metriques de carnot cartheodory et quasiisometries des espaces symetriques de rang un we exhibit a rigidity property of the simple groups sp n 1 and f7 20 which implies mostow rigidity this property does not extend to o n 1 and u n 1 the proof relies on quasiconformal theory applied in the cr setting extensions are given to a class of solvable lie groups as a byproduct a result on quasiisometries of infinite nilpotent groups is obtained dans cet article on etablit une propriete de rigidite des groupes simples de rang un sp n 1 n 2 2 et fqui entraine la rigidite de mostow 1 theoreme toute quasiisornwtrie de 1 espace hyperbolique quaternionien hhn n 2 resp du plan hyperbolique de cayley cah2 est d distance bornee d une isometrie i e differe d une isomnetrie par une application qui diplace les points d une distance bornee une application f entre espaces metriques est une quasiisometrie s il existe des constantes l et c telles que l image de f soit c dense et que pour tous x y c d x y d fx fy ld x y c l une quasiisometrie entre des g et g est une sorte d isomorphisme virtuel dans la categorie topologique en effet cela correspond a une action de g sur un fibre principal c de groupe g sur une base compacte cf ra un isomorphisme entre sous groupes cocompacts covolume fini suffit pour les groupes simples de rang un de groupes de lie s etend en une quasiisometrie des espaces symetriques ou des groupes de lie si celle ci est proche d une isometrie des espaces symetriques resp un isomorphisme des groupes de lie les sous groupes sont conjugues c est la rigidite de mostow m2 la propriete ci dessus ne s etend pas aux groupes o n 1 et u n 1 en effet paragraphe 11 7 ceux ci ont beaucoup de quasiisometries au moins this content downloaded from 207 46 13 92 on sun 20 nov 2016 04 26 46 utc all use subject to http about jstor org terms

the double algebraic view of finite quantum groupoids abstract double algebra is the structure modelled by the properties of the ordinary and the convolution product in hopf algebras weak hopf algebras and hopf algebroids if a frobenius integral is given the hopf algebroids possessing a frobenius integral are precisely the frobenius double algebras in which the two multiplications satisfy distributivity the double algebra approach makes it manifest that all comultiplications in such measured hopf algebroids are of the abrams kadison type i e they come from a frobenius algebra structure in some bimodule category antipodes for double algebras correspond to the connes moscovici deformed antipode as we show by discussing hopf and weak hopf algebras from the double algebraic point of view frobenius algebra extensions provide further examples that need not be distributive

on the spectral and combinatorial structure of 2d positive systems abstract the dynamics of a 2d positive system depends on the pair of nonnegative square matrices that provide the updating of its local states in this paper several spectral properties such as finite memory separability and property l which depend on the characteristic polynomial of the pair are investigated under the nonnegativity constraint and in connection with the combinatorial structure of the matrices some aspects of the perron frobenius theory are extended to the 2d case in particular conditions are provided guaranteeing the existence of a common maximal eigenvector for two nonnegative matrices with irreducible sum finally some results on 2d positive realizations are presented

singularities of j holomorphic curves let v j be an almost complex manifold recall that a map from a riemann surface to v is j holomorphic if its derivative is complex linear a j holomorphic curve or j curve for short is the image of such a map nowhere locally constant the study of singularities of j holomorphic curves began with the statement of m gromov g about the positivity of intersections in the case when v is of dimension 4 the homological intersection of two compact j curves is a sum of positive local contributions and a similar property holds for the self intersection of one curve adjunction formula this property is crucial for applications of the theory to symplectic topology in dimension 4 and contact topology in dimension 3 gromov mcduff eliashberg hofer the positivity of intersections was proved by d mcduff mcd1 see also chapter vi in al then in mcd2 she proved that in any dimension a germ of j holomorphic map c 0 v v0 is topologically equivalent to a standard holomorphic germ from c 0 to cn 0 this was then strengthened to a c 1equivalence by m micallef and b white mw as a special case of a much more general result on singularities of generalized minimal surfaces they proved also that it applies to reducible germs

the asymptotics of ech capacities in a previous paper the second author used embedded contact homology ech of contact three manifolds to define ech capacities of four dimensional symplectic manifolds in the present paper we prove that for a four dimensional liouville domain with all ech capacities finite the asymptotics of the ech capacities recover the symplectic volume this follows from a more general theorem relating the volume of a contact three manifold to the asymptotics of the amount of symplectic action needed to represent certain classes in ech the latter theorem was used by the first and second authors to show that every contact form on a closed three manifold has at least two embedded reeb orbits

finite energy foliations of tight three spheres and hamiltonian dynamics surfaces of sections are a classical tool in the study of 3 dimensional dynamical systems their use goes back to the work of poincare and birkhoff in the present paper we give a natural generalization of this concept by constructing a system of transversal sections in the complement of finitely many distinguished periodic solutions such a system is established for nondegenerate reeb flows on the tight 3 sphere by means of pseudoholomorphic curves the applications cover the nondegenerate geodesic flows on t1s2 rp3 via its double covering s3 and also nondegenerate hamiltonian systems in ir4 restricted to sphere like energy surfaces of contact type

the dynamics on three dimensional strictly convex energy surfaces we show that a hamiltonian flow on a three dimensional strictly convex energy surface s c r4 possesses a global surface of section of disc type it follows in particular that the number of its periodic orbits is either 2 or oc by a recent result of j franks on area preserving homeomorphisms of an open annulus in the plane the construction of this surface of section is based on partial differential equations of cauchy riemann type for maps from punctured riemann surfaces into r x s3 equipped with special almost complex structures

weak hopf bialgebras in this chapter the r bialgebroids and hopf algebroids of the previous chapter are investigated further in the particular case when the base algebra r carries a so called separable frobenius structure separable frobenius structures on some algebra r are shown to correspond to separable frobenius structures on the forgetful functor from the category of r bimodules to the category of vector spaces based on that the bijection of chap 5 is refined to bijections between three structures for any algebra a first monoidal structures on the category of a modules together with separable frobenius structures on the forgetful functor to the category of vector spaces second bialgebroid structures on a over some base algebra r together with separable frobenius structures on r finally weak bialgebra structures on a the r bialgebroid a is a hopf algebroid if and only if the corresponding weak bialgebra a is a weak hopf algebra

on the structure of finite memory and separable two dimensional systems abstract several characterizations of finite memory and separability properties for a two dimensional system are presented in terms of both the characteristic polynomial and the matrix pair that describes the state evolution necessary and sufficient conditions for a finite memory or separable two dimensional system to have an inverse with the same properties are given these involve only the structure of the transfer matrix

the unregularized gradient flow of the symplectic action the symplectic action can be defined on the space of smooth paths in a symplectic manifold p which join two lagrangian submanifolds of p to pursue a new approach to the variational theory of this function we define on a subset of the path space the flow whose trajectories are given by the solutions of the cauchy riemann equation with respect to a suitable almost complex structure on p in particular we prove compactness and transversality results for the set of bounded trajectories

on the structure of weak hopf algebras we study the group of group like elements of a weak hopf algebra and derive an analogue of radford s formula for the fourth power of the antipode s which implies that the antipode has a finite order modulo a trivial automorphism we find a sufficient condition in terms of tr s2 for a weak hopf algebra to be semisimple discuss relation between semisimplicity and cosemisimplicity and apply our results to show that a dynamical twisting deformation of a semisimple hopf algebra is cosemisimple

pseudoholomorphic curves in symplectizations with applications to the weinstein conjecture in dimension three l in t roduct ion 515 1 1 the weinstein conjecture 515 1 2 the main results 517 1 3 sketch of the proof 522 2 local f redho lm theory 525 2 1 similar i ty principle and consequences 525 2 2 n m as a lmost complex manifold and local fillings 529 2 3 local propert ies of ho lomorphic curves 531 3 bubbl ing off analysis and global fredholm theory 533 3 l grad ien t bounds imply ca bounds 533 3 2 bubbing off analysis of ho lomorphic planes 534 3 3 grad ien t bounds near the boundary 537 4 proof of the main results 544 4 1 set up for a global fredholm theory 544 4 2 bishop s theorem 545 4 3 proof of theorem 5 546 4 4 proof of theorems 8 and 11 547 4 5 uniqueness results for foliations 547 5 appendix 550 5 1 per turbing surfaces 550 5 2 ki l l ing crit ical points of hypersurfaces 552 5 3 adding a thin tube to a sphere 560

weak hopf algebras i integral theory and c structure we give an introduction to the theory of weak hopf algebras proposed as a coassociative alternative of weak quasi hopf algebras we follow an axiomatic approach keeping as close as possible to the classical theory of hopf algebras the emphasis is put on the new structure related to the presence of canonical subalgebras al and ar in any weak hopf algebra a that play the role of non commutative numbers in many respects a theory of integrals is developed in which we show how the algebraic properties of a such as the frobenius property or semisimplicity or innerness of the square of the antipode are related to the existence of non degenerate normalized or haar integrals in case of c weak hopf algebras we prove the existence of a unique haar measure h a and of a canonical grouplike element g a implementing the square of the antipode and factorizing into left and right elements g glg 1r gl al gr ar further discussion of the c case will be presented in part ii

transversality in elliptic morse theory for the symplectic action our goal in this paper is to settle some transversality question for the perturbed nonlinear cauchy riemann equations on the cylinder these results play a central role in the denition of symplectic floer homology and hence in the proof of the arnold conjecture there is currently no other reference to these transversality results in the open literature our approach does not require aronszajn s theorem instead we derive the unique continuation theorem from a generalization of the carleman similarity principle

a c 2 smooth counterexample to the hamiltonian seifert conjecture in r 4 we construct a proper c 2 smooth function on r 4 such that its hamiltonian flow has no periodic orbits on at least one regular level set this result can be viewed as a c 2 smooth counterexample to the hamiltonian seifert conjecture in dimension four

on the category of weak bialgebras abstract weak hopf bialgebras are described as hopf bimonoids in appropriate duoidal also known as 2 monoidal categories this interpretation is used to define a category wba of weak bialgebras over a given field as an application the free vector space functor from the category of small categories with finitely many objects to wba is shown to possess a right adjoint given by taking certain group like elements this adjunction is proven to restrict to the full subcategories of groupoids and of weak hopf algebras respectively as a corollary we obtain equivalences between the category of small categories with finitely many objects and the category of pointed cosemisimple weak bialgebras and between the category of small groupoids with finitely many objects and the category of pointed cosemisimple weak hopf algebras

larson sweedler theorem and the role of grouplike elements in weak hopf algebras abstract we extend the larson sweedler theorem amer j math 91 1969 75 to weak hopf algebras by proving that a finite dimensional weak bialgebra is a weak hopf algebra iff it possesses a non degenerate left integral we show that the category of modules over a weak hopf algebra is autonomous monoidal with semisimple unit and invertible modules we also reveal the connection of invertible modules to left and right grouplike elements in the dual weak hopf algebra defining distinguished left and right grouplike elements we derive the radford formula amer j math 98 1976 333 for the fourth power of the antipode in a weak hopf algebra and prove that the order of the antipode is finite up to an inner automorphism by a grouplike element in the trivial subalgebra a t of the underlying weak hopf algebra a

on the geometric rank of homogeneous spaces of nonpositive curvature in this paper we investigate differential geometric properties of homogeneous hadamard manifolds and prove a rigidity result in this class of spaces a hadamard manifold h is a simply connected complete riemannian manifold of nonpositive curvature if in addition h is homogeneous then by the work of all aw1 aw2 and hei we know that h is isometric to a solvable lie group s a n with lie algebra s of type nc see aw1 and a left invariant metric here a exp s s denotes an abelian complement to the nilpotent commutator subgroup n exp s s assuming that h has a trivial flat de rham factor the number k dim a only depends on h whereas s need not be uniquely determined and is called the aloebraic rank of h k is an upper bound for the oeometric rank r rk h dr3 i e the maximal integer such that every geodesic is contained in a r fiat a totally geodesic copy of in h the geometric meaning of rk m is well understood for a huge class of hadamard manifolds m thanks to the following remarkable result by ballmann and burns spatzier we quote a generalisation from e3 and eh

generalized counterexamples to the seifert conjecture using the theory of plugs and the self insertion construction due to the second author we prove that a foliation of any codimension of any manifold can be modified in a real analytic or piecewise linear fashion so that all minimal sets have codimension 1 in particular the 3 sphere s 3 has a real analytic dynamical system such that all limit sets are 2 dimensional we also prove that a 1 dimensional foliation of a manifold of dimension at least 3 can be modified in a piecewise linear fashion so that there are no closed leaves but all minimal sets are 1 dimensional these theorems provide new counterexamples to the seifert conjecture which asserts that every dynamical system on s 3 with no singular points has a periodic trajectory

on fusion categories using a variety of methods developed in the literature in particular the theory of weak hopf algebras we prove a number of general results about fusion categories in characteristic zero we show that the global dimension of a fusion category is always positive and that the s matrix of any not necessarily hermitian modular category is unitary we also show that the category of module functors between two module categories over a fusion category is semisimple and that fusion categories and tensor functors between them are undeformable generalized ocneanu rigidity in particular the number of such categories functors realizing a given fusion datum is finite finally we develop the theory of frobenius perron dimensions in an arbitrary fusion category at the end of the paper we generalize some of these results to positive characteristic

holomorphic disks and topological invariants for closed three manifolds the aim of this article is to introduce certain topological invariants for closed oriented three manifolds y equipped with a spiny structure given a heegaard splitting of y u 0o u u 1 these theories are variants of the lagrangian floer homology for the g fold symmetric product of relative to certain totally real subspaces associated to u 0 and u 1

algebraic aspects of two dimensional convolutional codes two dimensional 2d codes are introduced as linear shift invariant spaces of admissible signals on the discrete plane convolutional and in particular basic codes are characterized both in terms of their internal properties and by means of their input output representations the algebraic structure of the class of all encoders that correspond to a given convolutional code is investigated and the possibility of obtaining 2d decoders free from catastrophic errors as veil as efficient syndrome decoders is considered some aspects of the state space implementation of 2d encoders and decoders via finite memory 2d system are discussed

taubes s proof of the weinstein conjecture in dimension three does every smooth vector field on a closed three manifold for example the three sphere have a closed orbit no according to counterexamples by k kuperberg and others on the other hand there is a special class of vector fields called reeb vector fields which are associated to contact forms the three dimensional case of the weinstein conjecture asserts that every reeb vector field on a closed oriented three manifold has a closed orbit this conjecture was recently proved by taubes using seiberg witten theory we give an introduction to the weinstein conjecture the main ideas in taubes s proof and the bigger picture into which it fits taubes s proof of the weinstein conjecture is the culmination of a large body of work both by taubes and by others in an attempt to make this story accessible to nonspecialists much of the present article is devoted to background and context and taubes s proof itself is only partially explained hopefully this article will help prepare the reader to learn the full story from taubes s paper 62 more exposition of this subject which was invaluable in the preparation of this article can be found in the online video archive from the june 2008 msri hot topics workshop 44 and in the article by auroux 5 below in 1 3 we introduce the statement of the weinstein conjecture and discuss some examples in 4 6 we discuss a natural strategy for approaching the weinstein conjecture which proves it in many but not all cases and provides background for taubes s work in 7 we give an overview of the big picture surrounding taubes s proof of the weinstein conjecture readers who already have some familiarity with the weinstein conjecture may wish to start here in 8 9 we recall necessary material from seiberg witten theory in 10 we give an outline of taubes s proof and in 11 we explain some more details of it to conclude in 12 we discuss some further results and open problems related to the weinstein conjecture 1 statement of the weinstein conjecture the weinstein conjecture asserts that certain vector fields must have closed orbits before stating the conjecture at the end of this section we first outline its origins this is discussion is only semi historical because only a sample of the relevant works will be cited and not always in chronological order 1 1 closed orbits of vector fields let y be a closed manifold in this article all manifolds and all objects defined on them are smooth unless otherwise stated 2000 mathematics subject classification 57r17 57r57 53d40 partially supported by nsf grant dms 0806037

geometric flow and rigidity on symmetric spaces of noncompact type in this paper we show that under a suitable condition every nonsingular geometric flow on a manifold which is modeled on the furstenberg boundary of x where x is a symmetric space of non compact type induces a torus action and in particular if the manifold is a rational homology sphere then the flow has a closed orbit

holomorphic triangles and invariants for smooth four manifolds abstract in earlier articles the authors introduced invariants for closed oriented three manifolds defined using a variant of lagrangian floer homology in the symmetric products of riemann surfaces the aim of this article is to introduce invariants of oriented smooth four manifolds built using these floer homology groups this four dimensional theory also endows the corresponding three dimensional theories with additional structure an absolute grading of certain of its floer homology groups

an index for closed orbits in beltrami fields abstract we consider the class of beltrami fields eigenfields of the curl operator on three dimensional riemannian solid tori such vector fields arise as steady incompressible inviscid fluids and plasmas using techniques from contact geometry we construct an integer valued index for detecting closed orbits in the flow which are topologically inessential they have winding number zero with respect to the solid torus the most important feature of the index is its computability it can be rigorously determined from a c 1 approximation to the vector field on any meridional disc of the solid torus as a result we obtain a test for detecting the non existence of a global cross section to the 3d flow via purely 2d information