Let @math be a contact form on a manifold @math , and @math a closed Legendrian submanifold. I prove that @math intersects some characteristic for @math at least twice if all characteristics are closed and of the same period, and @math embeds nicely into the product of @math and an exact symplectic manifold. As an application of the method of proof, the minimal action of a regular closed coisotropic submanifold of complex projective space is at most @math . This yields an obstruction to presymplectic embeddings, and in particular to Lagrangian embeddings.
As explained in @cite_20 after Corollary 1.13, it follows that there exists no exact Lagrangian embedding into @math of a closed manifold @math of nonpositive curvature. (Corollary is a stabilized version of this without the nonpositive curvature assumption.)
@cite_20
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We use a neck stretching argument for holomorphic curves to produce symplectic disks of small area and Maslov class with boundary on Lagrangian submanifolds of nonpositive curvature. Applications include the proof of Audin's conjecture on the Maslov class of Lagrangian tori in linear symplectic space, the construction of a new symplectic capacity, obstructions to Lagrangian embeddings into uniruled symplectic manifolds, a quantitative version of Arnold's chord conjecture, and estimates on the size of Weinstein neighbourhoods. The main technical ingredient is transversality for the relevant moduli spaces of punctured holomorphic curves with tangency conditions. This book consists of two independent works: Part I is 'Solutions of the Einstein Vacuum Equations', by Lydia Bieri. Part II is 'Solutions of the Einstein-Maxwell Equations', by Nina Zipser. A famous result of Christodoulou and Klainerman is the global nonlinear stability of Minkowski spacetime. In this book, Bieri and Zipser provide two extensions to this result. In the first part, Bieri solves the Cauchy problem for the Einstein vacuum equations with more general, asymptotically flat initial data, and describes precisely the asymptotic behavior. In particular, she assumes less decay in the power of @math and one less derivative than in the Christodoulou-Klainerman result. She proves that in this case, too, the initial data, being globally close to the trivial data, yields a solution which is a complete spacetime, tending to the Minkowski spacetime at infinity along any geodesic. In contrast to the original situation, certain estimates in this proof are borderline in view of decay, indicating that the conditions in the main theorem on the decay at infinity on the initial data are sharp. In the second part, Zipser proves the existence of smooth, global solutions to the Einstein-Maxwell equations. A nontrivial solution of these equations is a curved spacetime with an electromagnetic field. To prove the existence of solutions to the Einstein-Maxwell equations, Zipser follows the argument and methodology introduced by Christodoulou and Klainerman. To generalize the original results, she needs to contend with the additional curvature terms that arise due to the presence of the electromagnetic field @math ; in her case the Ricci curvature of the spacetime is not identically zero but rather represented by a quadratic in the components of @math . In particular the Ricci curvature is a constant multiple of the stress-energy tensor for @math . Furthermore, the traceless part of the Riemann curvature tensor no longer satisfies the homogeneous Bianchi equations but rather inhomogeneous equations including components of the spacetime Ricci curvature. Therefore, the second part of this book focuses primarily on the derivation of estimates for the new terms that arise due to the presence of the electromagnetic field. A (tandem) duplication of length @math is an insertion of an exact copy of a substring of length @math next to its original position. This and related types of impairments are of relevance in modeling communication in the presence of synchronization errors, as well as in several information storage applications. We demonstrate that Levenshtein’s construction of binary codes correcting insertions of zeros is, with minor modifications, applicable also to channels with arbitrary alphabets and with duplication errors of arbitrary (but fixed) length @math . Furthermore, we derive bounds on the cardinality of optimal @math -ary codes correcting up to @math duplications of length @math and establish the following corollaries in the asymptotic regime of growing block length: 1) the presented family of codes is optimal for every @math , in the sense of the asymptotic scaling of code redundancy; 2) the upper bound, when specialized to @math , @math , improves upon Levenshtein’s bound for every @math ; and 3) the bounds coincide for @math , thus yielding the exact asymptotic behavior of the size of optimal single-duplication-correcting codes. In this note, we characterize the graphs (1-skeletons) of some piecewise Euclidean simplicial and cubical complexes having nonpositive curvature in the sense of Gromov's CAT(0) inequality. Each such cell complex K is simply connected and obeys a certain flag condition. It turns out that if, in addition, all maximal cells are either regular Euclidean cubes or right Euclidean triangles glued in a special way, then the underlying graph G(K) is either a median graph or a hereditary modular graph without two forbidden induced subgraphs. We also characterize the simplicial complexes arising from bridged graphs, a class of graphs whose metric enjoys one of the basic properties of CAT(0) spaces. Additionally, we show that the graphs of all these complexes and some more general classes of graphs have geodesic combings and bicombings verifying the 1- or 2-fellow traveler property. We consider a continuous space which models the set of all phylogenetic trees having a fixed set of leaves. This space has a natural metric of nonpositive curvature, giving a way of measuring distance between phylogenetic trees and providing some procedures for averaging or combining several trees whose leaves are identical. This geometry also shows which trees appear within a fixed distance of a given tree and enables construction of convex hulls of a set of trees. This geometric model of tree space provides a setting in which questions that have been posed by biologists and statisticians over the last decade can be approached in a systematic fashion. For example, it provides a justification for disregarding portions of a collection of trees that agree, thus simplifying the space in which comparisons are to be made. Introduction Preliminaries Special linear groups: @math Orthogonal groups: @math Symplectic groups: @math Unitary groups: @math Proofs of Theorems 1.1 and 1.1, and of corollaries 1.2-1.4 Permutation group algorithms Concluding remarks References. Consider the problem of choosing a string of actions to optimize an objective function that is string submodular. It was shown in previous papers that the greedy strategy, consisting of a string of actions that only locally maximizes the step-wise gain in the objective function, achieves at least a @math -approximation to the optimal strategy. This paper improves this approximation by introducing additional constraints on curvature, namely, total backward curvature, total forward curvature, and elemental forward curvature. We show that if the objective function has total backward curvature @math , then the greedy strategy achieves at least a @math -approximation of the optimal strategy. If the objective function has total forward curvature @math , then the greedy strategy achieves at least a @math -approximation of the optimal strategy. Moreover, we consider a generalization of the diminishing-return property by defining the elemental forward curvature. We also introduce the notion of string-matroid and consider the problem of maximizing the objective function subject to a string-matroid constraint. We investigate two applications of string submodular functions with curvature constraints: 1) choosing a string of actions to maximize the expected fraction of accomplished tasks; and 2) designing a string of measurement matrices such that the information gain is maximized. Erdős-Ko-Rado (EKR) type theorems yield upper bounds on the sizes of families of sets, subject to various intersection requirements on the sets in the family. Stability versions of such theorems assert that if the size of a family is close to the maximum possible size, then the family itself must be close (in some appropriate sense) to a maximum-sized family. In this paper, we present an approach to obtaining stability versions of EKR-type theorems, via isoperimetric inequalities for subsets of the hypercube. Our approach is rather general, and allows the leveraging of a wide variety of exact EKR-type results into strong stability versions of these results, without going into the proofs of the original results. We use this approach to obtain tight stability versions of the EKR theorem itself and of the Ahlswede-Khachatrian theorem on @math -intersecting families of @math -element subsets of @math (for @math ), and to show that, somewhat surprisingly, all these results hold when the intersection requirement is replaced by a much weaker requirement. Other examples include stability versions of Frankl's recent result on the Erdős matching conjecture, the Ellis-Filmus-Friedgut proof of the Simonovits-Sos conjecture, and various EKR-type results on @math -wise (cross)- @math -intersecting families. We present a monotonic expression for the Ricci flow, valid in all dimensions and without curvature assumptions. It is interpreted as an entropy for a certain canonical ensemble. Several geometric applications are given. In particular, (1) Ricci flow, considered on the space of riemannian metrics modulo diffeomorphism and scaling, has no nontrivial periodic orbits (that is, other than fixed points); (2) In a region, where singularity is forming in finite time, the injectivity radius is controlled by the curvature; (3) Ricci flow can not quickly turn an almost euclidean region into a very curved one, no matter what happens far away. We also verify several assertions related to Richard Hamilton's program for the proof of Thurston geometrization conjecture for closed three-manifolds, and give a sketch of an eclectic proof of this conjecture, making use of earlier results on collapsing with local lower curvature bound. We prove a conjecture raised by the work of Diaconis and Shahshahani (1981) about the mixing time of random walks on the permutation group induced by a given conjugacy class. To do this we exploit a connection with coalescence and fragmentation processes and control the Kantorovitch distance by using a variant of a coupling due to Oded Schramm. Recasting our proof in the language of Ricci curvature, our proof establishes the occurrence of a phase transition, which takes the following form in the case of random transpositions: at time @math , the curvature is asymptotically zero for @math and is strictly positive for @math . Let @math be a geometrically bounded symplectic manifold, @math be a closed, regular coisotropic submanifold, and @math be a Hamiltonian diffeomorphism. The main result of this article is that the number of leafwise fixed points of @math is bounded below by the sum of the Betti numbers of @math , provided that the Hofer distance between @math and the identity is small enough and the pair @math is non-degenerate. As an application, I prove a presymplectic non-embedding result. A version of the Arnold-Givental conjecture for coisotropic submanifolds is also discussed. Abstract We introduce and study rough (approximate) lower curvature bounds for discrete spaces and for graphs. This notion agrees with the one introduced in [J. Lott, C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. 169 (2009), in press] and [K.T. Sturm, On the geometry of metric measure spaces. I, Acta Math. 196 (2006) 65–131], in the sense that the metric measure space which is approximated by a sequence of discrete spaces with rough curvature ⩾ K will have curvature ⩾ K in the sense of [J. Lott, C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. 169 (2009), in press; K.T. Sturm, On the geometry of metric measure spaces. I, Acta Math. 196 (2006) 65–131]. Moreover, in the converse direction, discretizations of metric measure spaces with curvature ⩾ K will have rough curvature ⩾ K . We apply our results to concrete examples of homogeneous planar graphs. We study arithmetic proof systems @math and @math operating with arithmetic circuits and arithmetic formulas, respectively, and that prove polynomial identities over a field @math . We establish a series of structural theorems about these proof systems, the main one stating that @math proofs can be balanced: if a polynomial identity of syntactic degree @math and depth @math has a @math proof of size @math , then it also has a @math proof of size @math in which every circuit has depth @math . As a corollary, we obtain a quasi-polynomial simulation of @math by @math . Using these results we obtain the following: consider the identities @math where @math , and @math are @math square matrices and @math is a triangular matrix with $z_... The following theorem is due to Dilworth [8]: Let P be a partially ordered set. If the maximal number of elements in an independent subset (anti-chain) of P is k, then P is the union of k chains (cliques). In this article we formalize an elegant proof of the above theorem for finite posets by Perles [13]. The result is then used in proving the case of infinite posets following the original proof of Dilworth [8]. A dual of Dilworth’s theorem also holds: a poset with maximum clique m is a union of m independent sets. The proof of this dual fact is considerably easier; we follow the proof by Mirsky [11]. Mirsky states also a corollary that a poset of r × s + 1 elements possesses a clique of size r + 1 or an independent set of size s + 1, or both. This corollary is then used to prove the result of Erdős and Szekeres [9]. Instead of using posets, we drop reflexivity and state the facts about antisymmetric and transitive relations. We study undirected networks with edge costs that satisfy the triangle inequality. Let @math denote the number of nodes. We present an @math -approximation algorithm for a generalization of the metric-cost subset @math -node-connectivity problem. Our approximation guarantee is proved via lower bounds that apply to the simple edge-connectivity version of the problem, where the requirements are for edge-disjoint paths rather than for openly node-disjoint paths. A corollary is that, for metric costs and for each @math , there exists a @math -node connected graph whose cost is within a factor of @math of the cost of any simple @math -edge connected graph. Based on our @math -approximation algorithm, we present an @math -approximation algorithm for the metric-cost node-connectivity survivable network design problem, where @math denotes the maximum requirement over all pairs of nodes. Our results contrast with the case of edge costs of 0 or 1, where Kortsarz, Krauthgamer, and Lee. [SIAM J. Comput., 33 (2004), pp. 704-720] recently proved, assuming NP @math DTIME( @math ), a hardness-of-approximation lower bound of @math for the subset @math -node-connectivity problem, where @math denotes a small positive number. We define a new notion of total curvature, called net total curvature, for finite graphs embedded inR n , and investigate its properties. Two guiding principles ar e given by Milnor's way of measuring the local crookedness of a Jorda n curve via a Crofton-type formula, and by considering the double cover of a given graph as an Eulerian circuit. The strength of combining these ideas in defining the curvature f unctional is (1) it allows us to interpret the singular non-eulidean behavior at the vertices of the graph as a superposition of vertices of a 1-dimensional manifold, and thus (2) one can compute the total curva- ture for a wide range of graphs by contrasting local and global properties of the graph utilizing the integral geometric representation of the cur vature. A collection of results on upper lower bounds of the total curvature on isotopy homeomorphism classes of embed- dings is presented, which in turn demonstrates the effectiveness of net total curvature as a new functional measuring complexity of spatial graphs in differential-geometric terms. Let @math be a smooth scheme over an algebraically closed field @math of characteristic zero and @math a regular function, and write @math Crit @math , as a closed subscheme of @math . The motivic vanishing cycle @math is an element of the @math -equivariant motivic Grothendieck ring @math defined by Denef and Loeser math.AG 0006050 and Looijenga math.AG 0006220, and used in Kontsevich and Soibelman's theory of motivic Donaldson-Thomas invariants, arXiv:0811.2435. We prove three main results: (a) @math depends only on the third-order thickenings @math of @math . (b) If @math is another smooth scheme, @math is regular, @math Crit @math , and @math is an embedding with @math and @math an isomorphism, then @math equals @math "twisted" by a motive associated to a principal @math -bundle defined using @math , where now we work in a quotient ring @math of @math . (c) If @math is an "oriented algebraic d-critical locus" in the sense of Joyce arXiv:1304.4508, there is a natural motive @math , such that if @math is locally modelled on Crit @math , then @math is locally modelled on @math . Using results from arXiv:1305.6302, these imply the existence of natural motives on moduli schemes of coherent sheaves on a Calabi-Yau 3-fold equipped with "orientation data", as required in Kontsevich and Soibelman's motivic Donaldson-Thomas theory arXiv:0811.2435, and on intersections of oriented Lagrangians in an algebraic symplectic manifold. This paper is an analogue for motives of results on perverse sheaves of vanishing cycles proved in arXiv:1211.3259. We extend this paper to Artin stacks in arXiv:1312.0090. We introduce a new class of closed symplectic manifolds called subcritical. These manifolds are closed analogues of subcritical Stein manifolds. We study symplectic and Lagrangian embeddings into such manifolds and into their hyperplane sections. Let (Mn, g) be a compact Riemannian manifold with boundary and dimensionn⩾2. In this paper we discuss the first non-zero eigenvalue problem (1) Problem (1) is known as the Stekloff problem because it was introduced by him in 1902, for bounded domains of the plane. We discuss estimates of the eigenvalueν1in terms of the geometry of the manifold (Mn, g). In the two-dimensional case we generalize Payne's Theorem [P] for bounded domains in the plane to non-negative curvature manifolds. In this case we show thatν1⩾k0, wherekg⩾k0andkgrepresents the geodesic curvature of the boundary. In higher dimensionsn⩾3 for non-negative Ricci curvature manifolds we show thatν1>k0 2, wherek0is a lower bound for any eigenvalue of the second fundamental form of the boundary. We introduce an isoperimetric constant and prove a Cheeger's type inequality for the Stekloff eigenvalue. A set @math is called an antichain (resp. antichain) if it does not contain two distinct elements @math and @math satisfying @math (resp. @math ) for all @math . We show that the Hausdorff dimension of a weak antichain @math in the @math -dimensional unit cube @math is at most @math and that the @math -dimensional Hausdorff measure of @math is at most @math , which are the best possible bounds. This result is derived as a corollary of the following projection inequality , which may be of independent interest: The @math -dimensional Hausdorff measure of a (weak) antichain @math cannot exceed the sum of the @math -dimensional Hausdorff measures of the @math orthogonal projections of @math onto the facets of the unit @math -cube containing the origin. For the proof of this result we establish a discrete variant of the projection inequality applicable to weak antichains in @math and combine it with ideas from geometric measure theory.