shifted simplicial complexes are laplacian integral we show that the combinatorial laplace operators associated to the boundary maps in a shifted simplicial complex have all integer spectra we give a simple combinatorial interpretation for the spectra in terms of vertex degree sequences generalizing a theorem of merris for graphs we also conjecture a majorization inequality for the spectra of these laplace operators in an arbitrary simplicial complex with equality achieved if and only if the complex is shifted this generalizes a conjecture of grone and merris for graphs

deformations of associative submanifolds with boundary abstract let m be a topological g 2 manifold we prove that the space of infinitesimal associative deformations of a compact associative submanifold y with boundary in a coassociative submanifold x is the solution space of an elliptic problem for a connected boundary y of genus g the index is given by y c 1 x 1 g where x denotes the orthogonal complement of t y in t x y and c 1 x the first chern class of x with respect to its natural complex structure further we exhibit explicit examples of non trivial index

the geometry of three forms in six dimensions we study the special algebraic properties of alternating 3 forms in 6 dimensions and introduce a diffeomorphism invariant functional on the space of differential 3 forms on a closed 6 manifold m restricting the functional to a de rham cohomology class in h3 m r we find that a critical point which is generic in a suitable sense defines a complex threefold with trivial canonical bundle this approach gives a direct method of showing that an open set in h3 m r is a local moduli space for this structure and introduces in a natural way the special pseudo kahler structure on it

dirichlet duality and the nonlinear dirichlet problem we study the dirichlet problem for fully nonlinear degenerate elliptic equations of the form f hess u 0 on a smoothly bounded domain n in our approach the equation is replaced by a subset f sym2 n of the symmetric n n matrices with f f 0 we establish the existence and uniqueness of continuous solutions under an explicit geometric f convexity assumption on the boundary we also study the topological structure of f convex domains and prove a theorem of andreotti frankel type two key ingredients in the analysis are the use of subaffine functions and dirichlet duality associated to f is a dirichlet dual set f that gives a dual dirichlet problem this pairing is a true duality in that the dual of f is f and in the analysis the roles of f and f are interchangeable the duality also clarifies many features of the problem including the appropriate conditions on the boundary many interesting examples are covered by these results including all branches of the homogeneous monge ampere equation over and equations appearing naturally in calibrated geometry lagrangian geometry and p convex riemannian geometry and all branches of the special lagrangian potential equation 2008 wiley periodicals inc

combinatorial decompositions of a class of rings in this paper we introduce some combinatorial methods for the study of graded rings and use these methods to analyze a class of rings that are of importance in combinatorics our methods are based on a unique representation theorem which is shown section 2 to hold for any graded algebra and which in a sense is an extension of the direct sum decomposition for cohen macaulay algebras this unique representation is reminiscent of a similar result obtained by rees 20 both results yield a decomposition of the algebra however our result differs from that of rees in that it carries more information about the algebra our main results are concerned with a class of rings see section 3 for the definition associated to simplicial complexes there are several natural operations that can be performed on simplicial complexes to obtain new complexes which in turn have an interpretation for the corresponding rings we show sections 4 5 and 6 that our basic decompositions can be transferred during these operations from the original rings to the newly constructed ones the operations studied here are the chain transform or barycentric subdivision section 4 rank selection both for partially ordered sets and for simplicial complexes section s and localization section 6 as we develop these tools we give some applications in section 5 we give a new topological characterization of the cohen macaulay property for partially ordered sets an immediate consequence of which is the rankselection theorem baclawski 2 this characterization is closely related to those obtained by garsia in 14

minimal surfaces in minimally convex domains in this paper we prove that every conformal minimal immersion of a compact bordered riemann surface m into a minimally convex domain d r can be approximated uniformly on compacts in m m bm by proper complete conformal minimal immersions m d see theorems 1 1 1 7 and 1 9 we also obtain a rigidity theorem for complete immersed minimal surfaces of finite total curvature contained in a minimally convex domain in r see theorem 1 16 and we characterize the minimal surface hull of a compact set k in r for any n 3 by sequences of conformal minimal disks whose boundaries converge to k in the measure theoretic sense see corollary 5 6

a categorification of biclosed sets of strings abstract we consider a closure space known as biclosed sets of strings of a gentle algebra of finite representation type palu pilaud and plamondon proved that the collection of all biclosed sets of strings forms a lattice and moreover that this lattice is congruence uniform many interesting examples of finite congruence uniform lattices may be represented as the lattice of torsion classes of an associative algebra we introduce a generalization the lattice of torsion shadows and we prove that the lattice of biclosed sets of strings is isomorphic to a lattice of torsion shadows when every indecomposable module over the gentle algebra is a brick finite congruence uniform lattices admit an alternate partial order known as the core label order in many cases the core label order of a congruence uniform lattice is isomorphic to a lattice of wide subcategories of an associative algebra analogous to torsion shadows we introduce wide shadows and prove that the core label order of the lattice of biclosed sets is isomorphic to a lattice of wide shadows

a broad class of shellable lattices abstract we introduce a new class of lattices the modernistic lattices and their duals the comodernistic lattices we show that every modernistic or comodernistic lattice has shellable order complex we go on to exhibit a large number of examples of co modernistic lattices we show comodernism for two main families of lattices that were not previously known to be shellable the order congruence lattices of finite posets and a weighted generalization of the k equal partition lattices we also exhibit many examples of co modernistic lattices that were already known to be shellable to begin with the definition of modernistic is a common weakening of the definitions of semimodular and supersolvable we thus obtain a unified proof that lattices in these classes are shellable subgroup lattices of solvable groups form another family of comodernistic lattices that were already proven to be shellable we show not only that subgroup lattices of solvable groups are comodernistic but that solvability of a group is equivalent to the comodernistic property on its subgroup lattice indeed the definition of comodernistic exactly requires on every interval a lattice theoretic analogue of the composition series in a solvable group thus the relation between comodernistic lattices and solvable groups resembles in several respects that between supersolvable lattices and supersolvable groups

on the construction of some complete metrics with exceptional holonomy on construit explicitement trois metriques completes distinctes d holonomie egale a g 2 une metrique complete d holonomie egale a spin 7 et diverses metriques incompletes d holonomie exceptionnelle

quaternionic reduction and quaternionic orbifolds on considere un processus de reduction pour des varietes de kahler quaternioniques on montre qu il existe dans ce cas une certaine application impulsion qui se reduit a celle classique quand la courbure scalaire k de l espace est zero et les metriques deviennent hyperkahler quand k 0 cette application impulsion est definie de facon unique pour un groupe compact d automorphismes h

a characterization of solvability for finite groups in terms of their frame abstract the frame of a group is the poset of conjugacy classes of all its proper subgroups in this paper we will prove that a finite group is solvable if and only if every collection of maximal elements of its frame has a well defined meet and the poset consisting of all such meets including the meet of the empty set is a modular lattice

structure of the loday ronco hopf algebra of trees abstract loday and ronco defined an interesting hopf algebra structure on the linear span of the set of planar binary trees they showed that the inclusion of the hopf algebra of non commutative symmetric functions in the malvenuto reutenauer hopf algebra of permutations factors through their hopf algebra of trees and these maps correspond to natural maps from the weak order on the symmetric group to the tamari order on planar binary trees to the boolean algebra we further study the structure of this hopf algebra of trees using a new basis for it we describe the product coproduct and antipode in terms of this basis and use these results to elucidate its hopf algebraic structure in the dual basis for the graded dual hopf algebra our formula for the coproduct gives an explicit isomorphism with a free associative algebra we also obtain a transparent proof of its isomorphism with the non commutative connes kreimer hopf algebra of foissy and show that this algebra is related to non commutative symmetric functions as the commutative connes kreimer hopf algebra is related to symmetric functions

subsolution theorem and the dirichlet problem for the quaternionic monge ampere equation in this paper we obtain the existence of solution to the dirichlet problem for quaternionic monge ampere equation after showing a stability theorem we prove the subsolution theorem for quaternionic monge ampere equation by combining our stability theorem and kolodziej s approach for the complex case

topology of order complexes of intervals in subgroup lattices abstract we conjecture that the order complex of an open interval in the subgroup lattice of a finite group has the homotopy type of a wedge of spheres and prove that if h g is a minimal counterexample to this conjecture then either g is almost simple or g hn where n is the unique minimal normal subgroup of g n is non abelian and h n 1

a generalization of pde s from a krylov point of view abstract we introduce and investigate the notion of a generalized equation which extends nonlinear elliptic equations and which is based on the notions of subequations and dirichlet duality precisely a subset h sym 2 r n is a generalized equation if it is an intersection h e g where e and g are subequations and g is the subequation dual to g we utilize a viscosity definition of solution to h the mirror of h is defined by h g e one of the main results theorem 2 6 concerns the dirichlet problem on arbitrary bounded domains r n for solutions to h with prescribed boundary function c we prove that a uniqueness holds h has no interior and b existence holds h has no interior for b the appropriate boundary convexity of must be assumed many examples of generalized equations are discussed including the constrained laplacian the twisted monge ampere equation and the c 1 1 equation the closed sets h sym 2 r n which can be written as generalized equations are intrinsically characterized for such an h the set of subequation pairs e g with h e g is partially ordered see 1 10 if e g e g then any solution for the first is also a solution for the second furthermore in this ordered set there is a canonical least element contained in all others a general form of the main theorem which holds on any manifold is also established

an introduction to potential theory in calibrated geometry in this paper we introduce and study the notion of plurisubharmonic functions in calibrated geometry these functions generalize the classical plurisubharmonic functions from complex geometry and enjoy their important properties moreover they exist in abundance whereas the corresponding pluriharmonics are generally quite scarce a number of the results established in complex analysis via plurisubharmonic functions are extended to calibrated manifolds this paper introduces and investigates questions of pseudo convexity in the context of a general calibrated manifold x phi analogues of totally real submanifolds are introduced and used to construct enormous families of strictly phi convex spaces with every topological type allowed by morse theory specific calibrations are used as examples throughout in a sequel the duality between phi pluri sub harmonic functions and phi positive currents is investigated this study involves boundaries generalized jensen measures and other geometric objects on a calibrated manifold

perturbing gauge gravity duals by a romans mass we show how to produce algorithmically gravity solutions in massive iia as infinitesimal first order perturbations in the romans mass parameter dual to assigned conformal field theories we illustrate the procedure on a family of chern simons matter conformal field theories that we recently obtained from the theory by waiving the condition that the levels sum up to zero

quasicontinuity and maximality of quaternionic plurisubharmonic functions abstract in this paper we establish the quaternionic versions of several results in the complex pluripotential theory by using the quaternionic closed positive currents which were introduced in 24 we show that quasicontinuity one of the most important properties of complex plurisubharmonic functions holds also for quaternionic plurisubharmonic functions in h n moreover we prove an equivalent characterization of the maximality which is central to the pluripotential theory a locally bounded plurisubharmonic function u is maximal if and only if it satisfies the homogeneous quaternionic monge ampere equation u n 0

linear decision trees subspace arrangements and mobius functions let p be a set in rn given a point x we want to test if x e p our model of computation is a linear decision tree a rooted ternary tree t where with each interior node v is associated a linear function lv x l j ajxj b and the three edges connecting v to its descendants are labelled starting from the root we move down the tree at each internal node v we check whether iv x 0 and follow the appropriately labelled edge leaves are labelled yes and no and arriving at a leaf we read off the answer to the question is x e p linear decision trees are sometimes surprisingly powerful devices we mention here the result that some np complete problems like the knapsack problem have polynomial depth linear decision trees mh lower bound results on the size or depth of linear decision trees usually depend on counting the number of connected components of p or r n p this method was introduced by dobkin and lipton dl and further developed by steele and yao sy ben or bo and others in this paper we describe topological methods that can be applied to obtain lower bounds also if these sets are connected the basic idea of this paper was first explored in bly the euler characteristic of p or r n p was used to estimate the number of leaves in a linear decision tree for p here we improve and extend the topological methods of bly in particular we show that the euler characteristic can be replaced by the sum of the betti numbers of r n p in bl y the method was applied to the case when p is a subspace arrangement i e a union of a finite number of affine subspaces the euler characteristic of the union is essentially the mobius function of the intersection lattice of these subspaces in this paper using a formula of goresky and macpherson gm we work with the betti numbers of the complement of this union and thereby obtain better bounds the motivating problem in bly was the k equal problem given n real numbers xl xn decide if some k of them are equal it was shown that ev ry linear decision tree for the k equal problem has size at least 2njk 0 n and consequently depth at least q nlog 2njk which is best possible we

why the characteristic polynomial factors we survey three methods for proving that the characteristic polynomial of a finite ranked lattice factors over the nonnegative integers and indicate how they have evolved recently the first technique uses geometric ideas and is is based on zaslavsky s theory of signed graphs the second approach is algebraic and employs results of saito and terao about free hyperplane arrangements finally we consider a purely combinatorial theorem of stanley about supersolvable lattices and its generalizations

half flat structures and special holonomy it was proven by hitchin that any solution of his evolution equations for a half flat su 3 structure on a compact six manifold m defines an extension of m to a seven manifold with holonomy in g2 we give a new proof which does not require the compactness of m more generally we prove that the evolution of any half flat g structure on a six manifold m defines an extension of m to a ricci flat seven manifold n for any real form g of sl 3 c if g is noncompact then the holonomy group of n is a subgroup of the noncompact form gof g c similar results are obtained for the extension of nearly half flat structures by nearly parallel g2 or g structures as well as for the extension of cocalibrated g2 and g structures by parallel spin 7 and spin0 3 4 structures respectively as an application we obtain that any six dimensional homogeneous manifold with an invariant half flat structure admits a canonical extension to a seven manifold with a parallel g2 or g structure for the group h3 h3 where h3 is the three dimensional heisenberg group we describe all left invariant half flat structures and develop a method to explicitly determine the resulting parallel g2 or g structure without integrating in particular we construct three eight parameter families of metrics with holonomy equal to g2 and g moreover we obtain a strong rigidity result for the metrics induced by a half flat structure on h3 h3 satisfying z z 0 where z denotes the centre finally we describe the special geometry of the space of stable three forms satisfying a reality condition considering all possible reality conditions we find four different special kahler manifolds and one special para kahler manifold

generalized quotients in coxeter groups for w s a coxeter group we study sets of the form w v w e w i l wv 1 w i v for all v e v where v c w such sets w v here called generalized quotients are shown to have much of the rich combinatorial structure under bruhat order that has previously been known only for the case when v c s i e for minimal coset representatives modulo a parabolic subgroup we show that bruhat intervals in w v for general v c w are lexicographically shellable the mobius function on w v under bruhat order takes values in 1 0 1 for finite groups w generalized quotients are the same thing as lower intervals in the weak order this is however in general not true connections with the weak order are explored and it is shown that w v is always a complete meet semilattice and a convex order ideal as a subset of w under weak order descent classes di w e w il1 ws 1 w xt s e i for all s e s i c s are also analyzed using generalized quotients it is shown that each descent class as a poset under bruhat order or weak order is isomorphic to a generalized quotient under the corresponding ordering the latter half of the paper is devoted to the symmetric group and to the study of some specific examples of generalized quotients which arise in combinatorics for instance the set of standard young tableaux of a fixed shape or the set of linear extensions of a rooted forest suitably interpreted form generalized quotients we prove a factorization result for the quotients that come from rooted forests which shows that algebraically these quotients behave as a system of minimal coset representatives of a subset which is in general not a subgroup we also study the rank generating function for certain quotients in the symmetric group

l2 estimates on p convex riemannian manifolds abstract in this paper we establish various l 2 estimates for the exterior differential operator on p convex riemannian manifolds in the sense of harvey and lawson as applications we establish a carleman type estimate which is uniform with respect to both weight functions and domains and we also obtain topological restrictions for a riemannian manifold to be p convex

bibo stabilization for system with multiple mixed delays and nonlinear perturbations the problem of bibo stabilization for multiple mixed time delayed control system with nonlinear perturbations is studied in this paper the new delay dependent bibo stabilization criteria are derived by the lyapunov functional and given in terms of existence of a positive definite solution to an auxiliary algebraic riccati matrix equation the robust quadratic stability for such system is also discussed the work of this paper will extend the results of some references

characteristic polynomials of subspace arrangements and finite fields let a be any subspace arrangement in rndefined over the integers and let fqdenote the finite field withqelements letqbe a large prime we prove that the characteristic polynomial a q of a counts the number of points in fnqthat do not lie in any of the subspaces ofa viewed as subsets of fnq this observation which generalizes a theorem of blass and sagan about subarrangements of the bnarrangement reduces the computation of a q to a counting problem and provides an explanation for the wealth of combinatorial results discovered in the theory of hyperplane arrangements in recent years the basic idea has its origins in the work of crapo and rota 1970 we find new classes of hyperplane arrangements whose characteristic polynomials have simple form and very often factor completely over the nonnegative integers

eigenspace arrangements of reflection groups the lattice of intersections of reflecting hyperplanes of a complex reflection group w may be considered as the poset of 1 eigenspaces of the elements of w in this paper we replace 1 with an arbitrary eigenvalue and study the topology and homology representation of the resulting poset after posing the main question of whether this poset is shellable we show that all its upper intervals are geometric lattices and then answer the question in the affirmative for the infinite family g m p n of complex reflection groups and the first 31 of the 34 exceptional groups by constructing cl shellings in addition we completely determine when these eigenspaces of w form a k 1 resp free arrangement for the symmetric group we also extend the combinatorial model available for its intersection lattice to all other eigenvalues by introducing balanced partition posets presented as particular upper order ideals of dowling lattices study the representation afforded by the top co homology group and give a simple map to the posets of pointed d divisible partitions 1 main question and results let v denote an n dimensional c vector space a reflection in v is any nonidentity element r in gl v of finite order that fixes some hyperplane hr and a finite subgroup w of gl v is called a reflection group if it is generated by reflections example the action of the symmetric group sn on n 1 2 n gives rise to a faithful action on c via ei e i where e1 e2 en denote the standard basis vectors since the transpositions act as reflections and generate the group this representation realizes sn as a reflection group in gl c we shall refer to it as the defining representation for an element g w and root of unity let v g denote the eigenspace of g in v define e w to be the w poset partially ordered set of all such eigenspaces v g g w ordered by reverse inclusion with w action given by h v g v hgh the eigenvalue 1 gives the lattice lw of intersections of reflecting hyperplanes for w see 29 lemma 4 4 the minimal elements of e w i e the inclusion maximal eigenspaces forw are the focus of springer s theory of regular elements 35 and each has dimension equal to the number a d of degrees d1 d2 dn of w that are divisible by d the order of see proposition 3 7 below when w is crystallographic the poset e w itself appears in broue malle and michel s sylow theory 8 2010 mathematics subject classification primary 20f55 05e10 05e18 55p20 14n20

shellable nonpure complexes and posets ii this is a direct continuation of shellable nonpure complexes and posets i which appeared in transactions of the american mathematical society 348 1996 1299 1327 8 interval generated lattices and dominance order in this section and the following one we will continue exemplifying the applicability of lexicographic shellability to nonpure posets let f i1 i2 in be a family of intervals of integers by which is meant sets of the form a b a a 1 b a b we assume that there are no containments among these intervals and that they are ordered so that their left and right endpoints are increasing let l f be the lattice of all sets that are unions of subfamilies of f ordered by inclusion such interval generated lattices l f were introduced and studied by greene g define an edge labeling of l f as follows if a b is a covering and a max b a then a b a if a 1 a and a is the left endpoint of some i f

calibrations and t duality for a subclass of hitchin s generalised geometries we introduce and analyse the concept of a structured submanifold which encapsulates the classical notion of a calibrated submanifold under a suitable integrability condition on the ambient geometry these generalised calibrated submanifolds minimise a functional occurring as d brane energy in type ii string theories further we investigate the behaviour of calibrated cycles under t duality and construct non trivial examples

polar coordinates induced by actions of compact lie groups let g be a connected lie subgroup of the real orthogonal group 0 n for the action of g on r we construct linear subspaces a that intersect all orbits we determine for which g there exists such an a meeting all the g orbits orthogo nally groups that act transitively on spheres are obvious examples with few exceptions all possible g arise as the isotropy subgroups of riemannian symmetric spaces introduction let g be a compact lie group acting on a real vectorspace v and let k be a g invariant inner product on v having a linear cross section a c v of

on the holonomy of connections with skew symmetric torsion we investigate the holonomy group of a linear metric connection with skew symmetric torsion in case of the euclidian space and a constant torsion form this group is always semisimple it does not preserve any non degenerated 2 form or any spinor suitable integral formulas allow us to prove similar properties in case of a compact riemannian manifold equipped with a metric connection of skew symmetric torsion on the aloff wallach space n 1 1 we construct families of connections admitting parallel spinors furthermore we investigate the geometry of these connections as well as the geometry of the underlying riemannian metric finally we prove that any 7 dimensional 3 sasakian manifold admits 2 parameter families of linear metric connections and spinorial connections defined by 4 forms with parallel spinors

free resolutions of lex ideals over a koszul toric ring in this paper we study the minimal free resolution of lex ideals over a koszul toric ring in particular we study in which toric ring r all lex ideals are componentwise linear we give a certain necessity and sufficiency condition for this property and show that lex ideals in a strongly koszul toric ring are componentwise linear in addition it is shown that in the toric ring arising from the segre product p p every hilbert function of a graded ideal is attained by a lex ideal and that lex ideals have the greatest graded betti numbers among all ideals having the same hilbert function

group actions on arrangements of linear subspaces and applications to configuration spaces for an arrangement of linear subspaces in ri that is invariant under a finite subgroup of the general linear group gin ir we develop a formula for the g module structure of the cohomology of the complement m a our formula specializes to the well known goresky macpherson theorem in case g 1 but for g 1 the formula shows that the g module structure of the complement is not a combinatorial invariant as an application we are able to describe the free part of the cohomology of the quotient space ma4ig our motivating examples are arrangements in cn that are invariant under the action of sn by permuting coordinates a particular case is the k equal arrangement first studied by bj6rner lovasz and yao motivated by questions in complexity theory in these cases ma4 and ma sn are spaces of ordered and unordered point configurations in cn many of whose properties are reduced by our formulas to combinatorial questions in partition lattices more generally we treat point configurations in rd and provide explicit results for the k equal and the k divisible cases

kahler reduction of metrics with holonomy g2 a torsion free g 2 structure admitting an infinitesimal isometry such that the quotient is a kahler manifold is shown to give rise to a 4 manifold equipped with a complex symplectic structure and a 1 parameter family of functions and 2 forms linked by second order equations reversing the process in various special cases leads to the construction of explicit metrics with holonomy equal to g 2

stability and bifurcation analysis on a logistic model with discrete and distributed delays in this paper a logistic model with discrete and distributed delays is investigated we first consider the stability of the positive equilibrium and the existence of local hopf bifurcations in succession using the normal form theory and center manifold argument we derive the explicit formulas which determine the stability direction and other properties of bifurcating periodic solutions a numerical simulation for supporting the theoretical analysis is also given

su 3 instantons and g 2 spin 7 heterotic string solitons necessary and sufficient conditions to the existence of a hermitian connection with totally skew symmetric torsion and holonomy contained in su 3 are given a formula for the riemannian scalar curvature is obtained non compact solution to the supergravity type i equations of motion with non zero flux and non constant dilaton is found in dimension 6 non conformally flat non compact solutions to the supergravity type i equations of motion with non zero flux and non constant dilaton are found in dimensions 7 and 8 a riemannian metric with holonomy contained in g2 arises from our considerations and hitchin s flow equations which seems to be new compact examples of su 3 g2 and spin 7 instanton satisfying the anomaly cancellation conditions are presented

multi moment maps we introduce a notion of moment map adapted to actions of lie groups that preserve a closed three form we show existence of our multi moment maps in many circumstances including mild topological assumptions on the underlying manifold such maps are also shown to exist for all groups whose second and third lie algebra betti numbers are zero we show that these form a special class of solvable lie groups and provide a structural characterisation we provide many examples of multi moment maps for different geometries and use them to describe manifolds with holonomy contained in g2 preserved by a two torus symmetry in terms of tri symplectic geometry of four manifolds

the self duality equations on a riemann surface in this paper we shall study a special class of solutions of the self dual yang mills equations the original self duality equations which arose in mathematical physics were defined on euclidean 4 space the physically relevant solutions were the ones with finite action the so called instantons the same equations may be dimensionally reduced to euclidean 3 space by imposing invariance under translation in one direction these equations also have physical relevance the solutions which have finite action in three dimensions are the magnetic monopoles if we take the reduction process one step further and consider solutions which are invariant under two translations we obtain a set of equations in the plane here however there is no clear physical meaning and indeed attempts to find finite action solutions have failed nevertheless these are the equations we shall consider despite the lack of interesting solutions in r2 the equations have the important property conformal invariance which allows them to be defined on manifolds modelled on r2 by conformal maps namely riemann surfaces we shall consider here solutions of the self duality equations defined on a compact riemann surface there are in fact solutions as we shall show and the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure which will be the focus of our study it brings together in a harmonious way the subjects of riemannian geometry topology algebraic geometry and symplectic geometry illuminating all these facets of the same object accounts for the length of this paper the self duality equations are equations from gauge theory geometrically they are defined in terms of connections on principal bundles while the group of the principal bundle may be chosen arbitrarily for the equations to make sense we restrict attention here to the simplest case of su 2 or so 3 there are two reasons for this the first and most obvious is that it simplifies calculations and avoids the use of inductive processes which are inherent in the consideration of a general lie group of higher rank the second reason is that solutions for su 2 have an intimate relationship with the internal structure of the riemann surface as a consequence of results we shall prove about solutions to the self duality equations we learn something about the moduli space of complex structures on the surface itself namely teichmuller space

local classification of conformally einstein kahler metrics in higher dimensions the requirement that a non einstein kahler metric in any given complex dimension m 2 be almost everywhere conformally einstein turns out to be much more restrictive even locally than in the case of complex surfaces the local biholomorphic isometry types of such metrics depend for each m 2 on three real parameters along with an arbitrary kahler einstein metric h in complex dimension m 1 we provide an explicit description of all these local isometry types for any given h this result is derived from a more general local classification theorem for metrics admitting functions that we call special kahler ricci potentials

duality of positive currents and plurisubharmonic functions in calibrated geometry recently the authors showed that there is a robust potential theory attached to any calibrated manifold x phi in particular on x there exist phi plurisubharmonic functions phi convex domains phi convex boundaries etc all inter related and having a number of good properties in this paper we show that in a strong sense the plurisubharmonic functions are the polar duals of the phi submanifolds or more generally the phi currents studied in the original paper on calibrations in particular we establish an analogue of duval sibony duality which characterizes points in the phi convex hull of a compact set k subset x in terms of phi positive green s currents on x and jensen measures on k we also characterize boundaries of phi currents entirely in terms of phi plurisubharmonic functions specific calibrations are used as examples throughout analogues of the hodge conjecture in calibrated geometry are considered

ricci flow killing spinors and t duality in generalized geometry abstract we introduce a notion of ricci flow in generalized geometry extending a previous definition by gualtieri on exact courant algebroids special stationary points of the flow are given by solutions to first order differential equations the killing spinor equations which encompass special holonomy metrics with solutions of the hull strominger system our main result investigates a method to produce new solutions of the ricci flow and the killing spinor equations for this we consider t duality between possibly topologically distinct torus bundles endowed with courant structures and demonstrate that solutions of the equations are exchanged under this symmetry as applications we give a mathematical explanation of the dilaton shift in string theory and prove that the hull strominger system is preserved by t duality

spacetime singularity resolution by m theory fivebranes calibrated geometry anti de sitter solutions and special holonomy metrics the supergravity description of various configurations of supersymmetric m fivebranes wrapped on calibrated cycles of special holonomy manifolds is studied the description is provided by solutions of eleven dimensional supergravity which interpolate smoothly between a special holonomy manifold and an event horizon with anti de sitter geometry for known examples of anti de sitter solutions the associated special holonomy metric is derived one explicit anti de sitter solution of m theory is so treated for fivebranes wrapping each of the following cycles kahler cycles in calabi yau two three and four folds special lagrangian cycles in three and four folds associative three and co associative four cycles in g 2 manifolds complex lagrangian four cycles in sp 2 manifolds and cayley four cycles in spin 7 manifolds in each case the associated special holonomy metric is singular and is a hyperbolic analogue of a known metric the analogous known metrics are respectively eguchi hanson the resolved conifold and the four fold resolved conifold the deformed conifold and the stenzel four fold metric the bryant salamon gibbons page pope g 2 metrics on an mathbb r 4 bundle over s 3 and an mathbb r 3 bundle over s 4 or mathbb cp 2 the calabi hyper kahler metric on t mathbb cp 2 and the bryant salamon gibbons page pope spin 7 metric on an mathbb r 4 bundle over s 4 by the ads cft correspondence a conformal field theory is associated to each of the new singular special holonomy metrics and defines the quantum gravitational physics of the resolution of their singularities

uniqueness of tangent cones to positive p p integral cycles we prove that every positive p p integral cycle in an arbitrary almost complex manifold possesses at every point a unique tangent cone the argument relies on an algebraic blow up perturbed in order to face the analysis issues of this problem in the almost complex setting primary subjects 53c38 secondary subjects 49q05 49q20 35j99

metrics with exceptional holonomy it is proved that there exist metrics with holonomy g2 and spin 7 thus settling the remaining cases in berger s list of possible holonomy groups we first reformulate the holonomy h condition as a set of differential equations for an associated h structure on a given manifold we collect the needed algebraic facts about g2 and spin 7 we then apply the machinery of over determined partial differential equations in the form of the cartan kahler theorem to prove the existence of solutions whose holonomy is g2 or spin 7 we also provide explicit examples and some information about the generality of the space of

double hopf bifurcation in a container crane model with delayed position feedback in this paper we study dynamics in a container crane model with delayed position feedback with particular attention focused on non resonant double hopf bifurcation by using multiple time scales and center manifold reduction methods we obtain the equivalent normal forms near a double hopf critical point in this neutral delayed differential system moreover bifurcations are classified in a two dimensional parameter space near the critical point numerical simulations are presented to demonstrate the applicability of the theoretical results

plurisubharmonic functions in calibrated geometry and q convexity let m be a kahler manifold an integrable function varphi on m is called q plurisubharmonic if the current dd c varphi wedge omega q 1 is positive we prove that varphi is q plurisubharmonic if and only if varphi is subharmonic on all q dimensional complex subvarieties we prove that a q plurisubharmonic function is q convex and admits a local approximation by smooth q plurisubharmonic functions for any closed subvariety z subset m dim mathbb c z leq q 1 there exists a strictly q plurisubharmonic function in a neighbourhood of z this result is known for q convex functions this theorem is used to give a new proof of sibony s lemma on integrability of positive closed p p forms which are integrable outside of a complex subvariety of codimension p 1

rings with lexicographic straightening law the concept of a straightening law as a means of analyzing the structure of particular commutative algebras has appeared independently in the work of a number of authors the purpose of this paper is to introduce a systematic theory of algebras endowed with a structure of a lexicographic straightening law based on a partially ordered set or lexicographic ring for short much of the work in the literature dealing with such rings fits quite naturally in the context of our theory and we sketch some of this work moreover some of our later results in particular the betti number bound in section 6 may actually be new results even for the special cases of the known examples of lexicographic rings the concept of a lexicographic ring as an interesting area of study was suggested to this author by deconcini who conjectured that a lexicographic ring is cohen macaulay if the underlying partially ordered set is so we prove this result in two different ways it has recently come to our attention that this conjecture has also been proved by deconcini et al 131 using deformation theory methods the results of this paper are arranged in two parts the first part consisting of sections 2 through 4 is more elementary and uses the technique of combinatorial decompositions as developed by garsia and this author the second part sections 6 and 7 requires some knowledge of homological algebra methods in ring theory the intermediate section 5 discusses some of the known examples of lexicographic rings in this section we also sketch some of the ways that our theory may be extended to more general contexts the main result of the first part is the transfer theorem 4 3 which enables one to transfer combinatorial decompositions from the underlying partially ordered set or more precisely from the corresponding stanley reisner ring to the lexicographic ring as a result those concepts that are expressible in terms of combinatorial decompositions are also

existence of hkt metrics on hypercomplex manifolds of real dimension 8 abstract a hypercomplex manifold m is a manifold equipped with three complex structures i j k satisfying quaternionic relations such a manifold admits a canonical torsion free connection preserving the quaternion action called the obata connection a quaternionic hermitian metric is a riemannian metric which is invariant with respect to unitary quaternions such a metric is called hyperkahler with torsion hkt for short if it is locally obtained as the hessian of a function averaged with quaternions an hkt metric is a natural analogue of a kahler metric on a complex manifold we push this analogy further proving a quaternionic analogue of the result of buchdahl and of lamari that a compact complex surface m admits a kahler structure if and only if b 1 m is even we show that a hypercomplex manifold m with the obata holonomy contained in s l 2 h admits an hkt structure if and only if h 1 o m i is even dimensional

euler flag enumeration of whitney stratified spaces abstract the flag vector contains all the face incidence data of a polytope and in the poset setting the chain enumerative data it is a classical result due to bayer and klapper that for face lattices of polytopes and more generally eulerian graded posets the flag vector can be written as a c d index a non commutative polynomial which removes all the linear redundancies among the flag vector entries this result holds for regular cw complexes we relax the regularity condition to show the c d index exists for whitney stratified manifolds by extending the notion of a graded poset to that of a quasi graded poset this is a poset endowed with an order preserving rank function and a weighted zeta function this allows us to generalize the classical notion of eulerianness and obtain a c d index in the quasi graded poset arena we also extend the semi suspension operation to that of embedding a complex in the boundary of a higher dimensional ball and study the simplicial shelling components

on the linear algebra in the quaternionic pluripotential theory abstract we clarify the linear algebra used in the quaternionic pluripotential theory so that proofs of several results there can be greatly simplified in particular we characterize and normalize real 2 forms with respect to the quaternionic structure and show that the moore determinant of a quaternionic hyperhermitian matrix is the coefficient of the exterior product of the associated real 2 form as a corollary the quaternionic monge ampere operator is the coefficient of the exterior product of the baston operator

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

dirichlet duality and the nonlinear dirichlet problem on riemannian manifolds in this paper we study the dirichlet problem for fully nonlin ear second order equations on a riemannian manifold as in our previous paper hl4 we define equations via closed subsets of the 2 jet bundle where each equation has a natural dual equation basic existence and uniqueness theorems are established in a wide variety of settings however the emphasis is on starting with a constant coefficient equation as a model which then universally determines an equation on every riemannian manifold which is equipped with a topological reduction of the structure group to the invariance group of the model for example this covers all branches of the homogeneous complex monge ampere equation on an almost complex hermitian manifold x in general for an equation f on a manifold x and a smooth domain x we give geometric conditions which imply that the dirichlet problem on is uniquely solvable for all continuous boundary functions we begin by introducing a weakened form of comparison which has the advantage that local implies global we then introduce two fundamental concepts the first is the notion of a monotonicity cone m for f if x carries a global m subharmonic function then weak comparison implies full compar ison the second notion is that of boundary f convexity which is defined in terms of the asymptotics of f and is used to define bar riers in combining these notions the dirichlet problem becomes uniquely solvable as claimed this article also introduces the notion of local affine jet equiva lence for subequations it is used in treating the cases above but gives results for a much broader spectrum of equations on manifolds including inhomogeneous equations and the calabi yau equation on almost complex hermitian manifolds a considerable portion of the paper is concerned with specific examples they include a wide variety of equations which make sense on any riemannian manifold and many which hold univer sally on almost complex or quaternionic hermitian manifolds or topologically calibrated manifolds

on the shellability of the order complex of the subgroup lattice of a finite group we show that the order complex of the subgroup lattice of a finite group g is nonpure shellable if and only if g is solvable a by product of the proof that nonsolvable groups do not have shellable subgroup lattices is the determination of the homotopy types of the order complexes of the subgroup lattices of many minimal simple groups

a uniform estimate for general quaternionic calabi problem with appendix by daniel barlet abstract we prove a c 0 a priori estimate on a solution of the quaternionic calabi problem on an arbitrary compact connected hkt manifold this generalizes earlier results 9 and 7 where this result was proven under certain extra assumptions on the manifold

the homology representations of the k equal partition lattice we determine the character of the action of the symmetric group on the homology of the induced subposet of the lattice of partitions of the set 1 2 nr obtained by restricting block sizes to the set 1 k k 1 a plethystic formula for the generating function of the frobenius characteristic of the representation is given we combine techniques from the theory of nonpure shellability recently developed by bj6rner and wachs with symmetric function techniques developed by sundaram for determining representations on the homology of subposets of the partition lattice

a generalization of rota s nbc theorem abstract we generalize rota s theorem characterizing the mobius function of a geometric lattice in terms of subsets of atoms containing no broken circuit and give applications to the weak bruhat order of a finite coxeter group and the tamari lattices we also give a direct proof of the fact that in the geometric case any total order of the atoms can be used simple involutions are used in both proofs finally we show how involutions can be used in similar situations specifically in a special case of rota s crosscut theorem as well as in related proofs of walker on hall s theorem and reiner on characteristic and poincare polynomials

conformally parallel g2 structures on a class of solvmanifolds starting from a 6 dimensional nilpotent lie group n endowed with an invariant su 3 structure we construct a homogeneous conformally parallel g 2 metric on an associated solvmanifold we classify all half flat su 3 structures that endow the rank one solvable extension of n with a conformally parallel g 2 structure by suitably deforming the su 3 structures obtained we are able to describe the corresponding non homogeneous ricci flat metrics with holonomy contained in g 2 in the process we also find a new metric with exceptional holonomy

generalised g 2 manifolds we define new riemannian structures on 7 manifolds by a differential form of mixed degree which is the critical point of a possibly constrained variational problem over a fixed cohomology class the unconstrained critical points generalise the notion of a manifold of holonomy g 2 while the constrained ones give rise to a new geometry without a classical counterpart we characterise these structures by means of spinors and show the integrability conditions to be equivalent to the supersymmetry equations on spinors in type ii supergravity theory with bosonic background fields in particular this geometry can be described by two linear metric connections with skew symmetric torsion finally we construct explicit examples by introducing the device of t duality

decomposable clutters and a generalization of simon s conjecture abstract each equigenerated squarefree monomial ideal in the polynomial ring s k x 1 x n represents a family of subsets of n called a uniform clutter in this paper we introduce a class of uniform clutters called decomposable clutters whose associated ideal has linear quotients and hence linear resolution over all fields we show that chordality of these clutters guarantees the correctness of a conjecture raised by r s simon 23 on extendable shellebility of d skeletons of a simplex n for all d we then prove this conjecture for d n 3

geometric plurisubharmonicity and convexity an introduction this is an essay on potential theory for geometric plurisubharmonic functions it begins with a given closed subset g of the grassmann bundle g p tx of tangent p planes to a riemannian manifold x this determines a nonlinear partial differential equation which is convex but never uniformly elliptic p dimx a surprising number of results in complex analysis carry over to this more general setting the notions of a g submanifold an upper semi continuous g plurisubharmonic function a g convex domain a g harmonic function and a g free submanifold are defined results include a restriction theorem as well as the existence and uniqueness of solutions to the dirichlet problem for g harmonic functions on g convex domains

clusters coxeter sortable elements and noncrossing partitions we introduce coxeter sortable elements of a coxeter group for finite we give bijective proofs that coxeter sortable elements are equinumerous with clusters and with noncrossing partitions we characterize coxeter sortable elements in terms of their inversion sets and in the classical cases in terms of permutations

calibrations and spinors introduction i spinors in dimension 8m 1 1 spinors and pinors 1 2 constructing calibrations from spinors 1 3 triality and the octonians ii dimension 16 and es ii 1 a model for the clifford algebra cl 16 ii 2 the orbit structure of spin 16 on 8 16 ii 3 maximal isotropy subgroups of spin 16 ii 4 calibration on r 16 constructed by squaring spinors iii additional results in higher dimension iii 1 a differential system for each geometry iii 2 pure spinors and calibrations iii 3 spinors in dimensions 50 mod 8 references