self dual instantons and holomorphic curves a gradient flow of a morse function on a compact riemannian manifold is said to be of morse smale type if the stable and unstable manifolds of any two critical points intersect transversally for such a morse smale gradient flow there is a chain complex generated by the critical points and graded by the morse index the boundary operator has as its x y entry the number of gradient flow lines running from x to y counted with appropriate signs whenever the difference of the morse indices is 1 the homology of this chain complex agrees with the homology of the underlying manifold m and this can be used to prove the morse inequalities cf 33 26 around 1986 floer generalized this idea to infinite dimensional variational problems in which every critical point has infinite morse index but the moduli spaces of connecting orbits form finite dimensional manifolds for every pair of critical points the dimensions of these spaces give rise to a relative morse index and the boundary operator is defined by counting connecting

sharp systolic inequalities for reeb flows on the three sphere the systolic ratio of a contact form alpha on the three sphere is the quantity begin aligned rho mathrm sys alpha frac t min alpha 2 mathrm vol s 3 alpha wedge d alpha end aligned where t min alpha is the minimal period of closed reeb orbits on s 3 alpha a zoll contact form is a contact form such that all the orbits of the corresponding reeb flow are closed and have the same period our first main result is that rho mathrm sys le 1 in a neighbourhood of the space of zoll contact forms on s 3 with equality holding precisely at zoll contact forms this implies a particular case of a conjecture of viterbo a local middle dimensional non squeezing theorem and a sharp systolic inequality for finsler metrics on the two sphere which are close to zoll ones our second main result is that rho mathrm sys is unbounded from above on the space of tight contact forms on s 3

poincare birkhoff theorems in random dynamics we propose a generalization of the poincare birkhoff theorem on area preserving twist maps to area preserving twist maps f that are random with respect to an ergodic probability measure in this direction we will prove several theorems concerning existence density and type of the fixed points to this end first we introduce a randomized version of generalized generating functions and verify the correspondence between its critical points and the fixed points of f a fact which we successively exploit in order to prove the theorems the study we carry out needs to combine probabilistic techniques with methods from nonlinear pde and from differential geometry notably moser s method and conley zehnder theory our stochastic model in the periodic case coincides with the classical setting of the poincare birkhoff theorem

several expressions for the generalized drazin inverse of a block matrix in a banach algebra we present some new representations for the generalized drazin inverse of a block matrix with generalized schur complement being generalized drazin invertible in a banach algebra under conditions weaker than those used in recent papers on the subject

thin instantons in g_2 manifolds and seiberg witten invariants for two nearby disjoint coassociative submanifolds c and c in a g 2 manifold we construct thin instantons with boundaries lying on c and c from regular j holomorphic curves in c we explain their relationship with the seiberg witten invariants for c

drazin inverse of partitioned matrices in terms of banachiewicz schur forms abstract let m a b c d be a partitioned matrix where a and d are square matrices denote the drazin inverse of a by a d the purpose of this paper is twofold firstly we develop conditions under which the drazin inverse of m having generalized schur complement s d ca d b group invertible can be expressed in terms of a matrix in the banachiewicz schur form and its powers secondly we deal with partitioned matrices satisfying rank m rank a d rank s d and give conditions under which the group inverse of m exists and a formula for its computation

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

multiple closed geodesics on 3 spheres abstract this paper is devoted to a study on closed geodesics on finsler and riemannian spheres we call a prime closed geodesic on a finsler manifold rational if the basic normal form decomposition cf y long bott formula of the maslov type index theory pacific j math 187 1999 113 149 of its linearized poincare map contains no 2 2 rotation matrix with rotation angle which is an irrational multiple of or irrational otherwise we prove that if there exists only one prime closed geodesic on a d dimensional irreversible finsler sphere with d 2 it cannot be rational then we further prove that there exist always at least two distinct prime closed geodesics on every irreversible finsler 3 dimensional sphere our method yields also at least two geometrically distinct closed geodesics on every reversible finsler as well as riemannian 3 dimensional sphere we prove also such results hold for all compact simply connected 3 dimensional manifolds with irreversible or reversible finsler as well as riemannian metrics

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

periodic approximations of irrational pseudo rotations using pseudoholomorphic curves we prove that every c 1 smooth area preserving dieomorphism of the closed 2 disk having not more than one periodic point is the uniform limit of periodic c 1 smooth dieomorphisms in particular every smooth irrational pseudo rotation can be c 0 approximated by integrable systems this partially answers a long standing question of a katok regarding zero entropy hamiltonian systems in low dimensions our approach uses pseudoholomorphic curve techniques from symplectic geometry

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

full constraint satisfaction problems feder and vardi have conjectured that all constraint satisfaction problems to a fixed structure constraint language are polynomial or np complete this so called dichotomy conjecture remains open although it has been proved in a number of special cases most recently bulatov has verified the conjecture for conservative structures i e structures which contain all possible unary relations we explore three different implications of bulatov s result first the above dichotomy can be extended to so called inclusive structures corresponding to conservative constraint satisfaction problems in which each variable comes with its own domain this has also been independently observed by bulatov we prove a more general version extending the dichotomy to so called three inclusive structures i e structures which contain with any unary relation r all unary relations r for subsets r subseteq r with at most three elements for the constraint satisfaction problems in this generalization we must restrict the instances to so called 1 full structures in which each variable is involved in a unary constraint this leads to our second focus which is on restrictions to more general kinds of full input structures for any set w of positive integers we consider a restriction to w full input structures i e structures in which for each w in w any w variables are involved in a w ary constraint we identify a class of structures the so called w set full structures for which the restriction to w full input structures does not change the complexity of the constraint satisfaction problem and hence the family of these restricted problems also exhibits dichotomy the general family of three inclusive constraint satisfaction problems restricted to w full input structures contains examples which we cannot seem to prove either polynomial or np complete nevertheless we are able to use our result on the dichotomy for three inclusive constraint satisfaction problems to deduce the fact that all three inclusive constraint satisfaction problems restricted to w full input structures are np complete or quasi polynomial of order n o log n our third focus deals with bounding the number of occurrences of a variable which we call the degree we conjecture that the complexity classification of three inclusive constraint satisfaction problems extends to the case where all degrees are bounded by three using previous results we are able to verify this conjecture in a number of special cases conservative inclusive and three inclusive constraint satisfaction problems can be viewed as problems in which each variable is restricted to a list of allowed values this point of view of lists is frequently encountered in the study of graph colorings graph homomorphisms and graph partitions our results presented here in all three areas were strongly motivated by these results on graphs

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

a local existence result for poincare einstein metrics abstract given a closed riemannian manifold m g m of dimension n 3 we prove the existence of a conformally compact einstein metric g defined on a collar neighborhood m 0 1 whose conformal infinity is g m

morse theory for lagrangian intersections soit p une variete symplectique compacte et soit l p une sous variete lagrangienne avec 2 p l 0 pour un diffeomorphisme exact de p avec la propriete que l coupe l transversalement on demontre une inegalite de morse reliant l ensemble l l a la cohomologie de l

the second closed geodesic on finsler spheres of dimension n 2 we show the existence of at least two geometrically distinct closed geodesics on an n dimensional sphere with a bumpy and non reversible finsler metric for n 2 2000 msc classification 53c22 53c60 58e10

polyfolds cobordisms and the strong weinstein conjecture abstract we prove the strong weinstein conjecture for closed contact manifolds that appear as the concave boundary of a symplectic cobordism admitting an essential local foliation by holomorphic spheres

c genericity of positive topological entropy for geodesic flows on s2 we show that there is a c open and dense set of positively curved metrics on s2 whose geodesic flow has positive topological entropy and thus exhibits chaotic behavior the geodesic flow for each of these metrics possesses a horseshoe and it follows that these metrics have an exponential growth rate of hyperbolic closed geodesics the positive curvature hypothesis is required to ensure the existence of a global surface of section for the geodesic flow our proof uses a new and general topological criterion for a surface diffeomorphism to exhibit chaotic behavior very shortly after this manuscript was completed the authors learned about remarkable recent work by hofer wysocki and zehnder 14 15 on three dimensional reeb flows in the special case of geodesic flows on s2 they show that if the geodesic flow has no parabolic closed geodesics this holds for an open and c dense set of riemannian metrics on s2 then it possesses either a global surface of section or a heteroclinic orbit it then immediately follows from the proof of our main theorem that there is a c open and dense set of riemannian metrics on s2 whose geodesic flow has positive topological entropy this concludes a program to show that every orientable compact surface has a c open and dense set of riemannian metrics whose geodesic flow has positive topological entropy

generalized inverses of partitioned matrices in banachiewicz schur form abstract the problem of developing conditions under which generalized inverses of a partitioned matrix can be expressed in the so called banachiewicz schur form is reconsidered theorem of marsaglia and styan sankhy ser a 36 1974 437 concerning the class of all generalized inverses the class of reflexive generalized inverses and the moore penrose inverse is strengthened and new results are established for the classes of outer inverses least squares generalized inverses and minimum norm generalized inverses

the complexity of approximating bounded degree boolean csp the degree of a csp instance is the maximum number of times that any variable appears in the scopes of constraints we consider the approximate counting problem for boolean csp with bounded degree instances for constraint languages containing the two unary constant relations 0 and 1 when the maximum allowed degree is large enough at least 6 we obtain a complete classification of the complexity of this problem it is exactly solvable in polynomial time if every relation in the constraint language is affine it is equivalent to the problem of approximately counting independent sets in bipartite graphs if every relation can be expressed as conjunctions of 0 1 and binary implication otherwise there is no fpras unless np rp for lower degree bounds additional cases arise where the complexity is related to the complexity of approximately counting independent sets in hypergraphs

formulae for the generalized drazin inverse of a block matrix in terms of banachiewicz schur forms abstract we introduce new expressions for the generalized drazin inverse of a block matrix with the generalized schur complement being generalized drazin invertible in a banach algebra under some conditions we generalized some recent results for drazin inverse and group inverse of complex matrices

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

expression of the drazin and mp inverse of partitioned matrix and quotient identity of generalized schur complement in this paper we give representations of the drazin and mp inverse of a 2x2 block matrix and quotient identities for the generalized schur complement of a partitioned 3x3 matrix under conditions different than those used in recent papers on the subject we present numerical examples to illustrate our results

fredholm theory and transversality for noncompact pseudoholomorphic maps in symplectizations we study pseudoholomorphic maps from a punctured riemann surface into the symplectization of a contact manifold a fredholm theory yields the virtual dimension of the moduli spaces of such maps in terms of the euler characteristic of the riemann surface and the asymptotics data given by the periodic solutions of the reeb vector field associated to the contact form the transversality results establish the existence of additional structure for these spaces to be more precise we prove that these spaces are generically smooth manifolds and therefore their virtual dimension coincides with their actual dimension

an instanton invariant for 3 manifolds to an oriented closed 3 dimensional manifoldm withh1 m 0 we assign a 8 graded homology groupi m whose euler characteristic is twice casson s invariant the definition uses a construction on the space of instantons onm

geodesics on s2 and periodic points of annulus homeomorphisms we show that an area preserving homeomorphism of the open or closed annulus which has at least one periodic point must in fact have infinitely many interior periodic points a consequence is the theorem that every smooth riemannian metric ons2 with positive gaussian curvature has infinitely many distinct closed geodesics

on disjoint range operators in a hilbert space for a bounded linear operator m in a hilbert space h various relations among the ranges r m r m r m m and the null spaces n m n m are considered from the point of view of their relations to the known classes of operators such as ep co ep weak ep gp dr or sr particular attention is paid to the range projectors of the operators m m and some further characteristics of these projectors are derived as well

more on generalized inverses of partitioned matrices with banachiewicz schur forms abstract necessary and sufficient conditions are derived for a 2 by 2 partitioned matrix to have 1 1 2 1 3 1 4 inverses and the moore penrose inverse with banachiewicz schur forms as applications the banachiewicz schur forms of 1 1 2 1 3 1 4 inverses and the moore penrose inverse of a 2 by 2 partitioned hermitian matrix are also given

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

a dynamical characterization of universally tight lens spaces we give necessary and sufficient conditions for a closed connected co orientable contact 3 manifold m to be a standard lens space based on assumptions on the reeb flow associated to a defining contact form our methods also provide rational global surfaces of section for non degenerate reeb flows on l p q std with prescribed binding orbits

closure properties of constraints many combinatorial search problems can be expressed as constraint satisfaction problems and this class of problems is known to be np complete in general in this paper we investigate the subclasses that arise from restricting the possible constraint types we first show that any set of constraints that does not give rise to an np complete class of problems must satisfy a certain type of algebraic closure condition we then investigate all the different possible forms of this algebraic closure property and establish which of these are sufficient to ensure tractability as examples we show that all known classes of tractable constraints over finite domains can be characterized by such an algebraic closure property finally we describe a simple computational procedure that can be used to determine the closure properties of a given set of constraints this procedure involves solving a particular constraint satisfaction problem which we call an indicator problem

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

a note on the drazin inverses with banachiewicz schur forms let h be a hilbert space m the closed subspace of h with orthocomplement m according to the orthogonal decomposition email protected m every operator email protected b h can be written in a block form m abcd in this note we give necessary and sufficient conditions for a partitioned operator matrix m to have the drazin inverse with banachiewicz schur form in addition this paper investigates the relations among the drazin inverse the moore penrose inverse and the group inverse when they can be expressed in the banachiewicz schur forms

expressions for the moore penrose inverse of block matrices involving the schur complement abstract we present a formula for the moore penrose inverse of a matrix of the form m x n y where x and y are nonsingular based on this result we develop explicit expressions for the moore penrose inverse of a 2 2 complex block matrix m we recover the cases when a schur complement in m is either equal to zero or nonsingular some applications to banded matrices and bordered matrices are indicated

a dichotomy theorem for the approximate counting of complex weighted bounded degree boolean csps we determine the computational complexity of approximately counting the total weight of variable assignments for every complex weighted boolean constraint satisfaction problem or csp with any number of additional unary i e arity 1 constraints particularly when degrees of input instances are bounded from above by a fixed constant all degree 1 counting csps are obviously solvable in polynomial time when the instance s degree is more than two we present a dichotomy theorem that classifies all counting csps admitting free unary constraints into exactly two categories this classification theorem extends to complex weighted problems an earlier result on the approximation complexity of unweighted counting boolean csps of bounded degree the framework of the proof of our theorem is based on a theory of signature developed from valiant s holographic algorithms that can efficiently solve seemingly intractable counting csps despite the use of arbitrary complex weight our proof of the classification theorem is rather elementary and intuitive due to an extensive use of a novel notion of limited t constructibility for the remaining degree 2 problems in contrast they are as hard to approximate as holant problems which are a generalization of counting csps

strongly fillable contact manifolds and j holomorphic foliations we prove that every strong symplectic filling of a planar contact manifold admits a symplectic lefschetz fibration over the disk and every strong filling of t 3 similarly admits a lefschetz fibration over the annulus it follows that strongly fillable planar contact structures are also stein fillable and all strong fillings of t 3 are equivalent up to symplectic deformation and blowup these constructions result from a compactness theorem for punctured j holomorphic curves that foliate a convex symplectic manifold we use it also to show that the compact v supported symplectomorphism group on t t 2 is contractible and to define an obstruction to strong fillability that yields a non gauge theoretic proof of gay s recent nonfillability result g for contact manifolds with positive giroux torsion

new representations of the moore penrose inverse of 2 2 block matrices abstract this paper presents a full rank factorization of a 2 2 block matrix without any restriction applying this factorization we obtain an explicit representation of the moore penrose inverse in terms of four individual blocks of the partitioned matrix which simplifies results of hung and markham 1975 12 and miao 1991 16 we also derive some important coincidence theorems including in the expressions of the moore penrose inverse of a sum of matrices all these results extend earlier works by various authors

enumerative geometry of calabi yau 4 folds gromov witten theory is used to define an enumerative geometry of curves in calabi yau 4 folds the main technique is to find exact solutions to moving multiple cover integrals the resulting invariants are analogous to the bps counts of gopakumar and vafa for calabi yau 3 folds we conjecture the 4 fold invariants to be integers and expect a sheaf theoretic explanation

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

the computational structure of monotone monadic snp and constraint satisfaction a study through datalog and group theory this paper starts with the project of finding a large subclass of np which exhibits a dichotomy the approach is to find this subclass via syntactic prescriptions while the paper does not achieve this goal it does isolate a class of problems specified by monotone monadic snp without inequality which may exhibit this dichotomy we justify the placing of all these restrictions by showing essentially using ladner s theorem that classes obtained by using only two of the above three restrictions do not show this dichotomy we then explore the structure of this class we show that all problems in this class reduce to the seemingly simpler class csp we divide csp into subclasses and try to unify the collection of all known polytime algorithms for csp problems and extract properties that make csp problems np hard this is where the second part of the title a study through datalog and group theory comes in we present conjectures about this class which would end in showing the dichotomy

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

closed geodesics on positively curved finsler spheres in this paper we prove that for every finsler n sphere sn f for n3 with reversibility and flag curvature k satisfying either there exist infinitely many prime closed geodesics or there exists one elliptic closed geodesic whose linearized poincare map has at least one eigenvalue which is of the form exp pi with an irrational furthermore there always exist three prime closed geodesics on any s3 f satisfying the above pinching condition

fanout limitations on constraint systems we examine several families of constraint systems constraint satisfaction and network stability are well understood network stability was introduced in an attempt to understand the role of fanout we introduce two new families of constraint systems in a new attempt to limit fanout these two families are graph constraint satisfaction and bipartite constraint satisfaction we obtain the following results 1 an extension of schaefer s classification argument for boolean satisfiability to boolean graph constraint satisfaction and bipartite constraint satisfaction 2 a characterization of the graph constraint satisfaction problems that lack fanout in terms of a new problem the generalized matroid parity problem which gives rise to many new problems whose complexity is yet to be determined in particular the coindependent set problem is shown to be polynomial and 3 generalized matroid intersection characterizes the lack of fanout in the bipartite case with a family of matroid intersection problems as a special polynomially solvable subcase

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

stability of closed geodesics on finsler 2 spheres abstract in this paper we prove that on every finsler 2 dimensional sphere either there exist infinitely many prime closed geodesics or there exist at least two irrationally elliptic prime closed geodesics