an infinite dimensional morse theory with applications in this paper we construct an infinite dimensional extraordinary cohomology theory and a morse theory corresponding to it these theories have some special properties which make them useful in the study of critical points of strongly indefinite functionals by strongly indefinite we mean a functional unbounded from below and from above on any subspace of finite codimension several applications are given to hamiltonian systems the onedimensional wave equation of vibrating string type and systems of elliptic partial differential equations 0 introduction let e be a real hilbert space with an inner product and let be a twice continuously differentiable functional denote the frechet derivative and the gradient of at x by x and x respectively where as usual x y x y y e recall that a point x0 e is said to be critical if x0 0 or equivalently if x0 0 the level c r will be called regular if 1 c contains no critical points and critical if x0 0 for some x0 1 c let a b a b be two regular levels of denote m 1 a b and consider the restriction of to m in morse theory one is interested in the local topological structure of the level sets of m near a critical point and in the relation between this local structure and the topological structure of the set m to be more specific suppose that x0 m is an isolated critical point of then one defines a sequence of critical groups of at x0 by setting cq x0 hq c u u x0 q 0 1 2 0 1 where c x0 c x e x c hq is the q th singular homology group with coefficients in some field f and u is a neighbourhood of x0 define the morse index of x0 to be the maximal dimension of a subspace of e on which the quadratic form x0 y y is negative definite one shows that if x0 is a nondegenerate critical point i e if x0 e e is invertible then cq x0 f received by the editors march 20 1995 1991 mathematics subject classification primary 58e05 secondary 34c25 35j65 35l05 55n20 58f05

morse theory and asymptotic linear hamiltonian system on considere l existence de solutions 2 periodiques des systemes hamiltoniens asymptotiquement quadratiques jz h z t ou j 0 i i 0 i in la matrice identite de n n

applications of equivariant degree for gradient maps to symmetric newtonian systems abstract we consider g s 1 with being a finite group for which the complete euler ring structure in u g is described the multiplication tables for d 6 s 4 and a 5 are provided in the appendix the equivariant degree for g orthogonal maps is constructed using the primary equivariant degree with one free parameter we show that the g orthogonal degree extends the degree for g gradient maps in the case of g s 1 introduced by g ba in k g ba w krawcewicz j wu an equivariant degree with applications to symmetric bifurcation problems i construction of the degree bull london math soc 69 1994 377 398 the computational results obtained are applied to a symmetric autonomous newtonian system for which we study the existence of 2 periodic solutions for some concrete cases we present the symmetric classification of the solution set for the systems considered

the minkowski problem for polytopes abstract the traditional solution to the minkowski problem for polytopes involves two steps first the existence of a polytope satisfying given boundary data is demonstrated in the second step the uniqueness of that polytope up to translation is then shown to follow from the equality conditions of minkowski s inequality a generalized isoperimetric inequality for mixed volumes that is typically proved in a separate context in this article we adapt the classical argument to prove both the existence theorem of minkowski and his mixed volume inequality simultaneously thereby providing a new proof of minkowski s inequality that demonstrates the equiprimordial relationship between these two fundamental theorems of convex geometry

solutions of systems of elliptic differential equations on circular domains abstract we study global bifurcation of weak solutions of systems of elliptic differential equations considered on so 2 invariant domains we formulate sufficient conditions for the existence of unbounded continua of nontrivial solutions branching from the trivial ones as the main tool we use the degree for so 2 equivariant gradient maps defined by the second author in rybicki nonlinear anal tma 23 1 1994 83

semiclassical spectral invariants for schrodinger operators in this article we show how to compute the semi classical spectral measure associated with the schrodinger operator on r n and by examining the rst few terms in the asymptotic expansion of this measure obtain inverse spectral results in one and two dimensions in particular we show that for the schrodinger operator on r 2 with a radially symmetric electric potential v and magnetic potential b both v and b are spectrally determined we also show that in one dimension there is a very simple explicit identity relating the spectral measure of the schrodinger operator with its birkho canonical form

the complexity of degree anonymization by vertex addition motivated by applications in privacy preserving data publishing we study the problem of making an undirected graph k anonymous by adding few vertices together with some incident edges that is after adding these dummy vertices for every vertex degree d appearing in the resulting graph there shall be at least k vertices with degree d we explore three variants of vertex addition justified by real world considerations and study their parameterized computational complexity we derive mostly intractability results even for very restricted cases including trees and bounded degree graphs but also obtain some encouraging fixed parameter tractability results degree anonymization by vertex addition is computationally intractable in general posing structural restrictions on the edges connected to the new vertices seems to make the problem even harder there are some tractable special cases for example when the number of new edges is small

a refined complexity analysis of degree anonymization in graphs motivated by a strongly growing interest in graph anonymization we study the np hard degree anonymity problem asking whether a graph can be made k anonymous by adding at most a given number of edges herein a graph is k anonymous if for every vertex in the graph there are at least k 1 other vertices of the same degree our algorithmic results shed light on the performance quality of a popular heuristic due to liu and terzi acm sigmod 2008 in particular we show that the heuristic provides optimal solutions if many edges need to be added based on this we develop a polynomial time data reduction yielding a polynomial size problem kernel for degree anonymity parameterized by the maximum vertex degree in terms of parameterized complexity analysis this result is in a sense tight since we also show that the problem is already np hard for h index three implying np hardness for smaller parameters such as average degree and degeneracy

spectral flow and bifurcation of critical points of strongly indefinite functionals abstract our main results here are as follows let x be a family of 2 periodic hamiltonian vectorfields that depend smoothly on a real parameter in a b and has a known trivial branch s of 2 periodic solutions let p be the poincare map of the linearization of x at s if the conley zehnder index of the path p does not vanish then any neighborhood of the trivial branch of periodic solutions contains 2 periodic solutions not on the branch moreover if each solution s is constant and each linearization a of x at s is time independent then bifurcation of 2 periodic orbits from the branch of equilibria arises whenever i a b i a b where i a is the index of the linear hamiltonian system ju au

the complexity of degree anonymization by graph contractions abstract we study the computational complexity of k anonymizing a given graph by as few graph contractions as possible a graph is said to be k anonymous if for every vertex in it there are at least k 1 other vertices with exactly the same degree the general degree anonymization problem is motivated by applications in privacy preserving data publishing and was studied to some extent for various graph operations most notable operations being edge addition edge deletion vertex addition and vertex deletion we complement this line of research by studying the computational complexity of degree anonymization by graph contractions we consider several variants of graph contractions which are operations of interest for example in the contexts of social networks and clustering algorithms we show that the problem of degree anonymization by graph contractions is np hard even for some very restricted inputs and identify some fixed parameter tractable cases

semi classical weights and equivariant spectral theory abstract we prove inverse spectral results for differential operators on manifolds and orbifolds invariant under a torus action these inverse spectral results involve the asymptotic equivariant spectrum which is the spectrum itself together with very large weights of the torus action on eigenspaces more precisely we show that the asymptotic equivariant spectrum of the laplace operator of any toric metric on a generic toric orbifold determines the equivariant biholomorphism class of the orbifold we also show that the asymptotic equivariant spectrum of a t n invariant schrodinger operator on r n determines its potential in some suitably convex cases in addition we prove that the asymptotic equivariant spectrum of an s 1 invariant metric on s 2 determines the metric itself in many cases finally we obtain an asymptotic equivariant inverse spectral result for weighted projective spaces as a crucial ingredient in these inverse results we derive a surprisingly simple formula for the asymptotic equivariant trace of a family of semi classical differential operators invariant under a torus action

kahler metrics on toric orbifolds a theorem of e lerman and s tolman generalizing a result of t delzant states that compact symplectic toric orbifolds are classified by their moment polytopes together with a positive integer label attached to each of their facets in this paper we use this result and the existence of global actionangle coordinates to give an effective parametrization of all compatible toric complex structures on a compact symplectic toric orbifold by means of smooth functions on the corresponding moment polytope this is equivalent to parametrizing all toric kahler metrics and generalizes an analogous result for toric manifolds a simple explicit description of interesting families of extremal kahler metrics arising from recent work of r bryant is given as an application of the approach in this paper the fact that in dimension four these metrics are self dual and conformally einstein is also discussed this gives rise in particular to a one parameter family of self dual einstein metrics connecting the well known eguchi hanson and taub nut metrics

a cohomological conley index in hilbert spaces and applications to strongly indefinite problems abstract a cohomological conley index is defined for flows on infinite dimensional real hilbert spaces generated by vector fields of the form f h h f x lx k x where l h h is a bounded linear operator satisfying certain technical assumptions and k is a completely continuous perturbation generalized morse inequalities for morse decompositions of isolated invariant sets are proved simple examples are presented to show how the theory can be applied to strongly indefinite problems

on critical point theory for indefinite functionals in the presence of symmetries we consider functionals which are not bounded from above or from below even modulo compact perturbations and which exhibit certain symmetries with respect to the action of a compact lie group we develop a method which permits us to prove the existence of multiple critical points for such functionals the proofs are carried out directly in an infinite dimensional hilbert space and they are based on minimax arguments the applications given here are to hamiltonian systems of ordinary differential equations where the existence of multiple time periodic solutions is established for several classes of hamiltonians symmetry properties of these hamiltonians such as time translation invariancy or evenness are exploited introduction many variational problems which arise from physics or mathematical physics are indefinite in the sense that the functionals involved are not bounded from below or from above however some of these functionals defined in an appropriate function space are semidefinite in the sense that they are bounded from below or from above if perturbed with a weakly continuous functional this paper deals with functionals which are not semidefinite usually problems involving indefinite functionals are more difficult to handle and only very recently a method has been developed which permits us to treat such problems directly in an infinite dimensional function space 7 in 7 some theorems have been proved which establish the existence of at least one nontrivial critical point for such functionals in many physical situations there are problems which have symmetries with respect to the action of some lie group in this case we expect the existence of many critical points this has been established for semidefinite functionals cf e g 8 and 3 for even functionals 9 for a zp action with p prime numbers 15 16 10 and 6 for an s action in this paper we develop a method which allows us to estimate the number of critical points of indefinite functionals in the presence of symmetries more precisely received by the editors july 16 1980 and in revised form june 26 1981 1980 mathematics subject classification primary 58e05 70h05 secondary 34c25

equivariant inverse spectral theory and toric orbifolds abstract let o 2 n be a symplectic toric orbifold with a fixed t n action and with a toric kahler metric g in 10 we explored whether when o is a manifold the equivariant spectrum of the laplace operator g on c o determines o up to symplectomorphism in the setting of toric orbifolds we significantly improve upon our previous results and show that a generic toric orbifold is determined by its equivariant spectrum up to two possibilities this involves developing the asymptotic expansion of the heat trace on an orbifold in the presence of an isometry we also show that the equivariant spectrum determines whether the toric kahler metric has constant scalar curvature

integer programming with a fixed number of variables it is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable the proof depends on methods from geometry of numbers

global bifurcations of critical orbits of g invariant strongly indefinite functionals abstract let h be a separable hilbert space which is an orthogonal representation of a compact lie group g and let h r be a g invariant strongly indefinite functional of the class c 1 to study critical orbits of we have defined the degree for g invariant strongly indefinite functionals which is an element of the euler ring u g using this degree we have formulated the rabinowitz alternative for g invariant strongly indefinite functionals the abstract results are applied to the study of global bifurcations of weak solutions of non cooperative systems of elliptic differential equations

finding large degree anonymous subgraphs is hard a graph is said to be k anonymous for an integer k if for every vertex in the graph there are at least k 1 other vertices with the same degree we examine the computational complexity of making a given undirected graph k anonymous either through at most s vertex deletions or through at most s edge deletions the corresponding problem variants are denoted by anonym v del and anonym e del we present a variety of hardness results most of them hold for both problems the two variants are intractable from the parameterized as well as from the approximation point of view in particular we show that both variants remain np hard on very restricted graph classes like trees even if k 2 we further prove that both variants are w1 hard with respect to the combined parameter solutions size s and anonymity level k with respect to approximability we obtain hardness results showing that neither variant can be approximated in polynomial time within a factor better than n 1 2 unless p np furthermore for the optimization variants where the solution size s is given and the task is to maximize the anonymity level k this inapproximability result even holds if we allow a running time of f s n o 1 for any computable function f on the positive side we classify both problem variants as fixed parameter tractable with respect to the combined parameter solution size s and maximum degree

equivariant bifurcation index abstract we consider a bifurcation index bif g k 0 1 u g defined in terms of the degree for g equivariant gradient maps see g ba 1997 21 rybicki 1994 22 rybicki 2005 23 where g is a real compact connected lie group and u g is the euler ring of g see tom dieck 1977 29 tom dieck 1987 30 the main result of this article is the following bif g k 0 1 u g iff bif t k 0 1 u t where t g is a maximal torus of g it is also shown that all the bifurcation points of weak solutions of the following problem u f u in b n u 0 on s n 1 are global bifurcation points additionally the global symmetry breaking bifurcation points are characterised

the index change and global bifurcation for flows with a first integral the existence continuation and bifurcation of solutions of fixed period r of one parameter families of ordinary differential equations with a first integral may be studied using the topological degrees defined in 6 and 7 and the change of index formula given in 8 the purpose here is threefold first suppose that f x v x 0 so that v is a first integral for x f x we examine how the behaviour of f restricts that of v and vice versa and how the periodic spectra of linear ordinary differential equations are affected by the presence of a linear or non linear which occurs surprisingly naturally first integral second we give classes of one parameter families of ordinary differential equations where the change of index in 8 is non zero

spectral flow and bifurcation of critical points of strongly indefinite functionals part i general theory abstract spectral flow is a well known homotopy invariant of paths of selfadjoint fredholm operators we describe here a new construction of this invariant and prove the following theorem let i u rbe ac2function on the product of a real intervali a b with a neighborhood u of the origin of a real separable hilbert space h and such that for each in i 0is a critical point of the functional assume that the hessian l of at0is fredholm and moreover that laand lbare nonsingular if the spectral flow of the path l idoes not vanish then the interval i contains at least one bifurcation point from the trivial branch for solutions of the equation x x 0 equivalently every neighborhood of i 0 contains points of the form x wherex 0 is a critical point of

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

critical point theory for indefinite functionals with symmetries letxbe a hilbert space and c1 x r be strongly indefinite assume in addition that a compact lie groupgacts orthogonally onxand that is invariant in order to find critical points of we develop the limit relative category of fournieret al in the equivariant context we use this to prove two generalizations of the symmetric mountain pass theorem and a linking theorem in the case of the mountain pass theorem the mountain range is allowed to lie in a subspace of infinite codimension also other conditions of the classical symmetric mountain pass theorem forg z 2 due to ambrosetti and rabinowitz can be weakened considerably for example we are able to deal with infinite dimensional fixed point spaces the proofs consist of a direct reduction to a relative borsuk ulam type theorem this provides a new proof even for the classical mountain pass theorem the abstract critical point theorems are applied to an elliptic system with dirichlet boundary conditions we only need a weak version of the usual superquadraticity condition the linking theorem can be applied to asymptotically linear hamiltonian systems which are symmetric with respect to a generalized symplectic group action