morse theory for periodic solutions of hamiltonian systems and the maslov index in this paper we prove morse type inequalities for the contractible 1 periodic solutions of time dependent hamiltonian differential equations on those compact symplectic manifolds m for which the symplectic form and the first chern class of the tangent bundle vanish over q m the proof is based on a version of infinite dimensional morse theory which is due to floer the key point is an index theorem for the fredholm operator which plays a central role in floer homology the index formula involves the maslov index of nondegenerate contractible periodic solutions this maslov index plays the same role as the morse index of a nondegenerate critical point does in finite dimensional morse theory we shall use this connection between floer homology and maslov index to establish the existence of infinitely many periodic solutions having integer periods provided that every i periodic solution has at least one floquet multiplier which is not equal to 1

local properties of j complex curves in lipschitz continuous structures we prove the existence of primitive curves and positivity of intersections of j complex curves for lipschitz continuous almost complex structures these results are deduced from the comparison theorem for j holomorphic maps in lipschitz structures previously known for j of class mathcal c 1 lip we also give the optimal regularity of curves in lipschitz structures it occurs to be mathcal c 1 lnlip i e the first derivatives of a j complex curve for lipschitz j are log lipschitz continuous a simple example that nothing better can be achieved is given further we prove the genus formula for j complex curves and determine their principal puiseux exponents all this for lipschitz continuous j s

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

reeb orbits and the minimal discrepancy of an isolated singularity let a be an affine variety inside a complex n dimensional vector space which has an isolated singularity at the origin the intersection of a with a very small sphere turns out to be a contact manifold called the link of a any contact manifold contactomorphic to the link of a is said to be milnor fillable by a if the first chern class of our link is torsion then we can assign an invariant of our singularity called the minimal discrepancy which is an important invariant in birational geometry we define an invariant of the link up to contactomorphism using conley zehnder indices of reeb orbits and then we relate this invariant with the minimal discrepancy as a result we show that the standard contact five dimensional sphere has a unique milnor filling up to normalization proving a conjecture by seidel

factoring polynomials over algebraic number fields we show that if f x is a polynomial in z alpha x where alpha satisfies a monic irreducible polynomial over z then f x can be factored over q alpha x in polynomial time we also show that the splitting field of f x can be determined in time polynomial in splitting field of f x q log x

frobenius maps of abelian varieties and finding roots of unity in finite fields if twere done when tis done then twere well it were done quickly macbeth abstract we give a generalization to abelian varieties over finite fields of the algorithm of schoof for elliptic curves schoof showed that for an elliptic curve e over f given by a weierstrass equation one can compute the number of q f rational points of e in time 0 log our result is the following let a be an abelian variety over f then one can compute the characteristic polynomial of the frobenius endomorphism of a in time 0 o q where a and the implied constant depend only on the dimension of the embedding space of a the number of equations defining a and the addition law and their degrees the method generalizing that of schoof is to use the machinery developed by weil to prove the riemann hypothesis for abelian varieties by means of this theory the calculation is reduced to ideal theoretic computations in a ring of polynomials in several variables over f as applications we show how to count the rational points on the reductions modulo primes p of a fixed curve over q in time polynomial in logp we show also that for a fixed prime we can compute the th roots of unity mod p when they exist in polynomial time in logp this generalizes schoof s application of his algorithm to find square roots of a fixed integer x mod p in this paper we generalize to abelian varieties over finite fields the algorithm of schoof 19 for elliptic curves over finite fields and the application given by schoof for his algorithm schoof showed that for an elliptic curve e over a finite field f given by a weierstrass equation one can compute the number of f rational points of f in time polynomial in log q the algorithm computes the characteristic polynomial p t g z t of the frobenius endomorphism of f for an elliptic curve p t is a monic quadratic polynomial with constant term q and the number of fq rational points of e is f l our result is the following theorem a let a be an abelian variety over a finite field f given explicitly as a projective variety with an explicit addition law then one can compute

an exact sequence for contact and symplectic homology a symplectic manifold w with contact type boundary m w induces a linearization of the contact homology of m with corresponding linearized contact homology hc m we establish a gysin type exact sequence in which the symplectic homology sh w of w maps to hc m which in turn maps to hc m by a map of degree 2 which then maps to sh w furthermore we give a description of the degree 2 map in terms of rational holomorphic curves with constrained asymptotic markers in the symplectization of m

a comparison theorem for gromov witten invariants in the symplectic category abstract we exploit the geometric approach to the virtual fundamental class due to fukaya ono and li tian to compare gromov witten invariants of a symplectic manifold and a symplectic submanifold whenever all constrained stable maps to the former are contained in the latter to first order various special cases of the comparison theorem in this paper have long been used in the algebraic category some of them have also appeared in the symplectic setting combined with the inherent flexibility of the symplectic category the main theorem leads to a confirmation of pandharipande s gopakumar vafa prediction for gw invariants of fano classes in 6 dimensional symplectic manifolds the proof of the main theorem uses deformations of the cauchy riemann equation that respect the submanifold and carleman similarity principle for solutions of perturbed cauchy riemann equations in a forthcoming paper we apply a similar approach to relative gromov witten invariants and the absolute relative correspondence in genus 0

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

vanishing of rabinowitz floer homology on negative line bundles following frauenfelder rabinowitz action functional on very negative line bundles habilitationsschrift munich munchen 2008 albers and frauenfelder bubbles and onis 2014 arxiv 1412 4360 we construct rabinowitz floer homology for negative line bundles over symplectic manifolds and prove a vanishing result ritter adv math 262 1035 1106 2014 showed that symplectic homology of these spaces does not vanish in general thus the theorem mathrm sh 0 leftrightarrow mathrm rfh 0 ritter in j topol 6 2 391 489 2013 does not extend beyond the symplectically aspherical situation we give a conjectural explanation in terms of the cieliebak frauenfelder oancea long exact sequence cieliebak et al ann sci ec norm super 4 43 6 957 1015 2010

geometric composition in quilted floer theory abstract we prove that floer cohomology of cyclic lagrangian correspondences is invariant under transverse and embedded composition under a general set of assumptions

a quantum cup length estimate for symplectic fixed points a new lower bound for the number of fixed points of hamiltonian automorphisms of closed symplectic manifolds m is established the new estimate extends the previously known estimates to the class of weakly monotone symplectic manifolds we prove for arbitrary closed symplectic manifolds with rational symplectic class that the cup length estimate holds true if the hofer energy of the hamiltonian automorphism is sufficiently small for arbitrary energy and on weakly monotone symplectic manifolds we define an analogon to the cup length based on the quantum cohomology ring of m providing a quantum cup length estimate

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

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

from symplectic cohomology to lagrangian enumerative geometry abstract we build a bridge between floer theory on open symplectic manifolds and the enumerative geometry of holomorphic disks inside their fano compactifications by detecting elements in symplectic cohomology which are mirror to landau ginzburg potentials we also treat the higher maslov index versions of the potentials we discover a relation between higher disk potentials and symplectic cohomology rings of smooth anticanonical divisor complements themselves conjecturally related to closed string gromov witten invariants and explore several other applications to the geometry of liouville domains

floer cohomology and geometric composition of lagrangian correspondences abstract we prove an isomorphism of floer cohomologies under geometric composition of lagrangian correspondences in exact and monotone settings

two coverings of jacobians of curves of genus 2 given a curve c of genus 2 defined over a field k of characteristic different from 2 with a jacobian variety j we show that the two coverings corresponding to elements of a large subgroup of h1 gal ks k j 2 ks containing the selmer group when k is a global field can be embedded as an intersection of 72 quadrics in 15k just as the jacobian j itself moreover we actually give explicit equations for the models of these twists in the generic case extending the work of gordon and grant which applied only to the case when all weierstrass points are rational in addition we describe elegant equations of the jacobian itself and answer a question of cassels and flynn concerning a map from the kummer surface in 3 to the desingularized kummer surface in 5

a module structure on the symplectic floer cohomology the aim of this paper is to offer an affirmative answer to the floer conjectures in 2 p 589 which states that there is a module structure on the z 2 n graded symplectic floer cohomology for monotone symplectic manifolds by constructing a z graded symplectic floer cohomology as an integral lift of the z 2 n graded symplectic floer cohomology we gain control of the holomorphic bubbling spheres this makes a module structure on the z graded floer cohomology there is a spectral sequence with e 1 given by the z graded symplectic floer cohomology such a spectral sequence converges to the z 2 n graded symplectic floer cohomology hence we induce a module structure for the z 2 n graded symplectic floer cohomology by the spectral sequence and algebraic topology methods

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

reduced genus one gromov witten invariants in a previous paper we described a natural closed subset m 0 1 k x a j of the moduli space m1 k x a j of stable genus one j holomorphic maps into a symplectic manifold x in this paper we generalize the definition of the main component to moduli spaces of perturbed in a restricted way j holomorphic maps this generalization implies that m 0 1 k x a j just like m1 k x a j carries a virtual fundamental class and can be used to define symplectic invariants these truly genus one invariants constitute part of the standard genus one gromovwitten invariants which arise from the entire moduli space m1 k x a j the new invariants are more geometric and can be used to compute the genus one gw invariants of complete intersections as shown in a separate paper

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

on the arnold conjecture for weakly monotone symplectic manifolds we show the arnold conjecture concerning symplectic fixed points in the case that the symplectic manifold is weakly monotone and all the fixed points are non degenerate in particular the conjecture is true in dimension 2 4 6 if all the fixed points are non degenerate

witten s complex and infinite dimensional morse theory we investigate the relation between the trajectories of a finite dimensional gradient flow connecting two critical points and the cohomology of the surrounding space the results are applied to an infinite dimensional problem involving the symplect c action function

quivers floer cohomology and braid group actions we consider the derived categories of modules over a certain family a m of graded rings and floer cohomology of lagrangian intersections in the symplectic manifolds which are the milnor fibres of simple singularities of type a m we show that each of these two rather different objects encodes the topology of curves on an m 1 punctured disc we prove that the braid group b m 1 acts faithfully on the derived category of a m modules and that it injects into the symplectic mapping class group of milnor fibers the philosophy behind our results is as follows using floer cohomology one should be able to associate to the milnor fibre a triangulated category its construction has not been carried out in detail this triangulated category should contain a full subcategory which is equivalent up to a slight difference in the grading to the derived category of a m modules the full embedding would connect the two occurrences of the braid group thus explaining the similarity between them

a combinatorial description of knot floer homology given a grid presentation of a knot or link k in the three sphere we describe a heegaard diagram for the knot complement in which the heegaard surface is a torus and all elementary domains are squares using this diagram we obtain a purely combinatorial description of the knot floer homology of k

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

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

the role of the legendre transform in the study of the floer complex of cotangent bundles consider a classical hamiltonian h on the cotangent bundle t m of a closed orientable manifold m and let l tm r be its legendre dual lagrangian in a previous paper we constructed an isomorphism from the morse complex of the lagrangian action functional that is associated to l to the floer complex that is determined by h in this paper we give an explicit construction of a homotopy inverse of contrary to other previously defined maps going in the same direction is an isomorphism at the chain level and preserves the action filtration its definition is based on counting floer trajectories on the negative half cylinder that on the boundary satisfy half of the hamilton equations albeit not of lagrangian type such a boundary condition defines fredholm operators with good compactness properties we also present a heuristic argument which independently of any fredholm and compactness analysis explains why the spaces of maps that are used in the definition of and are the natural ones the legendre transform plays a crucial role both in our rigorous and in our heuristic arguments we treat with some detail the delicate issue of orientations and show that the homology of the floer complex is isomorphic to the singular homology of the loop space of m with a system of local coefficients which is defined by the pullback of the second stiefel whitney class of tm on 2 tori in m 2015 wiley periodicals inc

homological mirror symmetry for calabi yau hypersurfaces in projective space we prove homological mirror symmetry for a smooth d dimensional calabi yau hypersurface in projective space for any d 3 for example d 3 is the quintic threefold the main techniques involved in the proof are the construction of an immersed lagrangian sphere in the d dimensional pair of pants the introduction of the relative fukaya category and an understanding of its grading structure a description of the behaviour of this category with respect to branched covers via an orbifold fukaya cate gory a morse bott model for the relative fukaya category that allows one to make explicit computations and the introduction of certain graded categories of matrix factorizations mirror to the relative fukaya category

on one parameter proofs of almost sure convergence of multiparameter processes the martingale convergence theorem of r cairoli 4 was proved assuming that the filtration satisfies a conditional independence assumption usualy called f4 it belongs to the folklore of the subject see also 17 that f4 can be restated as a condition on commutation of conditional expectation operators this formulation will allow us to derive cairoli s theorem from a simple general argument about operators on orlicz spaces the advantage is that one obtains a unified proof of multi parameter versions of several other results the theorems of rota dunford schwartz akcoglu and stein h fnlmer was able to apply proposition 2 1 below to random fields

on multiparameter ergodic and martingale theorems in infinite measure spaces a unified proof is given of several ergodic and martingale theorems in infinite measure spaces

equivariant split generation and mirror symmetry of special isogenous tori abstract we prove a version of equivariant split generation of fukaya category when a symplectic manifold admits a free action of a finite group g combining this with some generalizations of seidel s algebraic frameworks from 35 we obtain new cases of homological mirror symmetry for some symplectic tori with non split symplectic forms which we call special isogenous tori this extends the work of abouzaid smith 2 we also show that derived fukaya categories are complete invariants of special isogenous tori

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

quantum riemann roch lefschetz and serre given a holomorphic vector bundle e over a compact kahler manifold x one defines twisted gromov witten invariants of x to be intersection numbers in moduli spaces of stable maps f x with the cap product of the virtual fundamental class and a chosen multiplicative invertible characteristic class of the virtual vector bundle h 0 f e h 1 f e using the formalism of quantized quadratic hamiltonians 25 we express the descendant potential for the twisted theory in terms of that for x this result theorem 1 is a consequence of mumford s grothendieck riemann roch theorem applied to the universal family over the moduli space of stable maps it determines all twisted gromov witten invariants of all genera in terms of untwisted invari ants when e is concave and the c equivariant inverse euler class is chosen as the characteristic class the twisted invariants of x give gromov witten invariants of the total space of e nonlinear serre duality 21 23 expresses gromov witten invariants of e in terms of those of the super manifold e it relates gromov witten invariants of x twisted by the inverse euler class and e to gromov witten invariants of x twisted by the euler class and e we derive from theorem 1 nonlinear serre duality in a very general form corollary 2 when the bundle e is convex and a submanifold y x is defined by a global section of e the genus zero gromov witten invariants of e coin cide with those of y we establish a quantum lefschetz hyperplane section principle theorem 2 expressing genus zero gromov witten invariants of a complete intersection y in terms of those of x this extends earlier results 4 9 18 29 33 and yields most of the known mirror formulas for toric complete intersections

integer points on curves of genus two and their jacobians let c be a curve of genus 2 defined over a number field and e the image of c embedded into its jacobian j we show that the heights of points of j which are integral with respect to 2 6 can be effectively bounded as a result if p is a point on c and p its image under the hyperelliptic involution then the heights of points on c which are integral with respect to p and p can be effectively bounded in such a way that we can isolate the dependence on p and show that if the height of p is bigger than some bound then there are no points which are s integral with respect to p and p we relate points on c which are integral with respect to p to points on j which are integral with respect to 6 and discuss approaches toward bounding the heights of the latter

floer cohomology and pencils of quadrics there is a classical relationship in algebraic geometry between a hyperelliptic curve and an associated pencil of quadric hypersurfaces we investigate symplectic aspects of this relationship with a view to applications in low dimensional topology

monodromy of monomially admissible fukaya seidel categories mirror to toric varieties abstract mirror symmetry for a toric variety involves laurent polynomials whose symplectic topology is related to the algebraic geometry of the toric variety we show that there is a monodromy action on the monomially admissible fukaya seidel categories of these laurent polynomials as the arguments of their coefficients vary that corresponds under homological mirror symmetry to tensoring by a line bundle naturally associated to the monomials whose coefficients are rotated in the process we introduce the monomially admissible fukaya seidel category as a new interpretation of the fukaya seidel category of a laurent polynomial on c n which has other potential applications and give evidence of homological mirror symmetry for non compact toric varieties

inversion of a general hyperelliptic integral and particle motion in hořava lifshitz black hole space times the description of many dynamical problems such as the particle motion in higher dimensional spherically and axially symmetric space times is reduced to the inversion of hyperelliptic integrals of all three kinds the result of the inversion is defined locally using the algebro geometric techniques of the standard jacobi inversion problem and the foregoing restriction to the divisor for a representation of the hyperelliptic functions the klein weierstras multi variable function is introduced it is shown that all parameters needed for the calculations such as period matrices and abelian images of branch points can be expressed in terms of the periods of holomorphic differentials and constants the cases of genus two three and four are considered in detail the method is exemplified by the particle motion associated with genus one elliptic and genus three hyperelliptic curves applications are for instance solutions to the geodesic equations in the space times of static spherically symmetric ho ava lifshitz black holes

cycles of quadratic polynomials and rational points on a genus 2 curve it has been conjectured that for n sufficiently large there are no quadratic polynomials in q z with rational periodic points of period n morton proved there were none with n 4 by showing that the genus 2 algebraic curve that classifies periodic points of period 4 is birational to x 1 16 whose rational points had been previously computed we prove there are none with n 5 here the relevant curve has genus 14 but it has a genus 2 quotient whose rational points we compute by performing a 2 descent on its jacobian and applying a refinement of the method of chabauty and coleman we hope that our computation will serve as a model for others who need to compute rational points on hyperelliptic curves we also describe the three possible galois stable 5 cycles and show that there exist galois stable n cycles for infinitely many n furthermore we answer a question of morton by showing that the genus 14 curve and its quotient are not modular finally we mention some partial results for n 6

punctured holomorphic curves and lagrangian embeddings we use a neck stretching argument for holomorphic curves to produce symplectic disks of small area and maslov class with boundary on lagrangian submanifolds of nonpositive curvature applications include the proof of audin s conjecture on the maslov class of lagrangian tori in linear symplectic space the construction of a new symplectic capacity obstructions to lagrangian embeddings into uniruled symplectic manifolds a quantitative version of arnold s chord conjecture and estimates on the size of weinstein neighbourhoods the main technical ingredient is transversality for the relevant moduli spaces of punctured holomorphic curves with tangency conditions

the conley conjecture we prove the conley conjecture for a closed symplectically aspherical symplectic manifold a hamiltonian diffeomorphism of such a manifold has infinitely many periodic points more precisely we show that a hamiltonian diffeomorphism with finitely many fixed points has simple periodic points of arbitrarily large period this theorem generalizes for instance a recent result of hingston establishing the conley conjecture for tori

computing l l isogenies in polynomial time on jacobians of genus 2 curves in this paper we compute l isogenies between abelian varieties over a field of characteristic different from 2 in polynomial time in l when l is an odd prime which is coprime to the characteristic we use level n symmetric theta structure where n 2 or n 4 in a second part of this paper we explain how to convert between mumford coordinates of jacobians of genus 2 hyperelliptic curves to theta coordinates of level 2 or 4 combined with the preceding algorithm this gives a method to compute l l isogenies in polynomial time on jacobians of genus 2 curves

symplectic homology of some brieskorn manifolds this paper consists of two parts in the first part we use symplectic homology to distinguish the contact structures on the brieskorn manifolds varsigma 2 ell 2 2 2 which contact homology cannot distinguish this answers a question from kwon and van koert brieskorn manifolds in contact topology preprint 2013 arxiv 1310 0343 in the second part we prove the existence of infinitely many exotic but homotopically trivial exotic contact structures on s 7 distinguished by the mean euler characteristic of s 1 equivariant symplectic homology apart from various connected sum constructions these contact structures can be taken from the brieskorn manifolds varsigma 78k 1 13 6 3 3 we end with some considerations about extending this result to higher dimensions

lusternik schnirelmann theory and closed reeb orbits we develop a variant of lusternik schnirelmann theory for the shift operator in equivariant floer and symplectic homology our key result is that the spectral invariants are strictly decreasing under the action of the shift operator when periodic orbits are isolated as an application we prove new multiplicity results for simple closed reeb orbits on the standard contact sphere the unit cotangent bundle to the sphere and some other contact manifolds we also show that the lower conley zehnder index enjoys a certain recurrence property and revisit and reprove from a different perspective a variant of the common jump theorem of long and zhu this is the second combinatorial ingredient in the proof of the multiplicity results

symplectic fixed points and holomorphic spheres letp be a symplectic manifold whose symplectic form integrated over the spheres inp is proportional to its first chern class on the loop space ofp we consider the variational theory of the symplectic action function perturbed by a hamiltonian term in particular we associate to each isolated invariant set of its gradient flow an abelian group with a cyclic grading it is shown to have properties similar to the homology of the conley index in locally compact spaces as an application we show that if the fixed point set of an exact diffeomorphism onp is nondegenerate then it satisfies the morse inequalities onp we also discuss fixed point estimates for general exact diffeomorphisms

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

hyperelliptic solutions of modified korteweg de vries equation of genus g essentials of the miura transformation explicit hyperelliptic solutions of the modified korteweg de vries equations without any ambiguous parameters were constructed in terms of only the hyperelliptic al functions over the non degenerate hyperelliptic curve y2 f x of arbitrary genus g in the derivation any functions or baker akhiezer functions were not essentially used then the miura transformation naturally appears as the connection between the hyperelliptic functions and hyperelliptic al functions

the kunneth formula in floer homology for manifolds with restricted contact type boundary we prove the kunneth formula in floer co homology for manifolds with restricted contact type boundary we use viterbo s definition of floer homology involving the symplectic completion by adding a positive cone over the boundary the kunneth formula implies the vanishing of floer co homology for subcritical stein manifolds other applications include the weinstein conjecture in certain product manifolds obstructions to exact lagrangian embeddings existence of holomorphic curves with lagrangian boundary condition as well as symplectic capacities

coherent orientations in symplectic field theory we study the coherent orientations of the moduli spaces of holomorphic curves in symplectic field theory generalizing a construction due to floer and hofer in particular we examine their behavior at multiple closed reeb orbits under change of the asymptotic direction the orientations are determined by a certain choice of orientation at each closed reeb orbit that is similar to the orientation of the unstable tangent spaces of critical points in finite dimensional morse theory

on the floer homology of cotangent bundles this paper concerns floer homology for periodic orbits and for a lagrangian intersection problem on the cotangent bundle t m of a compact orientable manifold m the first result is a new l estimate for the solutions of the floer equation which allows us to deal with a larger and more natural class of hamiltonians the second and main result is a new construction of the isomorphism between the floer homology and the singular homology of the free loop space of m in the periodic case or of the based loop space of m in the lagrangian intersection problem the idea for the construction of such an isomorphism is to consider a hamiltonian that is the legendre transform of a lagrangian on t m and to construct an isomorphism between the floer complex and the morse complex of the classical lagrangian action functional on the space of w1 2 free or based loops on m 2005 wiley periodicals inc

the symplectic arc algebra is formal this is the author accepted manuscript the final version is available from duke university press via http dx doi org 10 1215 00127094 3449459

computing the mordell weil rank of jacobians of curves of genus two we derive the equations necessary to perform a two descent on the jacobians of curves of genus two with rational weierstrass points we compute the mordell weil rank of the jacobian of some genus two curves defined over the rationals and discuss the practicality of using this method

factorization with genus 2 curves the elliptic curve method ecm is one of the best factorization methods available it is possible to use hyperelliptic curves instead of elliptic curves but it is in theory slower we use special hyperelliptic curves and kummer surfaces to reduce the complexity of the algorithm our implementation gmp hecm is faster than gmp ecm for factoring large numbers

framed bordism and lagrangian embeddings of exotic spheres in dimensions congruent to 1 modulo 4 we prove that the cotangent bundle of an exotic sphere which does not bound a parallelisable manifold is not symplectomorphic to the cotangent bundle of the standard sphere more precisely we prove that such an exotic sphere cannot embed as a lagrangian in the cotangent bundle of the standard sphere the main ingredients of the construction are 1 the fact that the graph of the hopf bration embeds the standard sphere and hence any lagrangian which embeds in its cotangent bundle as a displaceable lagrangian in the product a symplectic vector space of the appropriate dimension with its complex projective space and 2 a moduli space of solutions to a perturbed cauchyriemann equation introduced by gromov

quantum lefschetz hyperplane theorem the mirror theorem is generalized to any smooth projective variety x that is a fundamental relation between the gromov witten invariants of x and gromov witten invariants of complete intersections y in x is established

floer theory for negative line bundles via gromov witten invariants abstract we prove that the gw theory of negative line bundles m tot l b determines the symplectic cohomology indeed s h m is the quotient of q h m by the kernel of a power of quantum cup product by c 1 l we prove this also for negative vector bundles and the top chern class we calculate s h and q h for o n c p m for example for o 1 m is the blow up of c m 1 at the origin and s h m has rank m we prove kodaira vanishing for very negative l s h 0 and serre vanishing if we twist a complex vector bundle by a large power of l s h 0 observe s h m 0 iff c 1 l is nilpotent in q h m this implies oancea s result b 2 b 0 s h m 0 we prove the weinstein conjecture for any contact hypersurface surrounding the zero section of a negative line bundle for symplectic manifolds x conical at infinity we build a homomorphism from 1 ham l x to invertibles in s h x this is similar to seidel s representation for closed x except now they are not invertibles in q h x

the complexity of the word problems for commutative semigroups and polynomial ideals abstract any decision procedure for the word problems for commutative semigroups and polynomial deals inherently requires computational storage space growing exponentially with the size of the problem instance to which the procedure is applied this bound is achieved by a simple procedure for the semigroup problem