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

quantum cohomology of flag manifolds and toda lattices we discuss relations of vafa s quantum cohomology with floer s homology theory introduce equivariant quantum cohomology formulate some conjectures about its general properties and on the basis of these conjectures compute quantum cohomology algebras of the flag manifolds the answer turns out to coincide with the algebra of regular functions on an invariant lagrangian variety of a toda lattice

on the maslov index in this paper we give four definitions of maslov index and show that they all satisfy the same system of axioms and hence are equivalent to each other moreover relationships of several symplectic and differential geometric analytic and topological invariants including triple maslov indices eta invariants spectral flow and signatures of quadratic forms to the maslov index are developed and formulae relating them are given the broad presentation is designed with a view to applications both in geometry and in analysis 1994 john wiley sons inc

a symplectic fixed point theorem for cpn two symplectic diffeomorphisms 0 1 of a symplectic manifold x are said to be homologous if there exists a smooth homotopy 1 t 0 1 of symplectic diffeomorphisms between them such that the timedependent vector field t defined byd dt t t o t is a globally hamiltonian vector field for allt i e there exists a smooth real valued timedependent hamiltonian functionh x t onx x 0 1 such that t dh t whereht h x t

symplectic topology and hamiltonian dynamics on etudie des applications symplectiques non lineaires capacites symplectiques construction d une capacite symplectique problemes de plongement problemes de rigidite

on the obstructed lagrangian floer theory abstract lagrangian floer homology in a general case has been constructed by fukaya oh ohta and ono where they construct an a algebra or an a bimodule from lagrangian submanifolds they developed obstruction and deformation theories of the lagrangian floer homology theory but for obstructed lagrangian submanifolds the standard lagrangian floer homology cannot be defined we explore several well known homology theories on these a objects which are hochschild and cyclic homology for an a objects and chevalley eilenberg or cyclic chevalley eilenberg homology for their underlying l objects we show that these homology theories are well defined and invariant even in the obstructed cases due to the existence of m 0 the standard homological algebra does not work and we develop analogous homological algebra over novikov fields we provide computations of these homology theories in some cases we show that for an obstructed a algebra with a non trivial primary obstruction chevalley eilenberg floer homology vanishes whose proof is inspired by the comparison with cluster homology theory of lagrangian submanifolds by cornea and lalonde in contrast we also provide an example of an obstructed case whose cyclic floer homology is non vanishing

lagrangian floer theory on compact toric manifolds i we introduced the notion of weakly unobstructed lagrangian submanifolds and constructed their potential function po purely in terms of a model data in fooo3 in this article we carry out explicit calculations involving po on toric manifolds and study the relationship between this class of lagrangian submanifolds with the earlier work of givental g1 which advocates that the quantum cohomology ring is isomorphic to the jacobian ring of a certain function called the landau ginzburg superpotential combining this study with the results from fooo3 we also apply the study to various examples to illustrate its implications to symplectic topology of lagrangian fibers of toric manifolds in particular we relate it to the hamiltonian displacement property of lagrangian fibers and to entov polterovich s symplectic quasi states

stability conditions on an singularities we study the spaces of locally finite stability conditions on the derived categories of coherent sheaves on the minimal resolutions of a n singularities supported at the exceptional sets our main theorem is that they are connected and simply connected the proof is based on the study of spherical objects in a ishii and h uehara autoequivalences of derived categories on the minimal resolutions of a n singularities on surfaces j differential geom 71 3 385 435 2005 and the homological mirror symmetry for a n singularities

multiplicity of lagrangian orbits on symmetric star shaped hypersurfaces abstract in this paper the multiplicity of lagrangian orbits on c 2 smooth compact symmetric star shaped hypersurfaces with respect to the origin in r 2 n is studied we prove an infinitely many existence result via z 2 index theory this is a multiplicity result about the arnold chord in some sense

homotopy equivalence of nearby lagrangians and the serre spectral sequence we construct using relatively basic techniques a spectral sequence for exact lagrangians in cotangent bundles similar to the one constructed by fukaya seidel and smith that spectral sequence was used to prove that exact relative spin lagrangians in simply connected cotangent bundles with vanishing maslov class are homology equivalent to the base a similar result was also obtained by nadler the ideas in that paper were extended by abouzaid who proved that vanishing maslov class alone implies homotopy equivalence in this paper we present a short proof of the fact that any exact lagrangian with vanishing maslov class is homology equivalent to the base and that the induced map on fundamental groups is an isomorphism when the fundamental group of the base is pro finite this implies homotopy equivalence

unilateral contact of two solids subject to large deformations formulation and existence results the approach that is currently the most widely used for formulating the unilateral frictionless contact between two deformable solids involves two key elements the orthogonal projection of one potential contact material surface on another and the action reaction principle of contact forces in the present paper we investigate this approach in the general case of large deformations and study unilateral frictionless boundary value problems of elasticity formulated by means of it the conditions for the orthogonal projection to be unique or further a c 1 diffeomorphism are established an extended action reaction principle is explicitly given for the forces acting on a pair of potential contact surfaces assuming the solids under consideration can be described by polyconvex hyperelastic laws we formulate the associated unilateral frictionless boundary value problem as a minimum principle and show the existence of a solution to it

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

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 birkhoff lewis fixed point theorem and a conjecture of v i arnold abstract the following conjecture of v i arnold is proved every measure preserving diffeomorphism of the torus t2 which is homologeous to the identity and which leaves the center of mass invariant possesses at least 3 fixed points the proof of this global fixed point theorem does not make use of the generating function technique the theorem is a consequence of the statement that a hamiltonian vector field on a torus t2n which depends periodically on time possesses at least 2n 1 forced oscillations these periodic solutions are are found using the classical variational principle by means of two qualitative statements for general flows a second conjecture of v i arnold proved concerns a birkhoff lewis type fixed point theorem for symplectic maps additional keywords periodic functions variational principles equations reprints author

wavelet bi frames with uniform symmetry for curve multiresolution processing this paper is about the construction of univariate wavelet bi frames with each framelet being symmetric as bivariate filter banks are used for surface multiresolution processing it is required that the corresponding decomposition and reconstruction algorithms have high symmetry so that it is possible to design the corresponding multiresolution algorithms for extraordinary vertices for open surfaces special multiresolution algorithms are designed to process boundary vertices when the multiresolution algorithms derived from univariate wavelet bi frames are used as the boundary algorithms it is desired that not only the scaling functions but also all framelets be symmetric in addition the algorithms for curve surface multiresolution processing should be given by templates so that they can be easily implemented in this paper first by appropriately associating the lowpass and highpass outputs to the nodes of z we show that both biorthogonal wavelet multiresolution algorithms and bi frame multiresolution algorithms can be represented by templates then using the idea of the lifting scheme we provide frame algorithms given by several iterative steps with each step represented by a symmetric template finally with the given templates of algorithms we obtain the corresponding filter banks and construct bi frames based on their smoothness and vanishing moments two types of symmetric bi frames are studied in this paper in order to provide a clearer picture on the template based procedure for bi frame construction in this paper we also consider the template based construction of biorthogonal wavelets the approach of the template based bi frame construction introduced in this paper can be extended easily to the construction of bivariate bi frames with high symmetry for surface multiresolution processing

a morse complex for infinite dimensional manifolds part i abstract in this paper and in the forthcoming part ii we introduce a morse complex for a class of functions f defined on an infinite dimensional hilbert manifold m possibly having critical points of infinite morse index and co index the idea is to consider an infinite dimensional subbundle or more generally an essential subbundle of the tangent bundle of m suitably related with the gradient flow of f this part i deals with the following questions about the intersection w of the unstable manifold of a critical point x and the stable manifold of another critical point y finite dimensionality of w possibility that different components of w have different dimension orientability of w and coherence in the choice of an orientation compactness of the closure of w classification up to topological conjugacy of the gradient flow on the closure of w in the case dim w 2

morse theory for lagrange multipliers and adiabatic limits abstract given two morse functions f on a compact manifold m we study the morse homology for the lagrange multiplier function on m r which sends x to f x x take a product metric on m r and rescale its r component by a factor 2 we show that generically for large the morse smale witten chain complex is isomorphic to the one for f and the metric restricted to 1 0 with grading shifted by one on the other hand in the limit 0 we obtain another chain complex which is geometrically quite different but has the same homology as the singular homology of 1 0 the isomorphism between the chain complexes is provided by the homotopy obtained by varying our proofs use both the implicit function theorem on banach manifolds and geometric singular perturbation theory

cup length estimate for lagrangian intersections abstract in this paper we consider the arnold conjecture on the lagrangian intersections of some closed lagrangian submanifold of a closed symplectic manifold with its image of a hamiltonian diffeomorphism we prove that if the hofer s symplectic energy of the hamiltonian diffeomorphism is less than a topology number defined by the lagrangian submanifold then the arnold conjecture is true in the degenerated nontransversal case

differential algebra of legendrian links let the space r q p u be equipped with the standard contact form du pdq a link l r3 is called legendrian if the restriction of to l vanishes two legendrian links are said to be legendrian isotopic if they can be connected by a path in the space of legendrian embeddings consider the map r3 r2 q p u q p since the kernel of d is everywhere transverse to the kernel of the projection l of a legendrian link l is an immersed curve we say that a link l r3 is generic with respect to or generic if all self intersections of l are transverse double points given a generic legendrian link l its diagram is the curve l r2 at every crossing of which the overpassing branch the one with the larger value of u is marked it is well known that every smooth link type can be realized by a legendrian link it is shown in sect 11 that such a realization can be chosen to have a diagram with relatively few crossings the so called classical invariants of an oriented legendrian link l are defined as follows first of them is just the smooth isotopy type of l let l1 lk be the components of l the maslov number m li of li is twice the rotation number of li the thurston bennequin number li of li is the linking number between li and s li where s is a small shift along the u direction the orientation on r3 involved in the definition of the linking number is given by the form d dq dp du in terms of the diagram li is the number of double points of li counted with signs shown in fig 1 the thurston bennequin number does not depend on the orientation of li while the maslov number changes its sign when the orientation changes

exact lagrangian immersions with one double point revisited we study exact lagrangian immersions with one double point of a closed orientable manifold k into mathbb c n we prove that if the maslov grading of the double point does not equal 1 then k is homotopy equivalent to the sphere and if in addition the lagrangian gauss map of the immersion is stably homotopic to that of the whitney immersion then k bounds a parallelizable n 1 manifold the hypothesis on the gauss map always holds when n 2k or when n 8k 1 the argument studies a filling of k obtained from solutions to perturbed cauchy riemann equations with boundary on the image f k of the immersion this leads to a new and simplified proof of some of the main results of ekholm and smith exact lagrangian immersions with a single double point 2011 which treated lagrangian immersions in the case n 2k by applying similar techniques to a lagrange surgery of the immersion as well as to an extension of these results to the odd dimensional case

propagation in hamiltonian dynamics and relative symplectic homology the main result asserts the existence of noncontractible periodic orbits for compactly supported time dependent hamiltonian systems on the unit cotangent bundle of the torus or of a negatively curved manifold whenever the generating hamiltonian is sufficiently large over the zero section the proof is based on floer homology and on the notion of a relative symplectic capacity applications include results about propagation properties of sequential hamiltonian systems periodic orbits on hypersurfaces hamiltonian circle actions and smooth lagrangian skeletons in stein manifolds

hamiltonian delay equations examples and a lower bound for the number of periodic solutions abstract we describe a variational approach to a notion of hamiltonian delay equations our delay hamiltonians are of product form we consider several examples for closed symplectically aspherical symplectic manifolds m we prove that for generic delay hamiltonians the number of 1 periodic solutions of the hamiltonian delay equation is at least the sum of the betti numbers of m extending the proof of the arnold conjecture to the case with delay

an iterated graph construction and periodic orbits of hamiltonian delay equations abstract according to the arnold conjectures and floer s proofs there are non trivial lower bounds for the number of periodic solutions of hamiltonian differential equations on a closed symplectic manifold whose symplectic form vanishes on spheres we use an iterated graph construction and lagrangian floer homology to show that these lower bounds also hold for certain hamiltonian delay equations