recent progress on the poincare conjecture and the classification of 3 manifolds if m is a closed 3 manifold with trivial fundamental group then is m diffeomorphic to s the poincare conjecture is that the answer to this question is yes developing tools to attack this problem formed the basis for much of the work in 3 dimensional topology over the last century including for example the proof of dehn s lemma and the loop theorem and the study of surgery on knots and links in the 1980 s thurston developed another approach to 3 manifolds see 24 and 4 he considered 3 manifolds with riemannian metrics of constant negative curvature 1 these manifolds which are locally isometric to hyperbolic 3 space are called hyperbolic manifolds there are fairly obvious obstructions showing that not every 3 manifold can admit such a metric thurston formulated a general conjecture that roughly says that the obvious obstructions are the only ones should they vanish for a particular 3 manifold then that manifold admits such a metric his proof of various important special cases of this conjecture led him to formulate a more general conjecture about the existence of locally homogeneous metrics hyperbolic or otherwise for all manifolds this is called thurston s geometrization conjecture for 3 manifolds the statement of this conjecture is somewhat complicated so it is deferred until section 2 an important point is that thurston s geometrization conjecture includes the poincare conjecture as a very special case in addition thurston s conjecture has two advantages over the poincare conjecture it applies to all closed oriented 3 manifolds it posits a close relationship between topology and geometry in dimension three

minimal entropy and collapsing with curvature bounded from below we show that if a closed manifold m admits an structure not necessarily polarized possibly of rank zero then its minimal entropy vanishes in particular this is the case if m admits a non trivial s1 action as a corollary we obtain that the simplicial volume of a manifold admitting an structure is zero we also show that if m admits an structure then it collapses with curvature bounded from below this in turn implies that m collapses with bounded scalar curvature or equivalently its yamabe invariant is non negative we show that structures of rank zero appear rather frequently every compact complex elliptic surface admits one as well as any simply connected closed 5 manifold we use these results to study the minimal entropy problem we show the following two theorems suppose that m is a closed manifold obtained by taking connected sums of copies of s4 p2 overline mathbb c p 2 s2 s2and the k3 surface then m has zero minimal entropy moreover m admits a smooth riemannian metric with zero topological entropy if and only if m is diffeomorphic to s4 p2 s2 s2 p2 overline mathbb c p 2 or p2 p2 finally suppose that m is a closed simply connected 5 manifold then m has zero minimal entropy moreover m admits a smooth riemannian metric with zero topological entropy if and only if m is diffeomorphic to s5 s3 s2 then on trivial s3 bundle over s2 or the wu manifold su 3 so 3

subrepresentations of kronecker representations translated into the language of representations of quivers a challenge in matrix pencil theory is to find sufficient and necessary conditions for a kronecker representation to be a subfactor of another kronecker representation in terms of their kronecker invariants the problem is reduced to a numerical criterion for a kronecker representation to be a subrepresentation of another kronecker representation in terms of their kronecker invariants the key to the problem is the calculation of ranks of matrices over polynomial rings for this a generalization and specialization approach is introduced this approach is applied to provide a numerical criterion for a preprojective resp regular preinjective kronecker representation to be a subrepresentation of another preprojective resp regular preinjective kronecker representation in terms of their kronecker invariants

computing the extensions of preinjective and preprojective kronecker modules abstract let i i be preinjective kronecker modules i e all their indecomposable components are preinjective we describe the modules m for which there exists an exact sequence 0 i m i 0 by explicit easy to check numerical conditions resulting in an algorithm linear in the number of indecomposable components for the decision problem we also propose a method to generate all extensions of i by i and we give a different proof for a theorem in 13 providing numerical criteria in terms of kronecker invariants for the existence of a monomorphism f i i all these results apply dually to preprojective modules as well

limits and degenerations of unitary conformal field theories in the present paper degeneration phenomena in conformal field theories are studied for this purpose a notion of convergent sequences of cfts is introduced properties of the resulting limit structure are used to associate geometric degenerations to degenerating sequences of cfts which as familiar from large volume limits of non linear sigma models can be regarded as commutative degenerations of the corresponding quantum geometries as an application the large level limit of the a series of unitary virasoro minimal models is investigated in detail in particular its geometric interpretation is determined

geometric structures gromov norm and kodaira dimensions abstract we define the kodaira dimension for 3 dimensional manifolds through thurston s eight geometries along with a classification in terms of this kodaira dimension we show this is compatible with other existing kodaira dimensions and the partial order defined by non zero degree maps for higher dimensions we explore the relations of geometric structures and mapping orders with various kodaira dimensions and other invariants especially we show that a closed geometric 4 manifold has nonvanishing gromov norm if and only if it has geometry h 2 h 2 h 2 c or h 4

static displacement reanalysis of modified structues using the epsilon algorithm i n the optimization of structural systems it is very important to compute the static displacements of the structures when the parameters of the structures are changed one of themain obstacles is the high computational cost involved in the solution of large scale problems therefore the reanalysis problems excite the interest of many researchers and both approximate and exact reanalysis methods have been reported and reviewed 1 in general the following factors are considered in choosing an approximate behavior model for a specific optimal design problem 2 1 the accuracy of the calculations or the quality of the approximations 2 the computational effort involved or the efficiency of the method and 3 the ease of implementation at present the various approximations have been developed barthelemy kirsch and haftka et al 3 5 developed the global approximation also called multipoint approximations such as polynomial fitting or reduced basis methods these approximations are obtained by analyzing the structure at a number of design points and they are valid for the whole design space local approximation is called single point approximations such as the taylor series expansion or the binomial series expansion about a given point in the design space local approximations are based on information calculated at a single point thesemethods are effective only in cases of small changes in parameters of the structures for large changes in the design the accuracy of the approximations often deteriorates and they may become meaningless 6 kirsch 7 12 discussed the combined approximations which attempt to give global qualities to local approximations recently the pade approximation and shanks transformation were used to improve the accuracy of the reanalysis 13 14 it is shown that it can significantly improve the domain of convergence in this study we use the epsilon algorithm 15 17 to deal with the static displacement reanalysis the main objectives are to preserve the ease of implementation and to improve significantly the domain of convergence and the quality of the results such that the method can be used in problems with very large changes in the structural parameters a numerical example is demonstrated and the method is compared with the kirsch combined approximation

collapsing with a lower bound on the curvature operator cheeger and gromov showed that f structures are related to collapse with a double sided curvature bound we define fibered f structures and extend some of the cheeger gromov results to the setting of collapse with a lower bound on the curvature operator

on the upper estimate of the heat kernel of a complete riemannian manifold let m be a complete non compact riemannian manifold whose sectional curvature is bounded between two constants k and k then one expects that the heat diffusion in such a manifold behaves like the heat diffusion in euclidean space the purpose of this paper is to give a justification of such a statement in 5 j cheeger and the third author found a lower estimate of the heat kernel by comparing it with the heat kernel of the space form whose curvature is the lower bound of the curvature of the manifold this lower estimate is sharp if we insist the dependence should be on the lower bound of the ricci curvature alone it remains to give an upper estimate of the heat kernel one does not however expect to have a comparison theorem for the upper bound because it is more sensitive to the geometry of the manifold in fact the heat kernel of the upper hialf space and the heat kernel of the complete hyperbolic manifold with finite volume have quite different behavior this is reflected by the fact that the laplacian has no discrete spectrum in the first case while infinite number of discrete eigenvalues may exist in the latter case what we will prove here is that in any case the heat kernel has to decay in a manner similar to the euclidean heat kernel thus we will prove that for any constant c 4 there exists c1 depending on c t the bound of the curvature of m and x so that for all t e 0 t

three manifolds with many flat planes we discuss the rigidity or lack thereof imposed by dierent no tions of having an abundance of zero curvature planes on a complete riemann ian 3 manifold we prove a rank rigidity theorem for complete 3 manifolds showing that having higher rank is equivalent to having reducible universal covering we also study 3 manifolds such that every tangent vector is con tained in a at plane including examples with irreducible universal covering and discuss the eect of nite volume and real analiticity assumptions

generalized galerkin method for modal testing of structure with pseudomaterials this paper presents a generalized galerkin method based on the dynamic flexibility matrix expansion technique the main objective is to develop an analysis to predict the modes and frequencies of a real structure when the design verification methods used to conduct on the ground tests can only be performed on a pseudomaterial structure it is critical to develop analytical methods that correctly predict the dynamic behavior of real structure in order to guarantee a successful design of complex structures oftentimes a full scale spacecraft modal test cannot be performed on the real model due to hazardous payloads it carries stable and relatively inexpensive materials are often used to replace these hazardous materials during the test the basic idea for analyzing these test modes of pseudomaterial structures is based on eigenvalue reanalysis techniques this method is relatively easy to understand and can be implemented into the existing method it is numerically accurate and efficient and can be applied to assist in large complex space structure designs

splittings and cr structures for manifolds with nonpositive sectional curvature let m n denote the universal covering space of a compact riemannian manifold m n with sectional curvature 1 k mn 0 we show that a collection of deck transformations of m n satisfying certain metric dependent conditions determines an open dense subset of m n at every point of which there exists a local isometric splitting with nontrivial flat factor such a collection which we call an abelian structure also gives rise to an essentially canonical cr structure in the sense of buyalo i e an atlas for an injective f structure for which additional conditions hold it follows in particular that the minimal volume of m n vanishes we show that an abelian structure exists if the injectivity radius at all points of m n is less than e n 0 this yields a conjecture of buyalo as well as a strengthened version of the conclusion of gromov s gap conjecture in our special situation in addition we observe that abelian structures on nonpositively curved manifolds have certain stability properties under suitably controlled changes of metric

a boundary of the set of the riemannian manifolds with bounded curvatures and diameters on etudie la structure de la fermeture de l ensemble constitue par des classes d isometrie de varietes de riemann a courbures et diametres bornes

entropy and collapsing of compact complex surfaces we study the problem of existence of mathcal f structures in the sense of cheeger and gromov but not necessarily polarized on compact complex surfaces we give a complete classification of compact complex surfaces of kahler type admitting mathcal f structures in the non kahler case we give a complete classification modulo the gap in the classification of surfaces of class vii we then use these results to study the minimal entropy problem for compact complex surfaces we prove that modulo the gap in the classification of surfaces of class vii all compact complex surfaces of kodaira dimension less than or equal to 1 have minimal entropy zero and such a surface admits a smooth metric g with zero topological entropy if and only if it is mathbb c p 2 a ruled surface of genus 0 or 1 a hopf surface a complex torus a kodaira surface a hyperelliptic surface or a kodaira surface modulo a finite group the key result we use to prove this is a new topological obstruction to the existence of metrics with vanishing topological entropy finally we show that these results fit perfectly into wall s study of geometric structures on compact complex surfaces for instance we show that the minimal entropy problem can be solved for a minimal compact kahler surface s of kodaira dimension infty 0 or 1 if and only if s admits a geometric structure modelled on mathbb c p 2 s 2 times s 2 s 2 times mathbb e 2 or mathbb e 4

the second twisted betti number and the convergence of collapsing riemannian manifolds let mi overset d gh longrightarrow x denote a sequence of n manifolds converging to a compact metric space x in the gromov hausdorff topology such that the sectional curvature is bounded in absolute value and dim x n we prove the following stability result if the fundamental groups of mi are torsion groups of uniformly bounded exponents and the second twisted betti numbers of mi vanish then there is a manifold m and a sequence of diffeomorphisms from m to a subsequence of mi such that the distance functions of the pullback metrics converge to a pseudo metric in c0 norm furthermore m admits a foliation with leaves diffeomorphic to flat manifolds not necessarily compact such that a vector is tangent to a leaf if and only if its norm converges to zero with respect to the pullback metrics these results lead to a few interesting applications

reanalysis of linear and nonlinear structures using iterated shanks transformation reanalysis of structures is a common task in two important fields of structural engineering namely optimization and reliability it implies the solution of the kinematic equations for each new set of data in order to simplify this task several approximate methods have been developed in this paper a simple method for reanalysis applicable to either linear or nonlinear structures is proposed it is based on the improvement of the neumann resolvent series which diverges beyond a certain radius by means of the iterated shanks transformation the method is compared with two relevant proposals intended to improve the convergence of the neumann series namely the pade series and the scaled approximation methods the accuracy and simplicity of the proposed method is demonstrated through some examples concerning linear and nonlinear structures

volume collapsed three manifolds with a lower curvature bound in this paper we determine the topology of three dimensional complete orientable riemannian manifolds with a uniform lower bound of sectional curvature whose volume is sufficiently small

rationality of geometric signatures of complete 4 manifolds according to cg4 and cfg the complete manifolds with bounded sectional curvature and finite volume admit positive rank f structures near infinity in this paper we show that in dimension four if the manifolds also have bounded covering geometry near infinity then there exist f structures with special topological properties f structures with these properties cannot be constructed solely by means of the general methods in cg4 and cfg using these special properties we prove a conjecture of cheeger gromov on the rationality of the geometric signatures in the four dimensional case

a mass for alf manifolds we prove a positive mass theorem on alf manifolds i e complete noncompact manifolds that are asymptotic to a circle fibration over a euclidean base with fibers of asymptotically constant length

ends of finite volume nonpositively curved manifolds we study complete finite volume n manifolds m of bounded nonpositive sectional curvature a classical theorem of m gromov says that if such m has negative curvature then it is homeomorphic to the interior of a compact manifold with boundary and we denote this boundary partial m if n ge 3 we prove that the universal cover of the boundary widetilde partial m and also the pi 1m cover of the boundary partial widetilde m have vanishing n 2 dimensional homology for n 4 the first of these recovers a result of nguy tilde hat mathrm e n phan saying that each component of the boundary partial m is aspherical for any n ge 3 the second of these implies the vanishing of the first group cohomology group with group ring coefficients h 1 b pi 1m mathbb z pi 1m 0 a consequence is that pi 1m is freely indecomposable these results extend to manifolds m of bounded nonpositive curvature if we assume that m is homeomorphic to the interior of a compact manifold with boundary our approach is a form of homological collapse for ends of finite volume manifolds of bounded nonpositive curvature this paper is very much influenced by earlier yet still unpublished work of nguy tilde hat mathrm e n phan

collapsing riemannian manifolds while keeping their curvature bounded i soit y n une variete de riemann connexe complete et p y n le rayon d injectivite i p de l application exponentielle en p est defini comme etant le plus petit r tel que exp p b r p n est pas un diffeomorphisme sur son image on etudie la situation dans laquelle la taille du rayon d injectivite est petite relativement a la courbure

removing point singularities of riemannian manifolds we study the behavior of geodesics passing through a point singularity of a riemannian manifold in particular we show that if the curvature does not blow up too rapidly near the singularity then the singularity is at worst an orbifold singularity the idea is to construct the exponential map centered at a singularity since there is no tangent space at the singularity a surrogate is needed we show that the vector space of radially parallel vector fields is well defined and that there is a correspondence between unit radially parallel vector fields and geodesics emanating from the singular point

structural reanalysis for general layout modifications a procedure for reanalysis of structures subjected to various layout modifications is presented the procedure is based on exact analysis of the initial design it is suitable for all types of layout changes including the general case in which some members and joints are added other members and joints are deleted and some joint coordinates are modified in the particular and challenging case of addition of joints where the number of degrees of freedom is increased a simple reanalysis of a modified initial design is first carried out further changes are then applied to the latter design it is shown that for changes in a limited number of members of a truss structure an exact solution is readily achieved for changes in numerous members of a general structure an approximate solution can be efficiently obtained high quality approximations are demonstrated for various types of layout modifications

nilpotent structures and invariant metrics on collapsed manifolds let mn be a complete riemannian manifold of bounded curvature say iki 0 we put mn w n c u fn c where 1n e consists of those points at which the injectivity radius of the exponential map is e the complementary set n c is called the e collapsed part of mn if x e 4 n r e then the metric ball bx r is quasi isometric with small distortion to the flat ball b0 r in the euclidean space rn after slightly

assigning the kronecker invariants of a matrix pencil by row or column completions abstract the challenge consists in describing the relationships between the kronecker invariants of a matrix pencil and one of its subpencils for a given subpencil an algorithm for constructing a matrix pencil with prescribed kronecker invariants should also be proposed

collapsing of abelian fibered calabi yau manifolds we study the collapsing behavior of ricci flat kahler metrics on a projective calabi yau manifold which admits an abelian fibration when the volume of the fibers approaches zero we show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature and along the fibers the rescaled metrics become flat in the limit the limit metric on the base minus the critical locus is locally isometric to an open dense subset of any gromov hausdorff limit space of the ricci flat metrics we then apply these results to study metric degenerations of families of polarized hyperkahler manifolds in the large complex structure limit in this setting we prove an analogue of a result of gross and wilson for k3 surfaces which is motivated by the strominger yau zaslow picture of mirror symmetry