bi invariant metrics on the group of symplectomorphisms this paper studies the extension of the hofer metric and general finsler metrics on the hamiltonian symplectomorphism group ham m to the identity component symp 0 m of the symplectomorphism group in particular we prove that the hofer metric on ham m does not extend to a bi invariant metric on symp 0 m for many symplectic manifolds we also show that for the torus t 2n with the standard symplectic form no finsler metric on ham t 2n that satisfies a strong form of the invariance condition can extend to a bi invariant metric on symp 0 t 2 n another interesting result is that there exists no c 1 continuous bi invariant metric on symp 0 t 2n

higher homotopy operations and andre quillen cohomology there are two main approaches to the problem of realizing a algebra a graded group equipped with an action of the primary homotopy operations as the homotopy groups of a space x both involve trying to realize an algebraic free simplicial resolution g of by a simplicial space w and proceed by induction on the simplicial dimension the first provides a sequence of andre quillen cohomology classes in hn 2 n n 1 as obstructions to the existence of successive postnikov sections for w cf dwyer et al 1995 27 the second gives a sequence of geometrically defined higher homotopy operations as the obstructions cf blanc 1995 8 these were identified in blanc et al 2010 16 with the obstruction theory of dwyer et al 1989 25 there are also algebraic and geometric obstructions for distinguishing between different realizations of in this paper we a provide an explicit construction of the cocycles representing the cohomology obstructions b provide a similar explicit construction of certain minimal values of the higher homotopy operations which reduce to long toda brackets and c show that these two constructions correspond under an evident map

from taubes currents to almost kahler forms the taubes current is introduced by taubes in j symplectic geom 9 161 250 2011 as an intermediate step to construct almost kahler forms in this paper we prove that an almost complex structures being almost kahler structure in dimension four is the equivalent of the existence of taubes currents precisely we show that taubes currents could be regularized to almost kahler forms up to small perturbations of cohomology classes on any 4 dimensional almost complex manifold m j a similar result is established for higher dimensions under the assumption of almost kahler an application to donaldson s tamed to compatible question is provided

deconstructing hopf spaces we characterize hopf spaces with finitely generated cohomology as an algebra over the steenrod algebra we deconstruct the original space into an h space y with finite mod p cohomology and a finite number of p torsion eilenberg mac lane spaces we give a precise description of homotopy commutative h spaces in this setting

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

symplectic geometry and hilbert s fourth problem inspired by hofer s definition of a metric on the space of compactly supported hamiltonian maps on a symplectic manifold this paper exhibits an area length duality between a class of metric spaces and a class of symplectic manifolds using this duality it is shown that there is a twistor like correspondence between finsler metrics on pn whose geodesics are projective lines and a class of symplectic forms on the grassmannian of 2 planes in n 1

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

symplectic rigidity symplectic fixed points and global perturbations of hamiltonian systems in this paper we study a generalized symplectic fixed point problem first considered by j moser in 20 from the point of view of some relatively recently discovered symplectic rigidity phenomena this problem has interesting applications concerning global perturbations of hamiltonian systems 2007 wiley periodicals inc

fragility and persistence of leafwise intersections in this paper we study the question of fragility and robustness of leafwise intersections of coisotropic submanifolds namely we construct a closed hypersurface and a sequence of hamiltonians c 0 converging to zero such that the hypersurface and its images have no leafwise intersections showing that some form of the contact type condition on the hypersurface is necessary in several persistence results in connection with recent results in continuous symplectic topology we also show that c 0 convergence of hypersurfaces hamiltonian diffeomorphic to each other does not in general force c 0 convergence of the characteristic foliations

pseudo distances on symplectomorphism groups and applications to flux theory starting from a given norm on the vector space of exact 1 forms of a compact symplectic manifold we produce pseudo distances on its symplectomorphism group by generalizing an idea due to banyaga we prove that in some cases which include banyaga s construction their restriction to the hamiltonian diffeomorphism group is equivalent to the distance induced by the initial norm on exact 1 forms we also define genuine distances to the hamiltonian diffeomorphism group which we use to derive several consequences mainly in terms of flux groups

hamiltonian collapsing of irrational lagrangian submanifolds with small first betti number among the main symplectic invariants of a closed lagrange submanifoldl of the cotangent of n is the tubular radiusr l defined as the smallest tube d r n 1 of n t n in whichl can be pushed by an hamiltonian diffeotopy of n we show here using pseudo holomorphic techniques that such a submanifold cannot collapse if the first betti number ofl is smaller than 3 and if the maslov class ofl does not vanish in other words r l is then strictly positive and one can actually give an explicit lower bound in terms of the liouville and maslov classes ofl

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

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

commuting hamiltonians and hamilton jacobi multi time equations the aim of this paper is twofold first of all we show that the c 0 limit of a pair of commuting hamiltonians commute this means on one hand that if the limit of the hamiltonians is smooth the poisson bracket of their limit still vanishes and on the other hand that we may define commutation for c 0 functions the second part of the paper deals with solving multi time hamilton jacobi equations using variational solutions this extends the work of barles and tourin in the viscosity case to include the case of c 0 hamiltonians and removes their convexity assumption provided we are in the framework of variational solutions

k area hofer metric and geometry of conjugacy classes in lie groups given a closed symplectic manifold m we introduce a certain quantity associated to a tuple of conjugacy classes in the universal cover of the group ham m by means of the hofer metric on ham m we use pseudo holomorphic curves involved in the definition of the multiplicative structure on the floer cohomology of a symplectic manifold m to estimate this quantity in terms of actions of some periodic orbits of related hamiltonian flows as a corollary we get a new way to obtain agnihotri belkale woodward inequalities for eigenvalues of products of unitary matrices as another corollary we get a new proof of the geodesic property with respect to the hofer metric of hamiltonian flows generated by certain autonomous hamiltonians our main technical tool is k area defined for hamiltonian fibrations over a surface with boundary in the spirit of l polterovich s work on hamiltonian fibrations over s2

model driven segmentation of articulating humans in laplacian eigenspace we propose a general approach using laplacian eigenmaps and a graphical model of the human body to segment 3d voxel data of humans into different articulated chains in the bottom up stage the voxels are transformed into a high dimensional 6d or less laplacian eigenspace le of the voxel neighborhood graph we show that le is effective at mapping voxels on long articulated chains to nodes on smooth 1d curves that can be easily discriminated and prove these properties using representative graphs we fit 1d splines to voxels belonging to different articulated chains such as the limbs head and trunk and determine the boundary between splines using the spline fitting error a top down probabilistic approach is then used to register the segmented chains utilizing their mutual connectivity and individual properties our approach enables us to deal with complex poses such as those where the limbs form loops we use the segmentation results to automatically estimate the human body models while we use human subjects in our experiments the method is fairly general and can be applied to voxel based segmentation of any articulated object composed of long chains we present results on real and synthetic data that illustrate the usefulness of this approach

vanishing distance phenomena and the geometric approach to sqg in this article we study the induced geodesic distance of fractional order sobolev metrics on the groups of volume preserving diffeomorphisms and symplectomorphisms the interest in these geometries is fueled by the observation that they allow for a geometric interpretation for prominent partial differential equations in the field of fluid dynamics these include in particular the modified constantin lax majda and surface quasi geostrophic equations the main result of this article shows that both of these equations stem from a riemannian metric with vanishing geodesic distance

the diameter of the symplectomorphism group is infinite shnirelman has proved that the group of volume preversing diffeomorphisms of the cube in r 3 has finite diameter and has announced the result that this is false for the square despite the fact that shnirelman formulated his theorem only for the 3 dimensional cube his proof can be modified for the case of the group of volume preserving diffeomorphisms of any compact simply connected riemannian manifold of dimension 2 it turns out that the situation with the group of symplectic diffeomorphisms is completely different we prove that the diameter of the symplectomorphism group of any compact exact symplectic manifold necessarily with boundary is infinite

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

euler maclaurin with remainder for a simple integral polytope we give an euler maclaurin formula with remainder for the sum of the values of a smooth function on the integral points in a simple integral polytope we prove this formula by elementary methods

on the topological properties of symplectic maps in this paper we show that symplectic maps have surprising topological properties in particular we construct an interesting metric for the symplectic diffeomorphism groups which is related but not obviously to the topological properties of symplectic maps and phase space geometry we also prove a certain number of generalised symplectic fixed point theorems and give an application to a hamiltonian system

coisotropic displacement and small subsets of a symplectic manifold we prove a coisotropic intersection result and deduce the following a lower bounds on the displacement energy of a subset of a symplectic manifold in particular a sharp stable energy gromov width inequality b a stable non squeezing result for neighborhoods of products of unit spheres c existence of a badly squeezable set in mathbb r 2n of hausdorff dimension at most d for every n 2 and d n d existence of a stably exotic symplectic form on mathbb r 2n for every n 2 e non triviality of a new capacity which is based on the minimal action of a regular coisotropic submanifold of dimension d

face recognition using laplacianfaces we propose an appearance based face recognition method called the laplacianface approach by using locality preserving projections lpp the face images are mapped into a face subspace for analysis different from principal component analysis pca and linear discriminant analysis lda which effectively see only the euclidean structure of face space lpp finds an embedding that preserves local information and obtains a face subspace that best detects the essential face manifold structure the laplacianfaces are the optimal linear approximations to the eigenfunctions of the laplace beltrami operator on the face manifold in this way the unwanted variations resulting from changes in lighting facial expression and pose may be eliminated or reduced theoretical analysis shows that pca lda and lpp can be obtained from different graph models we compare the proposed laplacianface approach with eigenface and fisherface methods on three different face data sets experimental results suggest that the proposed laplacianface approach provides a better representation and achieves lower error rates in face recognition

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

the geometry of symplectic energy non squeezing theorem which says that it is impossible to embed a large ball symplectically into a thin cylinder of the form r2 x b2 where b2 is a 2 disc this led to hofer s discovery of symplectic capacities which give a way of measuring the size of subsets in symplectic manifolds recently hofer found a way to measure the size or energy of symplectic diffeomorphisms by looking at the total variation of their generating hamiltonians this gives rise to a bi invariant pseudo norm on the group ham m of compactly supported hamiltonian symplectomorphisms of the manifold m the deep fact is that this pseudo norm is a norm in other words the only symplectomorphism on m with zero energy is the identity map up to now this had been proved only for sufficiently nice symplectic manifolds and by rather complicated analytic arguments in this paper we consider a more geometric version of this energy which was first considered by eliashberg and hofer in connection with their study of the extent to which the interior of a region in a symplectic manifold determines its boundary we prove by a simple geometric argument that both versions of energy give rise to genuine norms on all symplectic manifolds roughly speaking we show that if there were a symplectomorphism of m which had too little energy one could embed a large ball into a thin cylinder m x

metric and isoperimetric problems in symplectic geometry introduction 411 1 some basic results in symplectic topology 413 2 capacity and symplectic reduction 414 3 volume estimates for lagrange submanifolds 416 3 1 the case of r 417 3 2 deformations of the zero section in cotangent bundles 419 3 3 generalization to the case of cp 422 4 an application to billiards 423 5 geometry of convex sets and periodic orbits 425 6 compensated compactness and closure of the symplectic group 426 appendix a a summary of symplectic geometry through generating functions 427 appendix b john s ellipsoid 428 references 430

periodic solutions on hypersurfaces and a result by c viterbo on considere un champ vectoriel hamiltonien x j h x x h x sur x r 2n h etant une fonction lisse dont le gradient h est defini par rapport a la metrique euclidienne on cherche des solutions periodiques sur une surface d energie donnee s x r 2n h x const supposee reguliere

laplacian eigenmaps for dimensionality reduction and data representation one of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data we consider the problem of constructing a representation for data lying on a low dimensional manifold embedded in a high dimensional space drawing on the correspondence between the graph laplacian the laplace beltrami operator on the manifold and the connections to the heat equation we propose a geometrically motivated algorithm for representing the high dimensional data the algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality preserving properties and a natural connection to clustering some potential applications and illustrative examples are discussed

schroedinger eigenmaps for the analysis of biomedical data we introduce schroedinger eigenmaps se a new semi supervised manifold learning and recovery technique this method is based on an implementation of graph schroedinger operators with appropriately constructed barrier potentials as carriers of labeled information we use our approach for the analysis of standard biomedical datasets and new multispectral retinal images

the krull filtration of the category of unstable modules over the steenrod algebra in the early 1990 s lionel schwartz gave a lovely characterization of the krull filtration of mathcal u the category of unstable modules over the mod p steenrod algebra soon after this filtration was used by the author as an organizational tool in posing and studying some topological nonrealization conjectures in recent years the krull filtration of mathcal u has been similarly used by castellana crespo and scherer in their study of h spaces with finiteness conditions recently gaudens and schwartz have given a proof of some of my conjectures in light of these topological applications it seems timely to better expose the algebraic properties of the krull filtration

displacement energy of unit disk cotangent bundles we give an upper bound of a hamiltonian displacement energy of a unit disk cotangent bundle d m in a cotangent bundle t m when the base manifold m is an open riemannian manifold our main result is that the displacement energy is not greater than c r m where r m is the inner radius of m and c is a dimensional constant as an immediate application we study symplectic embedding problems of unit disk cotangent bundles moreover combined with results in symplectic geometry our main result shows the existence of short periodic billiard trajectories and short geodesic loops

on topologically realizing modules over the steenrod algebra the mod p cohomology of a space or spectrum x is a module over the mod p steenrod algebra a if x is a space it has additional structure making it an unstable a algebra where we recall that an a module m is unstable if when p 2 sqi x 0 for all i ixi and x e m and when p 2 3epix 0 for all 2i e jxj e 0 or 1 and x e m questions about realizing particular a modules algebras and a algebras as the cohomology of spaces or spectra are among the most fundamental in algebraic topology the hopf invariant one problem a and the problem of realizing polynomial algebras dmw are two famous examples of ongoing interest if one regards the first of these as a problem about realizing finite a modules and the second of these as a problem about realizing rather large a modules it is the purpose of this paper to study the middle ground we begin with the following conjecture

isotopy of symplectic balls gromov s radius and the structure of ruled symplectic 4 manifolds there are various criteria enriques castelnuovo mori in algebraic geometry of complex surfaces that characterize rational or ruled surfaces up to birationality mori s criterion in particular states that a minimal compact algebraic surface is biholomorphic to cp 2 or to a ruled rational or irrational surface exactly when it contains a curve of strictly positive chern index this criterion has been partly generalized to symplectic 4 manifold by mcduff who showed the following any symplectic 4 manifold which contains an embedded symplectic sphere or immersed with positive normal double points of chern index larger than i is symplectomorphic to a manifold obtained by blowing up a finite number of symplectic balls either on p2 with its standard kahler structure or on a symplectic s2 fibration over a closed oriented real surface we need the answer to the following two problems in order to complete the classification 1 the classification of irrational symplectic s2 fibrations is not complete the statement of theorem 1 3 of 1 9 is not correct in the irrational case in particular it does not fully take into account the kahler structures coming from the projectivization of non split holomorphic rank 2 bundles the sharpest results on this problem are those of the recent article 1 11 when the area of the base of a symplectic s2 fibration v co s o over a surface of genus g is sufficiently large with respect to the area of the fiber the cohomology class of a symplectic form determines its isotopy class and therefore determines the symplectic type of the manifold see theorem 1 2 below actually under this cohomological hypothesis the symplectic ruled manifold v o may be obtained by cutting and pasting from a rational ruled

from symplectic measurements to the mahler conjecture in this note we link symplectic and convex geometry by relating two seemingly different open conjectures a symplectic isoperimetric type inequality for convex domains and mahler s conjecture on the volume product of centrally symmetric convex bodies more precisely we show that if for convex bodies of fixed volume in the classical phase space the hofer zehnder capacity is maximized by the euclidean ball then a hypercube is a minimizer for the volume product among centrally symmetric convex bodies

primitives and central detection numbers in group cohomology abstract fix a prime p given a finite group g let h g denote its mod p cohomology in the early 1990s henn lannes and schwartz introduced two invariants d 0 g and d 1 g of h g viewed as a module over the mod p steenrod algebra they showed that in a precise sense h g is respectively detected and determined by h d c g v for d d 0 g and d d 1 g with v running through the elementary abelian p subgroups of g the main goal of this paper is to study how to calculate these invariants we find that a critical role is played by the image of the restriction of h g to h c where c is the maximal central elementary abelian p subgroup of g a measure of this is the top degree e g of the finite dimensional hopf algebra h c h g f p a number that tends to be quite easy to calculate our results are complete when g has a p sylow subgroup p in which every element of order p is central using the benson carlson duality we show that in this case d 0 g d 0 p e p and a similar exact formula holds for d 1 as a bonus we learn that h e g p contains nontrivial essential cohomology reproving and sharpening a theorem of adem and karagueuzian in general we are able to show that d 0 g max e c g v v g if certain cases of benson s regularity conjecture hold in particular this inequality holds for all groups such that the difference between the p rank of g and the depth of h g is at most 2 when we look at examples with p 2 we learn that d 0 g 14 for all groups with 2 sylow subgroup of order up to 64 with equality realized when g su 3 4 en route we study two objects of independent interest if c is any central elementary abelian p subgroup of g then h g is an h c comodule and we prove that the subalgebra of h c primitives is always noetherian of krull dimension equal to the p rank of g minus the p rank of c if the depth of h g equals the rank of z g we show that the depth essential cohomology of g is nonzero reproving and extending a theorem of green and cohen macauley in a certain sense and prove related structural results