r here hf denotes the hessian of f and h f averages the frobenius norm of the hessian over m to define the hessian we use orthogonal coordinates on the tangent planes of m the key observation is that if m truly is locally isometric to an open connected subset of r d then h f has a d 1 dimensional null space consisting of the constant functions and a d dimensional space of functions spanned by the original isometric coordinates hence the isometric coordinates can be recovered up to a linear isometry our method may be viewed as a modification of locally linear embedding and our theoretical framework as a modification of the laplacian eigenmaps framework where we substitute a quadratic form based on the hessian in place of one based on the laplacian

mapping dynamic bayesian networks to alpha shapes application to human faces identification across ages we propose to map a dynamic bayesian network dbn to an ordered family of shapes to improve dbns classification power this mission is achieved by 1 embedding a dbn into a topological manifold and 2 applying the shape geometric constructor to build hierarchical structures assigned to the dbn this continuous representation of traditional dbns as shapes allows more information to be obtained about the objects to be classified these latter are viewed as hierarchies of geometrical objects with different levels of detail topological signatures are therefore unraveled and classification accuracy is enhanced we have applied the proposed formalism to the task of facial identification across ages preliminary results demonstrate that the proposed formalism is a powerful tool since it has outperformed some dbn models the k nn classifier and some recent approaches

some navigation rules for d brane monodromy we explore some aspects of monodromies of d branes in the kahler moduli space of calabi yau compactifications here a d brane is viewed as an object of the derived category of coherent sheaves we compute all the interesting monodromies in some nontrivial examples and link our work to recent results and conjectures concerning helices and mutations we note some particular properties of the 0 brane

reconstruction algebras of type a we introduce a new class of algebras called reconstruction algebras and present some of their basic properties these non commutative rings dictate in every way the process of resolving the cohen macaulay singularities c 2 g where g 1 r 1 a gl 2 c

on the algebras of bps states we define an algebra on the space of bps states in theories with extended supersymmetry we show that the algebra of perturbative bps states in toroidal compactification of the heterotic string is closely related to a generalized kac moody algebra we use d brane theory to compare the formulation of rr charged bps algebras in type ii compactification with the requirements of string string duality and find that the rr charged bps states should be regarded as cohomology classes on moduli spaces of coherent sheaves the equivalence of the algebra of bps states in heterotic iia dual pairs elucidates certain results and conjectures of nakajima and gritsenko nikulin on geometrically defined algebras and furthermore suggests nontrivial generalizations of these algebras in particular to any calabi yau 3 fold there are two canonically associated algebras exchanged by mirror symmetry

on the cn zm fractional branes we construct several geometric representatives for the cn zm fractional branes on either a partially or the completely resolved orbifold in the process we use large radius and conifold type monodromies and provide a strong consistency check in particular for c3 z5 we give three different sets of geometric representatives we also find the explicit seiberg duality which connects our fractional branes to the ones given by the mckay correspondence

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

strengthened monotonicity of relative entropy via pinched petz recovery map the quantum relative entropy between two states satisfies a monotonicity property meaning that applying the same quantum channel to both states can never increase their relative entropy it is known that this inequality is only tight when there is a recovery map that exactly reverses the effects of the quantum channel on both states in this paper we strengthen this inequality by showing that the difference of relative entropies is bounded below by the measured relative entropy between the first state and a recovered state from its processed version the recovery map is a convex combination of rotated petz recovery maps and perfectly reverses the quantum channel on the second state as a special case we reproduce recent lower bounds on the conditional mutual information such as the one proved by fawzi and renner our proof only relies on the elementary properties of pinching maps and the operator logarithm

computation of superpotentials for d branes we present a general method for the computation of tree level superpotentials for the world volume theory of b type d branes this includes quiver gauge theories in the case that the d brane is marginally stable the technique involves analyzing the a infinity structure inherent in the derived category of coherent sheaves this effectively gives a practical method of computing correlation functions in holomorphic chern simons theory as an example we give a more rigorous proof of previous results concerning 3 branes on certain singularities including conifolds we also provide a new example

proof of the bmv conjecture we prove the bmv bessis moussa villani 1 conjecture which states that the function t mapsto mathop rm tr exp a tb t geqslant 0 is the laplace transform of a positive measure on 0 if a and b are n times n hermitian matrices and b is positive semidefinite a semi explicit representation for this measure is given

monodromy of certain painleve vi transcendents and reflection groups we study the global analytic properties of the solutions of a particular family of painleve vi equations with the parameters 0 1 2 and 2 2 1 2 with arbitrary 2 we introduce a class of solutions having critical behaviour of algebraic type and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space the analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group we show that the finite orbits of this action correspond to the algebraic solutions of our painleve vi equation and use this result to classify all of them we prove that the algebraic solutions of our painleve vi equation are in one to one correspondence with the regular polyhedra or star polyhedra in the three dimensional space

multivariate trace inequalities we prove several trace inequalities that extend the golden thompson and the araki lieb thirring inequality to arbitrarily many matrices in particular we strengthen lieb s triple matrix inequality as an example application of our four matrix extension of the golden thompson inequality we prove remainder terms for the monotonicity of the quantum relative entropy and strong sub additivity of the von neumann entropy in terms of recoverability we find the first explicit remainder terms that are tight in the commutative case our proofs rely on complex interpolation theory as well as asymptotic spectral pinching providing a transparent approach to treat generic multivariate trace inequalities

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

mirror symmetry for del pezzo surfaces vanishing cycles and coherent sheaves we study homological mirror symmetry for del pezzo surfaces and their mirror landau ginzburg models in particular we show that the derived category of coherent sheaves on a del pezzo surface x k obtained by blowing up 2 at k points is equivalent to the derived category of vanishing cycles of a certain elliptic fibration w k m k with k 3 singular fibers equipped with a suitable symplectic form moreover we also show that this mirror correspondence between derived categories can be extended to noncommutative deformations of x k and give an explicit correspondence between the deformation parameters for x k and the cohomology class b i h 2 m k

monodromy of monomially admissible fukaya seidel categories mirror to toric varieties abstract mirror symmetry for a toric variety involves laurent polynomials whose symplectic topology is related to the algebraic geometry of the toric variety we show that there is a monodromy action on the monomially admissible fukaya seidel categories of these laurent polynomials as the arguments of their coefficients vary that corresponds under homological mirror symmetry to tensoring by a line bundle naturally associated to the monomials whose coefficients are rotated in the process we introduce the monomially admissible fukaya seidel category as a new interpretation of the fukaya seidel category of a laurent polynomial on c n which has other potential applications and give evidence of homological mirror symmetry for non compact toric varieties

t structures on some local calabi yau varieties abstract let z be a smooth fano variety satisfying the condition that the rank of the grothendieck group of z is one more than the dimension of z let z denote the total space of the canonical line bundle of z considered as a non compact calabi yau variety we use the theory of exceptional collections to describe t structures on the derived category of coherent sheaves on z the combinatorics of these t structures is determined by a natural action of an affine braid group closely related to the well known action of the artin braid group on the set of exceptional collections on z

mathbb c 2 mathbb z _ n fractional branes and monodromy we construct geometric representatives for the mathbb c 2 mathbb z n fractional branes in terms of branes wrapping certain exceptional cycles of the resolution in the process we use large radius and conifold type monodromies and also check some of the orbifold quantum symmetries we find the explicit seiberg duality which connects our fractional branes to the ones given by the mckay correspondence we also comment on the harvey moore bps algebras

on multivariate trace inequalities of sutter berta and tomamichel we consider a family of multivariate trace inequalities recently derived by sutter berta and tomamichel these inequalities generalize the golden thompson inequality and lieb s triple matrix inequality to an arbitrary number of matrices in a way that features complex matrix powers i e certain unitaries we show that their inequalities can be rewritten as an n matrix generalization of lieb s original triple matrix inequality the complex matrix powers are replaced by resolvents and appropriate maximally entangled states we expect that the technically advantageous properties of resolvents in particular for perturbation theory can be of use in applications of the n matrix inequalities e g for analyzing the performance of the rotated petz recovery map in quantum information theory and for removing the unitaries altogether

reconstruction algebras of type d i this is the second in a series of papers which give an explicit description of the reconstruction algebra as a quiver with relations these algebras arise naturally as geometric generalizations of preprojective algebras of extended dynkin diagrams this paper deals with dihedral groups g dn q for which all special cm modules have rank one and we show that all but four of the relations on such a reconstruction algebra are given simply as the relations arising from a reconstruction algebra of type a as a corollary the reconstruction algebra reduces the problem of explicitly understanding the minimal resolution g hilb to the same level of difficulty as the toric case

t duality and homological mirror symmetry for toric varieties abstract let x be a complete toric variety the coherent constructible correspondence of fang et al 2011 14 equates p erf t x with a subcategory sh c c m r of constructible sheaves on a vector space m r the microlocalization equivalence of nadler and zaslow 2009 27 and nadler 2009 25 relates these sheaves to a subcategory fuk t m r of the fukaya category of the cotangent t m r when x is nonsingular taking the derived category yields an equivariant version of homological mirror symmetry dcoh t x dfuk t m r which is an equivalence of triangulated tensor categories the nonequivariant coherent constructible correspondence of treumann preprint 33 embeds p erf x into a subcategory s h c t r of constructible sheaves on a compact torus t r when x is nonsingular the composition of and microlocalization yields a version of homological mirror symmetry dcoh x dfuk t t r which is a full embedding of triangulated tensor categories when x is nonsingular and projective the composition is compatible with t duality in the following sense an equivariant ample line bundle l has a hermitian metric invariant under the real torus whose connection defines a family of flat line bundles over the real torus orbits this data produces a t dual lagrangian brane l on the universal cover t m r of the dual real torus fibration we prove l l in fuk t m r thus equivariant homological mirror symmetry is determined by t duality

stability conditions on a non compact calabi yau threefold we study the space of stability conditions stab x on the non compact calabi yau threefold x which is the total space of the canonical bundle of mathbb p 2 we give a combinatorial description of an open subset of stab x and state a conjecture relating stab x to the frobenius manifold obtained from the quantum cohomology of mathbb p 2 we give some evidence from mirror symmetry for this conjecture

stokes matrices and monodromy of the quantum cohomology of projective spaces in this paper we compute stokes matrices and monodromy of the quantum cohomology of projective spaces this problem can be formulated in a classical framework as the problem of computation of stokes matrices and monodromy of differential equations with regular and irregular singularities we prove that the stokes matrix of the quantum cohomology coincides with the gram matrix in the theory of derived categories of coherent sheaves we also study the monodromy group of the quantum cohomology and we show that it is related to hyperbolic triangular groups

massless d branes on calabi yau threefolds and monodromy we analyze the link between the occurrence of massless b type d branes for specific values of moduli and monodromy around such points in the moduli space this allows us to propose a classification of all massless b type d branes at any point in the moduli space of calabi yau s this classification then justifies a previous conjecture due to horja for the general form of monodromy our analysis is based on using monodromies around points in moduli space where a single d brane becomes massless to generate monodromies around points where an infinite number become massless we discuss the various possibilities within the classification

weighted blowups and mirror symmetry for toric surfaces abstract this paper explores homological mirror symmetry for weighted blowups of toric varietes it will be shown that both the a model and b model categories have natural semi orthogonal decompositions an explicit equivalence of the right orthogonal categories will be shown for the case of toric surfaces

principal manifolds and nonlinear dimensionality reduction via tangent space alignment we present a new algorithm for manifold learning and nonlinear dimensionality reduction based on a set of unorganized data points sampled with noise from a parameterized manifold the local geometry of the manifold is learned by constructing an approximation for the tangent space at each data point and those tangent spaces are then aligned to give the global coordinates of the data points with respect to the underlying manifold we also present an error analysis of our algorithm showing that reconstruction errors can be quite small in some cases we illustrate our algorithm using curves and surfaces both in two dimensional three dimensional 2d 3d euclidean spaces and in higher dimensional euclidean spaces we also address several theoretical and algorithmic issues for further research and improvements

derived hall algebras the purpose of this work is to define a derived hall algebra mathcal dh t associated to any dg category t under some finiteness conditions our main theorem states that mathcal dh t is associative and unital it is shown that mathcal dh t contains the usual hall algebra mathcal h t when t is an abelian category we will also prove an explicit formula for the derived hall numbers purely in terms of invariants of the triangulated category associated to t as an example we describe the derived hall algebra of an hereditary abelian category

on deformations of triangulated models abstract this paper is the first part of a project aimed at understanding deformations of triangulated categories and more precisely their dg and a models and applying the resulting theory to the models occurring in the homological mirror symmetry setup in this first paper we focus on models of derived and related categories based upon the classical construction of twisted objects over a dg or a algebra for a hochschild 2 cocycle on such a model we describe a corresponding curvature compensating deformation which can be entirely understood within the framework of twisted objects we unravel the construction in the specific cases of derived a and abelian categories homotopy categories and categories of graded free qdg modules we identify a purity condition on our models which ensures that the structure of the model is preserved under deformation this condition is typically fulfilled for homotopy categories but not for unbounded derived categories