a spectral sequence for polyhedral products abstract the purpose of this paper is to exhibit fine structure for polyhedral products z k x a and polyhedral smash products z k x a where moment angle complexes are special cases in which x a d 2 s 1 there are three main parts to this paper 1 one part gives a natural filtration of the polyhedral product together with properties of the resulting spectral sequence in theorem 2 15 applications of this spectral sequence are given 2 the second part gives a homological decomposition of z k x a in case x a consists of cw pairs by using part 1 3 applications to the ring structure of z k x a are given for cw pairs x a satisfying suitable freeness conditions

local torus actions modeled on the standard representation abstract we introduce the notion of a local torus action modeled on the standard representation for simplicity we call it a local torus action it is a generalization of a locally standard torus action and also an underlying structure of a locally toric lagrangian fibration for a local torus action we define two invariants called a characteristic pair and an euler class of the orbit map and prove that local torus actions are classified topologically by them as a corollary we obtain a topological classification of locally standard torus actions which includes the topological classifications of quasi toric manifolds by davis and januszkiewicz and of effective t 2 actions on four dimensional manifolds without nontrivial finite stabilizers by orlik and raymond we discuss locally toric lagrangian fibrations from the viewpoint of local torus actions we also investigate the topology of a manifold equipped with a local torus action when the euler class of the orbit map vanishes

equivariant basic cohomology of riemannian foliations the basic cohomology of a riemannian foliation on a complete manifold with all leaves closed is the cohomology of the leaf space in this paper we introduce various methods to compute the basic cohomology in the presence of both closed and non closed leaves in the simply connected case or more generally for killing foliations we show that the total basic betti number of the union c of the closed leaves is smaller than or equal to the total basic betti number of the foliated manifold and we give sufficient conditions for equality if there is a basic morse bott function with critical set equal to c we can compute the basic cohomology explicitly another case in which the basic cohomology can be determined is if the space of leaf closures is a simple convex polytope our results are based on molino s observation that the existence of non closed leaves yields a distinguished transverse action on the foliated manifold with fixed point set c we introduce equivariant basic cohomology of transverse actions in analogy to equivariant cohomology of lie group actions enabling us to transfer many results from the theory of lie group actions to riemannian foliations the prominent role of the fixed point set in the theory of torus actions explains the relevance of the set c in the basic setting

rigidity of coxeter groups let w be a coxeter group acting properly discontinuously and cocompactly on manifolds n and m 9m 0 such that the fixed point sets of finite subgroups are contractible let f n n mx dk m xsk1 be a w homotopy equivalence which restricts to a w homeomorphism on the boundary under an assumption on the three dimensional fixed point sets we show that then f is w homotopic to a w homeomorphism

on the eigenspectrum of the gram matrix and the generalization error of kernel pca in this paper the relationships between the eigenvalues of the m spl times m gram matrix k for a kernel spl kappa spl middot spl middot corresponding to a sample x sub 1 x sub m drawn from a density p x and the eigenvalues of the corresponding continuous eigenproblem is analyzed the differences between the two spectra are bounded and a performance bound on kernel principal component analysis pca is provided showing that good performance can be expected even in very high dimensional feature spaces provided the sample eigenvalues fall sufficiently quickly

a non golod ring with a trivial product on its koszul homology abstract we present a monomial ideal a s such that s a is not golod even though the product in its koszul homology is trivial this constitutes a counterexample to a well known result by berglund and jollenbeck the error can be traced to a mistake in an earlier article by jollenbeck on the positive side we show that if r is a monomial ring such that the r ary massey product vanishes for all r max 2 reg r 2 then r is golod in particular if r is the stanley reisner ring of a simplicial complex of dimension at most 3 then r is golod if and only if the product in its koszul homology is trivial moreover we show that if is a triangulation of a k orientable manifold whose stanley reisner ring is golod then is 2 neighborly this extends a recent result of iriye and kishimoto

on simple polytopes letp be a simpled polytope ind dimensional euclidean space mathbb e d and let p be the subalgebra of the polytope algebra generated by the classes of summands ofp it is shown that the dimensions of the weight spaces r p of p are theh numbers ofp which describe the dehn sommerville equations between the numbers of faces ofp and reflect the duality between r p and d r p moreover p admits a lefschetz decomposition under multiplication by the element of 1 p corresponding top itself which yields a proof of the necessity of mcmullen s conditions in theg theorem on thef vectors of simple polytopes the lefschetz decomposition is closely connected with the new hodge riemann minkowski quadratic inequalities between mixed volumes which generalize minkowski s second inequality also proved are analogous generalizations of the aleksandrov fenchel inequalities a striking feature is that these are obtained without using brunn minkowski theory indeed the brunn minkowski theorem without characterization of the cases of equality can be deduced from them the connexion found between p and the face ring of the dual simplicial polytopep enables this ring to be looked at in two ways and a conjectured formulation of theg theorem in terms of a gale diagram ofp is also established

h shellings and h complexes we introduce and study a new class of shellings of simplicial complexes that we call h shellings these shellings allow the construction of a subcomplex of the original one with particularly nice enumerative and topological properties several infinite classes of such complexes are discussed included in these classes are coxeter complexes tits buildings and order complexes of many shellable posets

accurate error bounds for the eigenvalues of the kernel matrix the eigenvalues of the kernel matrix play an important role in a number of kernel methods in particular in kernel principal component analysis it is well known that the eigenvalues of the kernel matrix converge as the number of samples tends to infinity we derive probabilistic finite sample size bounds on the approximation error of individual eigenvalues which have the important property that the bounds scale with the eigenvalue under consideration reflecting the actual behavior of the approximation errors as predicted by asymptotic results and observed in numerical simulations such scaling bounds have so far only been known for tail sums of eigenvalues asymptotically the bounds presented here have a slower than stochastic rate but the number of sample points necessary to make this disadvantage noticeable is often unrealistically large therefore under practical conditions and for all but the largest few eigenvalues the bounds presented here form a significant improvement over existing non scaling bounds

comparison of symbolic and ordinary powers of ideals all given rings in this paper are commutative associative with identity and noetherian recently l ein r lazarsfeld and k smith els discovered a remarkable and surprising fact about the behavior of symbolic powers of ideals in affine regular rings of equal characteristic 0 if h is the largest height of an associated prime of i then i hn i n for all n 0 here if w is the complement of the union of the associated primes of i i t denotes the contraction of i t rw to r where rw is the localization of r at the multiplicative system w their proof depends on the theory of multiplier ideals including an asymptotic version and in particular requires resolution of singularities as well as vanishing theorems we want to acknowledge that without their generosity and quickness in sharing their research this manuscript would not exist our objective here is to give stronger results that can be proved by methods that are in some ways more elementary our results are valid in both equal characteristic 0 and in positive prime characteristic p but depend on reduction to characteristic p we use tight closure methods and in consequence we need neither resolution of singularities nor vanishing theorems that may fail in positive characteristic for the most basic form of the result all that we need from tight closure theory is the definition of tight closure and the fact that in a regular ring every ideal is tightly closed we note that the main argument here is closely related to a proof given in hu 5 14 16 p 45 that regular local rings in characteristic p are ufds

the homotopy types of moment angle complexes for flag complexes we study the homotopy types of moment angle complexes or equivalently of complements of coordinate subspace arrangements the overall aim is to identify the simplicial complexes k for which the corresponding moment angle complex z k has the homotopy type of a wedge of spheres or a connected sum of sphere products when k is flag we identify in algebraic and combinatorial terms those k for which z k is homotopy equivalent to a wedge of spheres and give a combinatorial formula for the number of spheres in the wedge this extends results of berglund and jollenbeck on golod rings and homotopy theoretical results of the first and third authors we also establish a connection between minimally non golod rings and moment angle complexes z k which are homotopy equivalent to a connected sum of sphere products we go on to show that for any flag complex k the loop space of z k is homotopy equivalent to a product of spheres and loops on spheres when localised rationally or at any prime not equal to 2

equivariant cohomology distinguishes toric manifolds the equivariant cohomology of a space with a group action is not only a ring but also an algebra over the cohomology ring of the classifying space of the acting group we prove that toric manifolds i e compact smooth toric varieties are isomorphic as varieties if and only if their equivariant cohomology algebras are weakly isomorphic we also prove that quasitoric manifolds which can be thought of as a topological counterpart to toric manifolds are equivariantly homeomorphic if and only if their equivariant cohomology algebras are isomorphic

homotopy groups of k contact toric manifolds contact toric manifolds of reeb type are a subclass of contact toric manifolds which have the property that they are classified by the images of the associated moment maps we compute their first and second homotopy group terms of the images of the moment map we also explain why they are k contact

a presentation for the unipotent group over rings with identity for a ring r with identity define unipn r to be the group of upper triangular matrices over r all of whose diagonal entries are 1 for i 1 2 n 1 let si denote the matrix whose only nonzero off diagonal entry is a 1 in the ith row and i 1 st column then for any integer m including m 0 it is easy to see that the si generate unipn z mz reiner gave relations among the si which he conjectured gave a presentation for unipn z 2z this conjecture was proven by biss comm algebra26 1998 2971 2975 and an analogous conjecture was made for unipn z mz in general we prove this conjecture as well as a generalization of the conjecture to unipotent groups over arbitrary rings

a geometric effective nullstellensatz the purpose of this paper is to present a geometric theorem which clarifies and extends in several directions work of brownawell kollar and others on the effective nullstellensatz specifically we work on an arbitrary smooth complex projective variety x with the previous classical results corresponding to the case when x is projective space in this setting we prove a local effective nullstellensatz for ideal sheaves and a corresponding global division theorem for adjoint type bundles we also make explicit the connection with the intersection theory of fulton and macpherson finally constructions involving products of prime ideals that appear in earlier work are replaced by geometrically more natural conditions involving orders of vanishing along subvarieties much of the previous activity in this area has been algebraic in nature and seems perhaps not well known in detail among geometers therefore we have felt it worthwhile to include here a rather extended introduction we start with an overview of the questions and earlier work on them then we present the set up and statement of our main theorem we conclude with a series of examples which can be read before the general result of what it yields in special cases

splitting multidimensional necklaces the well known splitting necklace theorem of alon n alon splitting necklaces adv math 63 1987 247 253 says that each necklace with kai beads of color i 1 n can be fairly divided between k thieves by at most n k 1 cuts alon deduced this result from the fact that such a division is possible also in the case of a continuous necklace 0 1 where beads of given color are interpreted as measurable sets ai 0 1 or more generally as continuous measures i we demonstrate that alon s result is a special case of a multidimensional consensus division theorem about n continuous probability measures 1 n on a d cube 0 1 d the dissection is performed by m1 md n k 1 hyperplanes parallel to the sides of 0 1 d dividing the cube into m1md elementary cuboids parallelepipeds where the integers mi are prescribed in advance

the signature of a toric variety we identify a combinatorial quantity the alternating sum of the h vector defined for any simple polytope as the signature of a toric variety this quantity was introduced by r charney and m davis in their work which in particular showed that its nonnegativity is closely related to a conjecture of h hopf on the euler characteristic of a nonpositively curved manifold we prove positive or nonnegative lower bounds for this quantity under geometric hypotheses on the polytope and in particular resolve a special case of their conjecture these hypotheses lead to ampleness or weaker conditions for certain line bundles on toric divisors and then the lower bounds follow from calculations using the hirzebruch signature formula moreover we show that under these hypotheses on the polytope the ith l class of the corresponding toric variety is 1 i times an effective class for any i

vector bundles over davis januszkiewicz spaces with prescribed characteristic classes for any n 1 dimensional simplicial complex we construct a particular n dimensional complex vector bundle over the associated davis januszkiewicz space whose chern classes are given by the elementary symmetric polynomials in the generators of the stanley reisner algebra we show that the isomorphism type of this complex vector bundle as well as of its realification are completely determined by its characteristic classes this allows us to show that coloring properties of the simplicial complex are reflected by splitting properties of this bundle and vice versa similar questions are also discussed for 2n dimensional real vector bundles with particular prescribed characteristic pontrjagin and euler classes we also analyze which of these bundles admit a complex structure it turns out that all these bundles are closely related to the tangent bundles of quasi toric manifolds and moment angle complexes 2012 american mathematical society

toda lattices with indefinite metric ii topology of the iso spectral manifolds abstract we consider the iso spectral real manifolds of tridiagonal hessenberg matrices with distinct real eigenvalues the manifolds are described by the iso spectral flows of indefinite toda lattice equations introduced by the authors physica d 91 1996 321 339 these toda lattices consist of 2n 1 different systems with hamiltonians h 1 2 k 1 n y k 2 k 1 n 1 s k s k 1 exp x k x k 1 where sk 1 which blow up in finite time except the case with all sksk 1 1 we compactify the manifolds by adding infinities according to the toda flows the resulting manifolds are shown to be nonorientable for n 2 and the symmetry group is the semi direct product of z 2 n 1 and the permutation group s n these properties identify themselves with small covers introduced by davis and januszkiewicz duke math j 62 1991 417 451 as a corollary of our construction we give a formula for the total number of zeros for a system of exponential polynomials generated as hankel determinants

the mod 2 cohomology of fixed point sets of anti symplectic involutions abstract let m be a compact connected symplectic manifold with a hamiltonian action of a compact n dimensional torus g tn suppose that is an anti symplectic involution compatible with the g action the real locus of m is x the fixed point set of duistermaat uses morse theory to give a description of the ordinary cohomology of x in terms of the cohomology of m there is a residual g r z 2 z n action on x and we can use duistermaat s result as well as some general facts about equivariant cohomology to prove an equivariant analogue to duistermaat s theorem in some cases we can also extend theorems of goresky kottwitz macpherson and goldin holm to the real locus

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

uniform bounds and symbolic powers on smooth varieties the purpose of this note is to show how one can use multiplier ideals toestablish effective uniform bounds on the multiplicative behavior of certainfamilies of ideal sheaves on a smooth algebraic variety in particular weprove a quick but rather surprising result concerning the symbolic powersof radical ideals on such a variety let

formulas for multiplier ideals on singular varieties we prove a generalization of demailly ein lazarsfeld s subadditivity formula and musta s summation formula for multiplier ideals to the case of singular varieties using characteristic p methods as an application of our formula we improve hochster huneke s result on the growth of symbolic powers of ideals in singular affine algebras

the theory of toric manifolds toric manifolds the topological analogue of toric varieties are determined by an n dimensional simple convex polytope and a function from the set of codimensionone faces into the primitive vectors of an integer lattice their cohomology was determined by davis and januszkiewicz in 1991 and corresponds with the theorem of danilovjurkiewicz in the toric variety case we use the adams spectral sequence to compute the ko theory of all toric manifolds and certain toric varieties

geometric nullstellensatz and symbolic powers on arbitrary varieties in recent years a multiplier ideal defined on arbitrary varieties so called mather jacobian multiplier ideal has been developed independently by ein ishii mustata and de fernex docampo with this new tool we have a chance of extending some classical results proved in nonsingular case to arbitrary varieties to establish their general forms in this paper we first extend a result of geometric nullstellensatz due to ein lazarsfeld in nonsingular case to any projective varieties then we prove a result on comparison of symbolic powers with ordinary powers on any varieties which extends results of ein lazarsfeld smith and hochster huneke

groups generated by reflections and aspherical manifolds not covered by euclidean space a coxeter system r v is a group r a coxeter group together with a set of generators v such that each element of v has order two and such that all relations in r are consequences of relations of the form vw m v w 1 where v w e v and m v w denotes the order of vw the m v w s which are positive integers or oo obviously determine the coxeter system up to isomorphism the list of finite coxeter groups is short and well known however general coxeter groups are much more flexible in fact the set v and the m v w s can be specified arbitrarily at least if v is finite subject only to the conditions m v v 1 and m v w m w v 2 if v w suppose that r v is a coxeter system that x is a hausdorff space and that xv v v is a locally finite family of closed subspaces indexed by v the xv are called the panels of x 2 there is a classical method for constructing a transformation group from these data for each x e x let v x denote the set of v in v such that x e xv for each subset s of v let rs be the subgroup generated by s and let xs be the face of x defined by

graph products of spheres associative graded algebras and hilbert series given a finite simple vertex weighted graph we construct a graded associative noncommutative algebra whose generators correspond to vertices and whose ideal of relations has generators that are graded commutators corresponding to edges we show that the hilbert series of this algebra is the inverse of the clique polynomial of the graph using this result it easy to recognize if the ideal is inert from which strong results on the algebra follow noncommutative grobner bases play an important role in our proof there is an interesting application to toric topology this algebra arises naturally from a partial product of spheres which is a special case of a generalized moment angle complex we apply our result to the loop space homology of this space

lie algebraic aspects of the finite nonperiodic toda flows in this paper we review various aspects of nonperiodic full and tridiagonal toda flow theory in particular we consider gradient structures in the dynamics and geometry of these systems and we compare and contrast a number of formulations of the nonperiodic toda equations we describe the structure of these systems on general semisimple lie algebras in the case of the full kostant asymmetric toda flow we describe some of the background behind its integrability and explain the role of noncommutative integrability in its qualitative behavior we describe the relationship between the asymmetric toda flows and the symmetric and indefinite toda flows and show how one may conjugate from the full kostant toda flows to the full symmetric toda flows via a poisson map we describe briefly related toda systems such as nonabelian toda and the peakon flows

the homotopy type of the polyhedral product for shifted complexes abstract we prove a conjecture of bahri bendersky cohen and gitler if k is a shifted simplicial complex on n vertices x 1 x n are pointed connected c w complexes and c x i is the cone on x i then the polyhedral product determined by k and the pairs c x i x i is homotopy equivalent to a wedge of suspensions of smashes of the x i s earlier work of the authors dealt with the special case where each x i is a loop space new techniques are introduced to prove the general case these have the advantage of simplifying the earlier results and of being sufficiently general to show that the conjecture holds for a substantially larger class of simplicial complexes we discuss connections between polyhedral products and toric topology combinatorics and classical homotopy theory

projectivities in simplicial complexes and colorings of simple polytopes for each strongly connected finite dimensional pure simplicial complex delta we construct a finite group pi delta the group of projectivities of delta which is a combinatorial but not a topological invariant of delta this group is studied for combinatorial manifolds and in particular for polytopal simplicial spheres the results are applied to a coloring problem for simplicial or dually simple polytopes which arises in the area of toric manifolds

on crystabelline deformation rings of mathrm gal overline mathbb q _p mathbb q _p with an appendix by jack shotton we prove that certain crystabelline deformation rings of two dimensional residual representations of mathrm gal overline mathbb q p mathbb q p are cohen macaulay as a consequence this allows to improve kisin s r 1 p mathbb t 1 p theorem to an r mathbb t theorem

characteristic classes of singular toric varieties in this paper we compute the motivic chern classes and homology hirzebruch characteristic classes of possibly singular toric varieties which in the context of complete toric varieties fit nicely with a generalized hirzebruch riemann roch theorem as important special cases we obtain new or recover well known formulae for the baum fulton macpherson todd or macpherson s chern classes of toric varieties as well as for the thom milnor l classes of simplicial projective toric varieties we present two different perspectives for the computation of these characteristic classes of toric varieties first we take advantage of the torus orbit decomposition and the motivic properties of the motivic chern and respectively homology hirzebruch classes to express the latter in terms of dualizing sheaves and respectively the dual todd classes of closures of orbits this method even applies to torus invariant subspaces of a given toric variety the obtained formula is then applied to weighted lattice point counting in lattice polytopes and their subcomplexes yielding generalized pick type formulae second in the case of simplicial toric varieties we compute our characteristic classes by using the lefschetz riemann roch theorem of edidin graham in the context of the geometric quotient description of such varieties in this setting we define mock hirzebruch classes of simplicial toric varieties which specialize to the mock chern mock todd and mock l classes of such varieties and investigate the difference between the actual homology hirzebruch class and the mock hirzebruch class we show that this difference is localized on the singular locus and we obtain a formula for it in which the contribution of each singular cone is identified explicitly finally the two methods of computing characteristic classes are combined for proving several characteristic class formulae originally obtained by cappell and shaneson in the early 1990s 2015 wiley periodicals inc

asymptotic hamiltonian dynamics the toda lattice the three wave interaction and the non holonomic chaplygin sleigh abstract in this paper we discuss asymptotic stability in energy preserving systems which have an almost poisson structure in particular we consider a class of poisson systems which includes the toda lattice in standard hamiltonian systems one of course does not expect asymptotic stability the key here is the structure of the phase space of the poisson or almost poisson systems and the nature of the equilibria our systems are in some sense generalizations of the integrable toda lattice system but are not integrable in general as a particular example we point out an interesting connection between three mechanical two degree of freedom systems that exhibit asymptotic stability two of them are classical hamiltonian systems while the third is a non holonomic system non holonomic systems are generally energy preserving but not hamiltonian but the system analyzed here turns out to have a phase space which is the union of hamiltonian ones we also discuss various higher dimensional examples

real quadrics in cn complex manifolds and convex polytopes in this paper we investigate the topology of a class of non kahler compact complex manifolds generalizing that of hopf and calabi eckmann manifolds these manifolds are diffeomorphic to special systems of real quadrics cn which are invariant with respect to the natural action of the real torus s1 n onto cn the quotient space is a simple convex polytope the problem reduces thus to the study of the topology of certain real algebraic sets and can be handled using combinatorial results on convex polytopes we prove that the homology groups of these compact complex manifolds can have arbitrary amount of torsion so that their topology is extremely rich we also resolve an associated wall crossing problem by introducing holomorphic equivariant elementary surgeries related to some transformations of the simple convex polytope finally as a nice consequence we obtain that affine non kahler compact complex manifolds can have arbitrary amount of torsion in their homology groups contrasting with the kahler situation

on relevant dimensions in kernel feature spaces we show that the relevant information of a supervised learning problem is contained up to negligible error in a finite number of leading kernel pca components if the kernel matches the underlying learning problem in the sense that it can asymptotically represent the function to be learned and is sufficiently smooth thus kernels do not only transform data sets such that good generalization can be achieved using only linear discriminant functions but this transformation is also performed in a manner which makes economical use of feature space dimensions in the best case kernels provide efficient implicit representations of the data for supervised learning problems practically we propose an algorithm which enables us to recover the number of leading kernel pca components relevant for good classification our algorithm can therefore be applied 1 to analyze the interplay of data set and kernel in a geometric fashion 2 to aid in model selection and 3 to denoise in feature space in order to yield better classification results

decompositions of the polyhedral product functor with applications to moment angle complexes and related spaces this article gives a natural decomposition of the suspension of a generalized moment angle complex or partial product space which arises as the polyhedral product functor described below the introduction and application of the smash product moment angle complex provides a precise identification of the stable homotopy type of the values of the polyhedral product functor one direct consequence is an analysis of the associated cohomology for the special case of the complements of certain subspace arrangements the geometrical decomposition implies the homological decomposition in earlier work of others as described below because the splitting is geometric an analogous homological decomposition for a generalized moment angle complex applies for any homology theory implied therefore is a decomposition for the stanley reisner ring of a finite simplicial complex and natural generalizations

real loci of symplectic reductions let m be a compact connected symplectic manifold with a hamiltonian action of a compact n dimensional torus t suppose that m is equipped with an anti symplectic involution a compatible with the t action the real locus of m is the fixed point set m of duistermaat introduced real loci and extended several theorems of symplectic geometry to real loci in this paper we extend another classical result of symplectic geometry to real loci the kirwan surjectivity theorem in addition we compute the kernel of the real kirwan map these results are direct consequences of techniques introduced by tolman and weitsman in some examples these results allow us to show that a symplectic reduction m t has the same ordinary cohomology as its real locus m t red with degrees halved this extends duistermaat s original result on real loci to a case in which there is not a natural hamiltonian torus action

hyperbolic 4 manifolds colourings and mutations we develop a way of seeing a complete orientable hyperbolic 4 manifold m as an orbifold cover of a coxeter polytope p h 4 that has a facet colouring we also develop a way of finding a totally geodesic sub manifold n in m and describing the result of mutations along n as an application of our method we construct an example of a complete orientable hyperbolic 4 manifold x with a single non toric cusp and a complete orientable hyperbolic 4 manifold y with a single toric cusp both x and y have twice the minimal volume among all complete orientable hyperbolic 4 manifolds

some combinatorial and algebraic properties of coxeter complexes and tits buildings a des groupes de coxeter et des groupes a bn paire sont associes certaines structures combinatoires les complexes de coxeter et les constructions de tits on etudie ces structures du point de vue des complexes de cohen macaulay et des complexes decorticables

conjugations on 6 manifolds with free integral cohomology in this article we show the existence of conjugations on many smooth simply connected spin 6 manifolds with free integral cohomology in a certain class the only condition on x6 to admit a conjugation with fixed point set m3 is the obvious one the existence of a degree halving ring isomorphism between the mathbb z 2 cohomologies of x and m as a consequence certain 6 manifolds for which puppe j fixed point theory appl 2 1 85 96 2007 proved the non existence of non trivial orientation preserving finite group actions do admit many involutions

gkm theory and hamiltonian non kahler actions in dimension 6 abstract using the classification of 6 dimensional manifolds by wall jupp and ubr we observe that the diffeomorphism type of simply connected compact 6 dimensional integer gkm t 2 manifolds is encoded in their gkm graph as an application we show that the 6 dimensional manifolds on which tolman and woodward constructed hamiltonian non kahler t 2 actions with finite fixed point set are diffeomorphic to eschenburg s twisted flag manifold su 3 t 2 in particular they admit a noninvariant kahler structure

universally consistent vertex classification for latent positions graphs in this work we show that using the eigen decomposition of the adjacency matrix we can consistently estimate feature maps for latent position graphs with positive definite link function provided that the latent positions are i i d from some distribution f we then consider the exploitation task of vertex classification where the link

configuration spaces and polyhedral products abstract this paper aims to find the most general combinatorial conditions under which a moment angle complex d 2 s 1 k is a co h space thus splitting unstably in terms of its full subcomplexes in this way we study to which extent the conjecture holds that a moment angle complex over a golod simplicial complex is a co h space our main tool is a certain generalisation of the theory of labelled configuration spaces

colorings of simplicial complexes and vector bundles over davis januszkiewicz spaces we show that coloring properties of a simplicial complex k are reflected by splitting properties of a bundle over the associated davis januszkiewicz space whose chern classes are given by the elementary symmetric polynomials in the generators of the stanley reisner algebra of k

completely integrable gradient flows in this paper we exhibit the toda lattice equations in a double bracket form which shows they are gradient flow equations on their isospectral set on an adjoint orbit of a compact lie group representations for the flows are given and a convexity result associated with a momentum map is proved some general properties of the double bracket equations are demonstrated including a discussion of their invariant subspaces and their function as a lie algebraic sorter

torus graphs and simplicial posets for several important classes of manifolds acted on by the torus the information about the action can be encoded combinatorially by a regular n valent graph with vector labels on its edges which we refer to as the torus graph by analogy with the gkm graphs we introduce the notion of equivariant cohomology of a torus graph and show that it is isomorphic to the face ring of the associated simplicial poset this extends a series of previous results on the equivariant cohomology of torus manifolds as a primary combinatorial application we show that a simplicial poset is cohen macaulay if its face ring is cohen macaulay this completes the algebraic characterisation of cohen macaulay posets initiated by stanley we also study blow ups of torus graphs and manifolds from both the algebraic and the topological points of view

decompositions of polyhedral products for shifted complexes abstract the conjecture of bahri bendersky cohen and gitler 2010 3 on wedge decompositions of polyhedral products of shifted complexes is settled affirmatively as a corollary it is proved that the homotopy type of the complement of a coordinate subspace arrangement associated with a shifted complex tensored with r r for any r 1 has the homotopy type of a wedge of spheres