using sampling and simplex derivatives in pattern search methods in this paper we introduce ways of making a pattern search more efficient by reusing previous evaluations of the objective function based on the computation of simplex derivatives e g simplex gradients at each iteration one can attempt to compute an accurate simplex gradient by identifying a sampling set of previously evaluated points with good geometrical properties this can be done using only past successful iterates or by considering all past function evaluations the simplex gradient can then be used to reorder the evaluations of the objective function associated with the directions used in the poll step or to update the mesh size parameter according to a sufficient decrease criterion neither of which requires new function evaluations we present these procedures in detail and apply them to a set of problems from the cuter collection numerical results show that these procedures can enhance significantly the practical performance of pattern search methods

associativity of quantum multiplication we proved the associativity of the multiplication of quantum cohomology for a monotone compact symplectic manifold v for which c 1 a 1 for any effective class the same proof also works for any positive compact symplectic manifold with c 1 a 1

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

superlinear convergence and implicit filtering in this paper we show how the implicit filtering algorithm can be coupled with the bfgs quasi newton update to obtain a superlinearly convergent iteration if the noise in the objective function decays sufficiently rapidly as the optimal point is approached in this way we give insight into the observations of good performance in practice of quasi newton methods when they are coupled with implicit filtering we also report on numerical experiments that show how an implementation of implicit filtering that exploits these new results can improve the performance of the algorithm

pattern search algorithms for bound constrained minimization we present a convergence theory for pattern search methods for solving bound constrained nonlinear programs the analysis relies on the abstract structure of pattern search methods and an understanding of how the pattern interacts with the bound constraints this analysis makes it possible to develop pattern search methods for bound constrained problems while only slightly restricting the flexibility present in pattern search methods for unconstrained problems we prove global convergence despite the fact that pattern search methods do not have explicit information concerning the gradient and its projection onto the feasible region and consequently are unable to enforce explicitly a notion of sufficient feasible decrease

examples of non kahler hamiltonian torus actions an important question with a rich history is the extent to which the symplectic category is larger than the kahler category many interesting examples of non kahler symplectic manifolds have been constructed t m g however sufficiently large symmetries can force a symplectic manifolds to be kahler d kn in this paper we solve several outstanding problems by constructing the first symplectic manifold with large non trivial symmetries which does not admit an invariant kahler structure the proof that it is not kahler is based on the atiyah guillemin sternberg convexity theorem at gs using the ideas of this paper c woodward shows that even the symplectic analogue of spherical varieties need not be kahler w

frames and grids in unconstrained and linearly constrained optimization a nonsmooth approach this paper describes a class of frame based direct search methods for unconstrained and linearly constrained optimization a template is described and analyzed using clarke s break nonsmooth calculus this provides a unified and simple approach to earlier results for grid and frame based methods and also provides partial convergence results when the objective function is not smooth undefined in some places or both the template also covers many new methods which combine elements of previous ideas using frames and grids these new methods include grid based simple descent algorithms which allow moving to points off the grid at every iteration and can automatically control the grid size provided function values are available the concept of a grid is also generalized to that of an admissible set which allows sets for example with circular symmetries the method is applied to linearly constrained problems using a simple barrier approach

normal crossings singularities for symplectic topology abstract we introduce topological notions of normal crossings symplectic divisor and variety and establish that they are equivalent in a suitable sense to the desired geometric notions our proposed concept of equivalence of associated topological and geometric notions fits ideally with important constructions in symplectic topology this partially answers gromov s question on the feasibility of defining singular symplectic sub varieties and lays foundation for rich developments in the future in subsequent papers we establish a smoothability criterion for symplectic normal crossings varieties in the process providing the multifold symplectic sum envisioned by gromov and introduce symplectic analogues of logarithmic structures in the context of normal crossings symplectic divisors

a stability property of symplectic packing we prove that for any closed symplectic 4 manifold m with h 2 m q there exists a number n 0 such that for every n n 0 m admits full symplectic packing by n equal balls we also indicate how to compute this n 0 our approach is based on donaldson s symplectic submanifold theorem and on tools from the framework of taubes theory of gromov invariants

on the convergence of pattern search algorithms we introduce an abstract definition of pattern search methods for solving nonlinear unconstrained optimization problems our definition unifies an important collection of optimization methods that neither compute nor explicitly approximate derivatives we exploit our characterization of pattern search methods to establish a global convergence theory that does not enforce a notion of sufficient decrease our analysis is possible because the iterates of a pattern search method lie on a scaled translated integer lattice this allows us to relax the classical requirements on the acceptance of the step at the expense of stronger conditions on the form of the step and still guarantee global convergence

parallel space decomposition of the mesh adaptive direct search algorithm this paper describes a parallel space decomposition psd technique for the mesh adaptive direct search mads algorithm mads extends a generalized pattern search for constrained nonsmooth optimization problems the objective of the present work is to obtain good solutions to larger problems than the ones typically solved by mads the new method psd mads is an asynchronous parallel algorithm in which the processes solve problems over subsets of variables the convergence analysis based on the clarke calculus is essentially the same as for the mads algorithm a practical implementation is described and some numerical results on problems with up to 500 variables illustrate the advantages and limitations of psd mads

on the convergence of grid based methods for unconstrained optimization the convergence of direct search methods for unconstrained minimization is examined in the case where the underlying method can be interpreted as a grid or pattern search over successively refined meshes an important aspect of the main convergence result is that translation rotation scaling and shearing of the successive grids are allowed

regularity and estimates for j holomorphic discs attached to a maximal totally real submanifold we prove that pseudo holomorphic discs attached to a maximal totally real submanifold inherit their regularity to the boundary from the regularity of the submanifold and of the almost complex structure the proof is based on the computation of an explicit lower bound for the kobayashi metric in almost complex manifolds which also yields explicit estimates of holderian norms of such discs

a cotangent fibre generates the fukaya category abstract we prove that the algebra of chains on the based loop space recovers the derived wrapped fukaya category of the cotangent bundle of a closed smooth oriented manifold the main new idea is the proof that a cotangent fibre generates the fukaya category using a version of the map from symplectic cohomology to the homology of the free loop space introduced by cieliebak and latschev

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

convergence of yang mills higgs fields in this paper we study the convergence of yang mills higgs ymh fields defined on fiber bundles over riemann surfaces where the fiber is a compact symplectic manifold and the conformal structure of the underlying surface is allowed to vary we show that away from the nodes the ymh fields converges up to gauge to a smooth ymh field modulo finitely many harmonic spheres while near the nodes where the conformal structure degenerates the ymh fields converges to a pair consisting of a flat connection and a twisted geodesic with potential in particular we generalize the recent compactness results on both harmonic maps from surfaces and twisted holomorphic curves to general ymh fields

pattern search methods for linearly constrained minimization we extend pattern search methods to linearly constrained minimization we develop a general class of feasible point pattern search algorithms and prove global convergence to a karush kuhn tucker point as in the case of unconstrained minimization pattern search methods for linearly constrained problems accomplish this without explicit recourse to the gradient or the directional derivative of the objective key to the analysis of the algorithms is the way in which the local search patterns conform to the geometry of the boundary of the feasible region

unconstrained derivative free optimization by successive approximation we present an algorithmic framework for unconstrained derivative free optimization based on dividing the search space in regions partitions every partition is assigned a representative point the representative points form a grid a piecewise constant approximation to the function subject to optimization is constructed using a partitioning and its corresponding grid the convergence of the framework to a stationary point of a continuously differentiable function is guaranteed under mild assumptions the proposed framework is appropriate for upgrading heuristics that lack mathematical analysis into algorithms that guarantee convergence to a local minimizer a convergent variant of the nelder mead algorithm that conforms to the given framework is constructed the algorithm is compared to two previously published convergent variants of the nm algorithm the comparison is conducted on the more garbow hillstrom set of test problems and on four variably dimensional functions with dimension up to 100 the results of the comparison show that the proposed algorithm outperforms both previously published algorithms

stationarity results for generating set search for linearly constrained optimization we present a new generating set search gss approach for minimizing functions subject to linear constraints gss is a class of direct search optimization methods that includes generalized pattern search one of our main contributions in this paper is a new condition to define the set of conforming search directions that admits several computational advantages for continuously differentiable functions we also derive a bound relating a measure of stationarity which is equivalent to the norm of the gradient of the objective in the unconstrained case and a parameter used by gss algorithms to control the lengths of the steps with the additional assumption that the derivative is lipschitz we obtain a big o bound as a consequence of this relationship we obtain subsequence convergence to a kkt point even though gss algorithms lack explicit gradient information numerical results indicate that the bound provides a reasonable estimate of stationarity

on the dynamics of laguerre s iteration z n 1 abstract the laguerre algorithm for computing the zeros of polynomial equations is well known many proofs of the global convergence of this method for the class of real polynomials with all real roots and starting from real initial conditions exist in this article the global behavior of the laguerre iteration is studied when applied to zn 1 for this class of polynomials an almost complete description of the dynamics of laguerre is rendered

geometry of interpolation sets in derivative free optimization we consider derivative free methods based on sampling approaches for nonlinear optimization problems where derivatives of the objective function are not available and cannot be directly approximated we show how the bounds on the error between an interpolating polynomial and the true function can be used in the convergence theory of derivative free sampling methods these bounds involve a constant that reflects the quality of the interpolation set the main task of such a derivative free algorithm is to maintain an interpolation sampling set so that this constant remains small and at least uniformly bounded this constant is often described through the basis of lagrange polynomials associated with the interpolation set we provide an alternative more intuitive definition for this concept and show how this constant is related to the condition number of a certain matrix this relation enables us to provide a range of algorithms whilst maintaining the interpolation set so that this condition number or the geometry constant remain uniformly bounded we also derive bounds on the error between the model and the function and between their derivatives directly in terms of this condition number and of this geometry constant

frankel s theorem in the symplectic category we prove that if an n 1 dimensional torus acts symplectically on a 2n dimensional symplectic manifold then the action has a fixed point if and only if the action is hamiltonian one may regard it as a symplectic version of frankel s theorem which says that a kahler circle action has a fixed point if and only if it is hamiltonian the case of n 2 is the well known theorem by mcduff

calibrated fibrations on noncompact manifolds via group actions in this paper we use lie group actions on noncompact riemannian manifolds with calibrations to construct calibrated submanifolds in particular if we have an n 1 torus acting on a noncompact calabi yau n fold with a trivial first cohomology then we have a special lagrangian fibration on that n fold we produce several families of examples for this construction and give some applications to special lagrangian geometry on compact almost calabi yau manifolds we also use group actions on noncompact g2 manifolds to construct coassociative submanifolds and we exhibit some new examples of coassociative submanifolds via this setup

complexity one hamiltonian su 2 and so 3 actions we consider compact connected six dimensional symplectic manifolds with hamilto nian su 2 or so 3 actions with cyclic principal stabilizers we classify such manifolds up to equivariant symplectomorphisms 1 introduction let m be a symplectic manifold and g be a compact connected lie group that acts effectively on m by symplectic transformations a moment map m g is a g equivariant map such that for every in the lie algebra g of g m d where m m tm denotes the induced vector eld of on m if there is a mo ment map we say that the action is hamiltonian the triple m is called a hamiltonian g manifold or hamiltonian g action an isomorphism between two such manifolds is an equivariant symplectomorphism that respects the moment maps we usually assume that m is connected that g acts effectively on m and that the moment map is proper for a point in the dual of the lie algebra of g the symplectic quotient or reduced space at is the topological space m 1 g g 1 g where g is the coadjoint orbit through and g denotes the stabilizer of i f is a regular value of the moment map the reduced space m is a symplectic orbifold in general the reduced space is a symplectic strati ed space ls bl the complexity of m is half the dimension of the reduced space m at a generic value in the moment image m suppose a torus t acts on a symplectic manifold m in a hamiltonian fashion its complexity is 1 dimm dimt in particular the complexity is zero exactly if the dimension of the torus is half the dimension of the manifold the space is then called a toric manifold these spaces are classi ed by their moment images d1 karshon and tolman have studied complexity one torus actions in arbitrary dimensions kt1 kt2 special cases are compact symplectic four manifolds with circle actions which are classi ed by karshon k2 also see audin and ahara hattori au1 au2 ah

generalized pattern search methods for a class of nonsmooth optimization problems with structure we propose a generalized pattern search gps method to solve a class of nonsmooth minimization problems where the set of nondifferentiability is included in the union of known hyperplanes and therefore is highly structured both unconstrained and linearly constrained problems are considered at each iteration the set of poll directions is enforced to conform to the geometry of both the nondifferentiability set and the boundary of the feasible region near the current iterate this is the key issue to guarantee the convergence of certain subsequences of iterates to points which satisfy first order optimality conditions numerical experiments on some classical problems validate the method

nonsingular affine k surfaces nonsingular affine k surfaces are classified as certain invariant open subsets of projective k surfaces a graph is defined which is an equivariant isomorphism invariant of an affine k surface over the complex numbers it is proved that the only acyclic affine surface which admits an effective action of the group c is c 2 which admits only linear actions of c

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

a heuristic iterated subspace minimization method with pattern search for unconstrained optimization recently an increasing attention was paid on different procedures for an unconstrained optimization problem when the information of the first derivatives is unavailable or unreliable in this paper we consider a heuristic iterated subspace minimization method with pattern search for solving such unconstrained optimization problems the proposed method is designed to reduce the total number of function evaluations for the implementation of high dimensional problems meanwhile it keeps the advantages of general pattern search algorithm i e the information of the derivatives is not needed at each major iteration of such a method a low dimensional manifold the iterated subspace is constructed and an approximate minimizer of the objective function in this manifold is then determined by a pattern search method numerical results on some classic test examples are given to show the efficiency of the proposed method in comparison with a conventional pattern search method and a derivative free method

solvability of dirac type equations abstract this paper develops a weighted l 2 method for the half dirac equation for dirac bundles over closed riemann surfaces we give a sufficient condition for the solvability of the half dirac equation in terms of a curvature integral applying this to the dolbeault dirac operator we establish an automatic transversality criteria for holomorphic curves in kahler manifolds on compact riemannian manifolds we give a new perspective on some well known results about the first eigenvalue of the dirac operator and improve the estimates when the dirac bundle has a z 2 grading on riemannian manifolds with cylindrical ends we obtain solvability in the l 2 spaces with suitable exponential weights while allowing mild negativity of the curvature

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

generalizations of laguerre s method higher order methods laguerre s method is an efficient and reliable method for finding zeros of polynomials and certain other functions a new derivation and motivation of laguerre s method is given which allows it to be included in a class of methods as general as methods of order three or more based on direct generalized hermite or hyperosculatory interpolation members of this new class share with laguerre s method the property of being globally convergent to zeros of polynomials with only real zeros and have the same order of convergence at simple zeros as the classic methods based on generalized hermite interpolation methods of order 4 and 3 303 are investigated and numerical results indicate that for large 100 times 100 eigenvalue problems the method of order 3 303 is as efficient and reliable as laguerre s method

tight not semi fillable contact circle bundles extending our earlier results we prove that certain tight contact structures on circle bundles over surfaces are not symplectically semi fillable thus confirming a conjecture of ko honda

open gromov witten disk invariants in the presence of an anti symplectic involution abstract for a symplectic manifold with an anti symplectic involution having non empty fixed locus we construct a model of the moduli space of real sphere maps out of moduli spaces of decorated disk maps and give an explicit expression for its first stiefel whitney class as a corollary we obtain a large number of examples which include all odd dimensional projective spaces and many complete intersections for which many types of real moduli spaces are orientable for these manifolds we define open gromov witten invariants with no restriction on the dimension of the manifolds or the type of the constraints if there are no boundary marked points a wdvv type recursion obtained in a sequel computes these invariants for many real symplectic manifolds if there are boundary marked points we define the invariants under some restrictions on the allowed boundary constraints even though the moduli spaces are not orientable in these cases

automorphisms of multiplicity free hamiltonian manifolds let m be a multiplicity free hamiltonian manifold m for a connected compact lie group k not necessarily abelian let p be the momentum polytope of m we calculate the automorphism of m as a sheaf over p and show that all higher cohomology groups of this sheaf vanish from this and a recent theorem of losev we deduce a conjecture of delzant the momentum polytope and the principal isotropy group determine m up to isomorphism moreover we give a criterion for when a polytope and a group are afforded by a multiplicity free manifold

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

centered complexity one hamiltonian torus actions we consider symplectic manifolds with hamiltonian torus actions which are almost but not quite completely integrable the dimension of the torus is one less than half the dimension of the manifold we provide a complete set of invariants for such spaces when they are centered and the moment map is proper in particular this classifies the preimages under the moment map of all sufficiently small open sets which is an important step towards global classification as an application we construct a full packing of each of the grassmannians gr 2 r5 and gr 2 r6 by two equal symplectic balls

on the iteration of a rational function computer experiments with newton s method using newton s method to look for roots of a polynomial in the complex plane amounts to iterating a certain rational function this article describes the behavior of newton iteration for cubic polynomials after a change of variables these polynomials can be parametrized by a single complex parameter and the newton transformation has a single critical point other than its fixed points at the roots of the polynomial we describe the behavior of the orbit of the free critical point as the parameter is varied the julia set points where newton s method fail to converge is also pictured these sets exhibit an unexpected stability of their gross structure while the changes in small scale structure are intricate and subtle

an implicit filtering algorithm for optimization of functions with many local minima in this paper we describe and analyze an algorithm for certain box constrained optimization problems that may have several local minima a paradigm for these problems is one in which the function to be minimized is the sum of a simple function such as a convex quadratic and high frequency low amplitude terms that cause local minima away from the global minimum of the simple function our method is gradient based and therefore the performance can be improved by use of quasi newton methods

new sequential and parallel derivative free algorithms for unconstrained minimization this paper presents sequential and parallel derivative free algorithms for finding a local minimum of smooth and nonsmooth functions of practical interest it is proved that under mild assumptions a sufficient decrease condition holds for a nonsmooth function based on this property the algorithms explore a set of search directions and move to a point with a sufficiently lower functional value if the function is strictly differentiable at its limit points a sub sequence of points generated by the algorithm converges to a first order stationary point nabla f x 0 if the function is convex around its limit points convergence of a subsequence to a point with nonnegative directional derivatives on a set of search directions is ensured preliminary numerical results on sequential algorithms show that they compare favorably with the recently introduced pattern search methods

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 maslov class of the lagrange surfaces and gromov s pseudo holomorphic curves for an immersed lagrange submanifold w c t x one can define a nonnegative integer topologic invariant m w such that the image of h1 w z under the maslov class is equal to m w z in this paper the value of m w is calculated for the case of a two dimensional oriented manifold x with the universal cover homeomorphic to r2 and an embedded lagrange torus w it is proved that if x t2 and w is homologic to the zero section then m w 0 in all the other cases m w 2 the last result is true also for a wide class of oriented properly embedded lagrange surfaces in t r the proof is based on the gromov s theory of pseudo holomorphic curves some applications to the hamiltonian mechanics are mentioned

irreversibility from staircases in symplectic embeddings abstract we present an argument whose goal is to trace the origin of the macroscopically irreversible behavior of hamiltonian systems of many degrees of freedom we use recent flexibility and rigidity results of symplectic embeddings quantified via the stabilized fibonacci and pell staircases to encode the underlying breadth of the possible initial conditions which alongside the multitude of degrees of freedom of the underlying system give rise to time irreversibility

analysis of generalized pattern searches this paper contains a new convergence analysis for the lewis and torczon generalized pattern search gps class of methods for unconstrained and linearly constrained optimization this analysis is motivated by a desire to understand the successful behavior of the algorithm under hypotheses that are satisfied by many practical problems specifically even if the objective function is discontinuous or extended valued the methods find a limit point with some minimizing properties simple examples show that the strength of the optimality conditions at a limit point depends not only on the algorithm but also on the directions it uses and on the smoothness of the objective at the limit point in question the contribution of this paper is to provide a simple convergence analysis that supplies detail about the relation of optimality conditions to objective smoothness properties and to the defining directions for the algorithm and it gives previous results as corollaries

on the variation in the cohomology of the symplectic form of the reduced phase space is called the momentum mapping of the hamiltonian t action given 1 1 the condition 1 2 just means that t acts along the fibers of j for the basic definitions and properties of non commutative hamiltonian group actions see am the results of this paper can easily be extended to hamiltonian actions of arbitrary compact connected lie groups by applying our results to the action of its maximal toms and using the equivariance of the momentum mapping for some more details see the remarks at the end of sect 2 we will assume throughout this paper that the momentum map is proper that is j i u is compact for each compact subset u of t now let r be a regular value of j that is t j tmm t is surjective for all meye j 1 4 then ye is a smooth submanifold of m compact because

convexity properties of the moment mapping re examined abstract consider a hamiltonian action of a compact lie group on a compact symplectic manifold a theorem of kirwan s says that the image of the momentum mapping intersects the positive weyl chamber in a convex polytope i present a new proof of kirwan s theorem which gives explicit information on how the vertices of the polytope come about and on how the shape of the polytope near any point can be read off from infinitesimal data on the manifold it also applies to some interesting classes of noncompact or singular hamiltonian spaces such as cotangent bundles and complex affine varieties

existence and classification of overtwisted contact structures in all dimensions we establish a parametric extension h principle for overtwisted contact structures on manifolds of all dimensions which is the direct generalization of the 3 dimensional result from 12 it implies in particular that any closed manifold admits a contact structure in any given homotopy class of almost contact structures

are there any julia sets for the laguerre iteration function abstract for polynomials some of whose zeros are complex little is known about the overall convergence properties of the laguerre s method the existence of free critical points of the laguerre function as applied to one parameter families of quadratic and cubic polynomials is examined with the help of microcomputer plots we investigate the julia sets of the laguerre function in the special case wherein this function is constructed to converge to the nth roots of unity

convexity properties of the moment mapping ii be its associated momen t mapping see w for definitions the set x is the union of co adjoint orbits the main result of this paper is a description of the orbit structure of this set to describe this result we first note that the coadjoint orbits in q can be parametr ized as follows let t be a caf tan subalgebra of g and let b be a positive definite g invariant bilinear form on by means of b we get a g equivariant identification q let t be the subspace of q corresponding to t then every co adjoint orbit intersects i in a weyl group orbit so there is a one one correspondence between co adjoint orbits in q and weyl group orbits in t to get an even sharper parametrization let t be a weyl chamber in then every weyl group orbit intersects t in a single point so t is a modul i space for the co adjoint orbits two orbits in x are said to be of the same type if they are isomorphic as homogeneous g spaces since x is compact there are at most a finite number of different types of orbits see 1113 let h be a closed subgroup of t of codimension r and let cit be the set of all orbits in x of type g h

examples of symplectic structures in this paper we construct symplectic forms tilde omega k k geqq 0 on a compact manifold tilde y which have the same homotopy theoretic invariants but which are not diffeomorphic

stratified symplectic spaces and reduction let m w be a hamiltonian g space with proper momentum map j m g it is well known that if zero is a regular value of j and g acts freely on the level set j 0 then the reduced space mo j 0 g is a symplectic manifold we show that if the regularity assumptions are dropped the space mo is a union of symplectic manifolds i e it is a stratified symplectic space arms et al 2 proved that mo possesses a natural poisson bracket using their result we study hamiltonian dynamics on the reduced space in particular we show that hamiltonian flows are strata preserving and give a recipe for lifting a reduced hamiltonian flow to the level set j 0 finally we give a detailed description of the stratification of mo and prove the existence of a connected

lagrangian non squeezing and a geometric inequality we prove that if the unit codisc bundle of a closed riemannian manifold embeds symplectically into a symplectic cylinder of radius one then the length of the shortest non trivial closed geodesic is at most half the area of the unit disc

addendum to on the variation in the cohomology of the symplectic form of the reduced phase space here we have used the notation rlx for the inner product of the vector field x with the form a and ix m x j m for the x component of a in an earlier paper 4 it was shown that the push forward j dm of the liouville measure dm on m under the momentum mapping j is a piecewise polynomial measure on moreover in case x has isolated isolated zeros on m an explicit formula for the integral

hamiltonian group actions on exact symplectic manifolds with proper momentum maps are standard we give a complete characterization of hamiltonian actions of compact lie groups on exact symplectic manifolds with proper momentum maps we deduce that every hamiltonian action of a compact lie group on a contractible symplectic manifold with a proper momentum map is globally linearizable

periodic fourth order cubic nls local well posedness and non squeezing property abstract in this paper we consider the cubic fourth order nonlinear schrodinger equation 4nls under the periodic boundary condition we prove two results one is the local well posedness in h s t with 1 3 s 0 for the cauchy problem of the wick ordered 4nls the other one is the non squeezing property for the flow map of 4nls in the symplectic phase space l 2 t to prove the former we used the ideas introduced in 36 and 27 and to prove the latter we used the ideas in 8

complex analytic dynamics on the riemann sphere dichotomie dynamique de fatou et julia points periodiques consequences du theoreme de montel l ensemble de julia est la fermeture de l ensemble des points periodiques repulseurs resultats classiques sur l ensemble de fatou classification de sullivan de l ensemble de fatou une condition pour le developpement sur l ensemble de julia la dynamique des polynomes l ensemble de mandelbrot et le travail de douady et hubbard le theoreme d application de riemann mesurable et la dynamique analytique

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

moment polytopes for symplectic manifolds with monodromy abstract a natural way of generalising hamiltonian toric manifolds is to permit the presence of generic isolated singularities for the moment map for a class of such almost toric 4 manifolds which admits a hamiltonian s 1 action we show that one can associate a group of convex polygons that generalise the celebrated moment polytopes of atiyah guillemin sternberg as an application we derive a duistermaat heckman formula demonstrating a strong effect of the possible monodromy of the underlying integrable system