deformations of the braid arrangement and trees abstract we establish general counting formulas and bijections for deformations of the braid arrangement precisely we consider real hyperplane arrangements such that all the hyperplanes are of the form x i x j s for some integer s classical examples include the braid catalan shi semiorder and linial arrangements as well as graphical arrangements we express the number of regions of any such arrangement as a signed count of decorated plane trees the characteristic and coboundary polynomials of these arrangements also have simple expressions in terms of these trees we then focus on certain well behaved deformations of the braid arrangement that we call transitive this includes the catalan shi semiorder and linial arrangements as well as many other arrangements appearing in the literature for any transitive deformation of the braid arrangement we establish a simple bijection between regions of the arrangement and a set of labeled plane trees defined by local conditions this answers a question of gessel

the connection matrix theory for morse decompositions the connection matrix theory for morse decompositions is introduced the connection matrices are matrices of maps between the homology indices of the sets in the morse decomposition the connection matrices cover in a natural way the homology index braid of the morse decomposition and provide information about the structure of the morse decomposition the existence of connection matrices of morse decompositions is established and examples illustrating applications of the connection matrix are provided

remarks on non compact gradient ricci solitons in this paper we show how techniques coming from stochastic analysis such as stochastic completeness in the form of the weak maximum principle at infinity parabolicity and lp liouville type results for the weighted laplacian associated to the potential may be used to obtain triviality rigidity results and scalar curvature estimates for gradient ricci solitons under lp conditions on the relevant quantities

the conjugate heat equation and ancient solutions of the ricci flow abstract we prove gaussian type bounds for the fundamental solution of the conjugate heat equation evolving under the ricci flow as a consequence for dimension 4 and higher we show that the backward limit of type i solutions of the ricci flow must be a non flat gradient shrinking ricci soliton this extends perelman s previous result on backward limits of solutions in dimension 3 in which case the curvature operator is nonnegative it follows from hamilton ivey curvature pinching estimate as an application this also addresses an issue left in naber 2010 23 where naber proves the interesting result that there exists a type i dilation limit that converges to a gradient shrinking ricci soliton but that soliton might be flat the gaussian bounds that we obtain on the fundamental solution of the conjugate heat equation under evolving metric might be of independent interest

perelman s entropy functional at type i singularities of the ricci flow we study blow ups around fixed points at type i singularities of the ricci flow on closed manifolds using perelman s w functional first we give an alternative proof of the result obtained by naber 18 and enders muller topping 8 that blow up limits are non flat gradient shrinking ricci solitons our second and main result relates a limit w density at a type i singular point to the entropy of the limit gradient shrinking soliton obtained by blowing up at this point in particular we show that no entropy is lost at infinity during the blow up process

labeled binary trees subarrangements of the catalan arrangements and schur positivity abstract in 1995 the first author introduced a multivariate generating function g that tracks the distribution of ascents and descents in labeled binary trees in addition to proving that g is symmetric he conjectured that g is schur positive we prove this conjecture by expanding g positively in terms of ribbon schur functions we obtain this expansion using a weight preserving bijection whose inverse is inspired by the push glide algorithm of preville ratelle and viennot in fact this weight preserving bijection allows us to establish a stronger version of the first author s conjecture showing that the generating function restricted to labeled binary trees with a fixed canopy is still schur positive we also discuss applications in the setting of hyperplane arrangements we show that a certain specialization of g equals the frobenius characteristic of the natural s n action on regions of the semiorder arrangement which we then expand in terms of the frobenius characteristics of foulkes characters we also construct an s n action on regions of the linial arrangement using a set of trees studied by bernardi and subsequently compute the character of this action by employing lagrange inversion the resulting expression generalizes postnikov s formula for the number of regions in the linial arrangement as a final application we prove nonnegativity for the distribution of the number of right edges over local binary search trees

multiderivations of coxeter arrangements let v be an l dimensional euclidean space let g o v be a finite irreducible orthogonal reflection group let be the corresponding coxeter arrangement let s be the algebra of polynomial functions on v for h choose h v such that h ker h for each nonnegative integer m define the derivation module d m ders h s mh the module is known to be a free s module of rank l by k saito 1975 for m 1 and l solomon h terao 1998 for m 2 the main result of this paper is that this is the case for all m moreover we explicitly construct a basis for d m their degrees are all equal to mh 2 when m is even or are equal to m 1 h 2 mi 1 i l when m is odd here m1 ml are the exponents of g and h ml 1 is the coxeter number the construction heavily uses the primitive derivation d which plays a central role in the theory of flat generators by k saito or equivalently the frobenius manifold structure for the orbit space of g some new results concerning the primitive derivation d are obtained in the course of proof of the main result

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

simple root bases for shi arrangements abstract in his affirmative answer to the edelman reiner conjecture yoshinaga proved that the logarithmic derivation modules of the cones of the extended shi arrangements are free modules however we know very little about the bases themselves except their existence in this article we prove the unique existence of two distinguished bases which we call the simple root basis plus srb and the simple root basis minus srb they are characterized by nice divisibility properties relative to the simple roots

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

affine hecke algebras and generalized standard young tableaux abstract this paper introduces calibrated representations for affine hecke algebras and classifies and constructs all finite dimensional irreducible calibrated representations the primary technique is to provide indexing sets for controlling the weight space structure of finite dimensional modules for the affine hecke algebra using these indexing sets we show that 1 irreducible calibrated representations are indexed by skew local regions 2 the dimension of an irreducible calibrated representation is the number of chambers in the local region 3 each irreducible calibrated representation is constructed explicitly by formulas which describe the action of the generators of the affine hecke algebra on a specific basis in the representation space the indexing sets for weight spaces are generalizations of standard young tableaux and the construction of the irreducible calibrated affine hecke algebra modules is a generalization of a young s seminormal construction of the irreducible representations of the symmetric group in this sense young s construction has been generalized to arbitrary lie type

witten s complex and infinite dimensional morse theory we investigate the relation between the trajectories of a finite dimensional gradient flow connecting two critical points and the cohomology of the surrounding space the results are applied to an infinite dimensional problem involving the symplect c action function

hyperplane arrangements interval orders and trees abstract a hyperplane arrangement is a finite set of hyperplanes in a real affine space an especially important arrangement is the braid arrangement which is the set of all hyperplanes xi xj 1 1 i j n in rn some combinatorial properties of certain deformations of the braid arrangement are surveyed in particular there are unexpected connections with the theory of interval orders and with the enumeration of trees for instance the number of labeled interval orders that can be obtained from n intervals i1 in of generic lengths is counted there is also discussed an arrangement due to n linial whose number of regions is the number of alternating or intransitive trees as defined by gelfand graev and postnikov gelfand i m graev m i and postnikov a 1995 preprint finally a refinement is given related to counting labeled trees by number of inversions of a result of shi shi j y 1986 lecture notes in mathematics no 1179 springer verlag that a certain deformation of the braid arrangement has n 1 n 1 regions

on trajectories of analytic gradient vector fields abstract let f r n 0 r 0 be an analytic function defined in a neighbourhood of the origin having a critical point at 0 we show that the set of non trivial trajectories of the equation x dot f x attracted by the origin has the same cech alexander cohomology groups as the real milnor fibre of f

lagrangian intersections and the serre spectral sequence for a transversal pair of closed lagrangian submanifolds l lof a sym plectic manifold m such that 1 l 1 l 0 c1 2 m 2 m and for a generic almost complex structure j we construct an invariant with a high homotopical content which consists in the pages of order 2o f as pec tral sequence whose differentials provide an algebraic measure of the high dimensional moduli spaces of pseudo holomorpic strips of finite energy that join l and l when l and lare hamiltonian isotopic we show that the pages of the spectral sequence coincide up to a horizontal translation with the terms of the serre spectral sequence of the path loop fibration l pl l and we deduce some applications

connected simple systems and the conley index of isolated invariant sets the object of this paper is to present new and simplified proofs for most of the basic results in the index theory for flows simple explicit formulae are derived for the maps which play a central role in the theory the presentation is self contained

homotopy invariants of repeller attractor pairs ii continuation of r a pairs on considere une famille continue a 1 parametre de flots dans le plan parametrise par 0 1 on etablit un cadre qui rend precise la notion de famille continue de paires de repulseur attracteurs on montre qu il y a des equivalences naturelles entre les suites d indices de 2 paires r a le long d un parcours de paires r a

characterization of a free arrangement and conjecture of edelman and reiner we consider a hyperplane arrangement in a vector space of dimension four or higher in this case the freeness of the arrangement is characterized by properties around a fixed hyperplane as an application we prove the freeness of cones over certain truncated affine weyl arrangements which was conjectured by edelman and reiner

non hamiltonian actions with isolated fixed points we construct a non hamiltonian symplectic circle action on a closed connected six dimensional symplectic manifold with exactly 32 fixed points

free and non free multiplicities on the a3 arrangement abstract we give a complete classification of free and non free multiplicities on the a 3 braid arrangement namely we show that all free multiplicities on a 3 fall into two families that have been identified by abe terao wakefield 2007 and abe nuida numata 2009 the main tool is a new homological obstruction to freeness derived via a connection to multivariate spline theory

global topological properties of the hopf bifurcation we study the homotopical and homological properties of the attractors evolving from a generalized hopf bifurcation we consider the lorenz equations for parameter values near the hopf bifurcation and study a natural morse decomposition of the global attractor calculating the cech homotopy type of the lorenz attractor the shape indexes of the morse sets and the morse equation of the decomposition

on complete gradient shrinking ricci solitons in this paper we derive optimal growth estimates on the potential functions of complete noncompact shrinking solitons based on this we prove that a complete noncompact gradient shrinking ricci soliton has at most euclidean volume growth this latter result can be viewed as an analog of the well known volume comparison theorem of bishop that a complete noncompact riemannian manifold with nonnegative ricci curvature has at most euclidean volume growth

lagrangian intersections critical points and qcategory fundacao para a ciencia e a tecnologia fct programa operacional ciencia tecnologia inovacao pocti

on the variation in the cohomology of the symplectic form of the reduced phase space is called the momentum mapping of the hamiltonian t action given 1 1 the condition 1 2 just means that t acts along the fibers of j for the basic definitions and properties of non commutative hamiltonian group actions see am the results of this paper can easily be extended to hamiltonian actions of arbitrary compact connected lie groups by applying our results to the action of its maximal toms and using the equivariance of the momentum mapping for some more details see the remarks at the end of sect 2 we will assume throughout this paper that the momentum map is proper that is j i u is compact for each compact subset u of t now let r be a regular value of j that is t j tmm t is surjective for all meye j 1 4 then ye is a smooth submanifold of m compact because

homotopical dynamics iii real singularities and hamiltonian flows on the space of nondepraved see real isolated singularities we consider the stable equivalence relation induced by smooth deformations whose asymptotic behaviour is controlled by the palais smale condition it is shown that the resulting space of equivalence classes admits a canonical semiring structure and is isomorphic to the semiring of stable homotopy classes of cw complexes in an application to hamiltonian dynamics we relate the existence of bounded and periodic orbits on noncompact level hypersurfaces of palais smale hamiltonians with just one singularity that is nondepraved to the lack of self duality in the sense of e spanier and j whitehead of the sublink of the singularity

deforming the metric on complete riemannian manifolds soit m g ij x une variete de riemann a n dimensions complete non compacte de tenseur de complexe riemannien r ijkl satisfaisant r ijkl 2 k 0 sur m ou 0 0 dependant seulement de n et de k 0 telle que l equation d evolution g ij x t t 2r ij x t sur m g ij x o g ij x x m ait une solution lisse g ij x t 0 pour un temps court 0 t t u k 0 et satisfait l estimation suivante pour un entier m 0 il existe des constantes c m 0 dependant seulement de n m et k 0 telles que sup x m v m r ijkl x t 2 c m t m 0 t t u k 0

random walks and hyperplane arrangements let be the set of chambers of a real hyperplane arrangement we study a random walk on introduced by bidigare hanlon and rockmore this includes various shuffling schemes used in computer science biology and card games it also includes random walks on zonotopes and zonotopal tilings we find the stationary distributions of these markov chains give good bounds on the rate of convergence to stationarity and prove that the transition matrices are diagonalizable the results are extended to oriented matroids

cohomologically symplectic spaces toral actions and the gottlieb group aspects of symplectic geometry are explored from a homotopical viewpoint in particular the question of whether or not a given toral action is hamiltonian is shown to be independent of geometry rather a new homotopical obstruction is described which detects when an action is hamiltonian this new entity the ae invariant allows many results of symplectic geometry to be generalized to manifolds which are only cohomologically symplectic in the sense that there is a degree 2 cohomology class which cups to a top class furthermore new results in symplectic geometry also arise from this homotopical approach