cohomogeneity one manifolds with positive ricci curvature one of the central problems in riemannian geometry is to determine how large the classes of manifolds with positive nonnegative sectional ricci or scalar curvature are see gr for scalar curvature the situation is fairly well understood by comparison special surgery constructions as in sy wr and bundle constructions as in na have resulted in a large number of interesting manifolds with positive ricci curvature so far the only known obstructions to have positive ricci curvature come from obstructions to have positive scalar curvature see li and rs and from the classical bonnet myers theorem which implies that a closed manifold with positive ricci curvature must have finite fundamental group it is well known that among homogeneous manifolds g h this is also a sufficient condition see e g the proof of corollary 3 5 or br in this paper we prove that this is true as well when the manifold admits an action by a compact lie group g with orbits of codimension one

curvature diameter homotopy groups and cohomology rings we establish two topological results a if m is a 1 connected compact n manifold and q 2 then the minimal number of generators for the qth homotopy group q m mng q m is bounded above by a number depending only on mng h m and q where h m is the homology group c let mathscr m n y be the collection of compact orientable n manifolds whose oriented bundles admit so n invariant fibrations over y with fiber compact nilpotent manifolds such that the induced so n actions on y are equivalent then q m finitely generated m mathscr m n y contains only finite isomorphism classes depending only on n y q together with the results of cg and gr1 from a we conclude that i if m is a complete n manifold of nonnegative curvature then mng q m is bounded above by a number depending only on n and q 2 together with the results of ch and cfg from c we conclude that ii if m is a compact n manifold whose sectional curvature and diameter satisfy secm and diamm d then q m has a finite number of possible isomorphism classes depending on n d q 2 provided q m is finitely generated we also show that b if m is a compact n manifold with secm and diam m d then the cohomology ring h m may have infinitely many isomorphism classes in particular b answers some questions raised by k grove gro

topological obstructions to nonnegative curvature we find new obstructions to the existence of complete riemannian metric of nonnegative sectional curvature on manifolds with infinite fundamental groups in particular we construct many examples of vector bundles whose total spaces admit no nonnegatively curved metrics

machine learning and nonparametric bandit theory in its most basic form bandit theory is concerned with the design problem of sequentially choosing members from a given collection of random variables so that the regret i e r sub n spl sigma sub j spl mu spl mu sub j et sub n j grows as slowly as possible with increasing n here spl mu sub j is the expected value of the bandit arm i e random variable indexed by j t sub n j is the number of times arm j has been selected in the first n decision stages and spl mu sup sup sub j spl mu sub j the present paper contributes to the theory by considering the situation in which observations are dependent to begin with the dependency is presumed to depend only on past observations of the same arm but later we allow that it may be with respect to the entire past and that the set of arms is infinite this brings queues and more generally controlled markov processes into our purview thus our black box methodology is suitable for the case when the only observables are cost values and in particular the probability structure and loss function are unknown to the designer the conclusion of the analysis is that under lenient conditions using algorithms prescribed herein risk growth is commensurate with that in the simplest i i d cases our methods represent an alternative to stochastic approximation perturbation analysis ideas for tuning queues

some group actions on homotopy spheres of dimension seven and fifteen 0 introduction this paper is based on the simple observation that every 4 plane bundle over s4 admits a natural action of 50 3 by bundle maps and that every 8 plane bundle over 58 admits a natural action of the compact lie group g2 by bundle maps these actions are easy to see once one remembers that so 3 is the group of automorphisms of the quaternions and that g2 is the group of automorphisms of the cayley numbers the study of these actions is closely connected to several well known phenomena in differential topology and compact transformation groups the most obvious connection is to milnor s original construction of exotic 7 spheres as 3 sphere bundles over 54 26 milnor proved that if the euler class of such a bundle is a generator of h4 s4 z then its total space is homeomorphic to 57 he also defined a numerical invariant of the diffeomorphism type and used it to detect an exotic differential structure on some of these sphere bundles subsequently eells and kuiper 12 introduced a refinement of this invariant called the it invariant using the u invariant they proved that the sphere bundle m7 milnor s notation is a generator for the group of homotopy 7 spheres these constructions also work for 7 sphere bundles over 58 32 and the manifold m15 is a generator for bp16 the group of homotopy 15 spheres which bound wn manifolds it is a routine matter to check that milnor s arguments and their subsequent refinements work g equivariantly where g 5so 3 or g2 and we shall do this in sections 2 and 3 in particular each sphere bundle with the correct euler class is g homeomorphic to an orthogonal action on s2 1 where 2n 1 7 or 15 moreover distinct sphere bundles have distinct oriented g diffeomorphism types the proof of the second fact uses an equivariant version of the it invariant as originally defined this invariant takes values in q z however it is well known that in the presence of a g action with s c g its value in q is well defined using

homeomorphism classification of positively curved manifolds with almost maximal symmetry rank we show that a closed simply connected 8 manifold 9 manifold of positive sectional curvature on which a 3 torus 4 torus acts isometrically is homeomorphic to a sphere a complex projective space or a quaternionic projective plane sphere we show that a closed simply connected 2m manifold m 5 of positive sectional curvature on which an m 1 torus acts isometrically is homeomorphic to a complex projective space if and only if its euler characteristic is not 2 by wi these results imply a homeomorphism classification for positively curved n manifolds n 8 of almost maximal symmetry rank open image in new window

isoparametric functions on exotic spheres abstract this paper extends widely the work in 11 existence and non existence results of isoparametric functions on exotic spheres and eells kuiper projective planes are established in particular every homotopy n sphere n 4 carries an isoparametric function with certain metric with 2 points as the focal set in strong contrast to the classification of cohomogeneity one actions on homotopy spheres 26 only exotic kervaire spheres admit cohomogeneity one actions besides the standard spheres as an application we improve a beautiful result of berard bergery 2 see also pp 234 235 of 3

a survey of some results in stochastic adaptive control some results in discrete time stochastic adaptive control are surveyed the survey divides itself into two parts bayesian and non bayesian adaptive control in the former area the problems of converting an incompletely observed system into a completely observed one multi armed bandit processes bayesian adaptive control of markov chains and bayesian adaptive control of linear systems are exposed and surveyed in the latter area non bayesian adaptive control of markov chains and the self tuning regulator are dealt with proofs are given where appropriate to illustrate the methods involved

concavity and rigidity in non negative curvature we show that for a manifold with non negative curvature one obtains a collection of concave functions special cases of which are the concavity of the length of a jacobi field in dimension 2 and the concavity of the volume in general we use these functions to show that there are many cohomogeneity one manifolds which do not carry an analytic invariant metric with non negative curvature this implies in particular that one of the candidates in positively curved cohomogeneity one manifolds and 3 sasakian geometry does not carry an invariant metric with positive curvature

cohomogeneity one topological manifolds revisited we prove a structure theorem for closed topological manifolds of cohomogeneity one this result corrects an oversight in the literature we complete the equivariant classification of closed simply connected cohomogeneity one topological manifolds in dimensions 5 6 and 7 and obtain topological characterizations of these spaces in these dimensions these manifolds are homeomorphic to smooth manifolds

non negative pinching moduli spaces and bundles with infinitely many souls we show that in each dimension n 10 there exist infinite sequences of homotopy equivalent but mutually non homeomorphic closed simply connected riemannian n manifolds with 0 sec 1 positive ricci curvature and uniformly bounded diameter we also construct open manifolds of fixed diffeomorphism type which admit infinitely many complete nonnegatively pinched metrics with souls of bounded diameter such that the souls are mutually non homeomorphic finally we construct examples of noncompact manifolds whose moduli spaces of complete metrics with sec 0 have infinitely many connected components

the skew energy of a digraph we are interested in the energy of the skew adjacency matrix of a directed graph d which is simply called the skew energy of d in this paper properties of the skew energy of d are studied in particular a sharp upper bound for the skew energy of d is derived in terms of the order of d and the maximum degree of its underlying undirected graph an infinite family of digraphs attaining the maximum skew energy is constructed moreover the skew energy of a directed tree is independent of its orientation and interestingly it is equal to the energy of the underlying undirected tree skew energies of directed cycles under different orientations are also computed some open problems are presented

bayesian prediction of deterministic functions with applications to the design and analysis of computer experiments abstract this article is concerned with prediction of a function y t over a multidimensional domain t given the function values at a set of sites t 1 t 2 t n in t and with the design that is with the selection of those sites the motivating application is the design and analysis of computer experiments where t determines the input to a computer model of a physical or behavioral system and y t is a response that is part of the output or is calculated from it following a bayesian formulation prior uncertainty about the function y is expressed by means of a random function y which is taken here to be a gaussian stochastic process the mean of the posterior process can be used as the prediction function t and the variance can be used as a measure of uncertainty this kind of approach has been used previously in bayesian interpolation and is strongly related to the kriging methods used in geostatistics here emphasis is placed on product linear and product cubic correlation func

lifting group actions and nonnegative curvature we examine the question when a group acting by cohomogeneity one on the base of a principal g bundle can be lifted to the total space and commutes with the action by g we answer this question completely when the base of the principle bundle is 4 dimensional

on the integral weighted oriented unicyclic graphs with minimum skew energy abstract let g be a weighted oriented graph which is obtained from a simple weighted undirected graph by assigning an orientation to each of its edges for a weighted oriented graph g the undirected graph obtained from g by removing the orientation and the weight of each of its arcs is called the underlying graph of g denoted by g by u n m m n we denote the set of all connected integral weighted oriented unicyclic graphs with order n in which each arc is assigned a positive integral weight and the sum of the weights of all arcs is equal to the specified integer m in this paper we investigate the minimal skew energies of integral weighted unicyclic oriented graphs showing that the underlying graph of the oriented graph with minimum skew energy among all graphs over u n m n 6 is s n 3 the graph obtained from a triangle by attaching n 3 pendent edges in exactly one of its vertices moreover we show that its weight sequence has form w 1 a a a k a 1 a 1 a 1 n 3 k w 2 w 3 in which the arc lying on the cycle c 3 and incident to no pendent arcs has weight w 1 and the two largest weights correspond the other two arcs of c 3 where positive integer numbers w 1 w 2 w 3 a and k satisfy 1 w 1 a a 1 w 2 w 3 a w 1 and m w 1 ka n k 3 a 1 w 2 w 3 in addition we determine the weight sequence above for n m 3 n 1

on the curvature of biquotients as a means to better understanding manifolds with positive curvature there has been much recent interest in the study of non negatively curved manifolds which contain either a point or an open dense set of points at which all 2 planes have positive curvature we study infinite families of biquotients defined by eschenburg and bazaikin from this viewpoint together with torus quotients of s3 s3

nonconnected moduli spaces of positive sectional curvature metrics for a closed manifold m let 9 m resp 9 ic m be the space of riemannian metrics on m with positive sectional resp ricci cur vature and let diff m be the diffeomorphism group of m which acts on these spaces we construct examples of 7 dimensional manifolds for which the moduli space 9 m diff m is not connected and others for which 9 c m diff m has infinitely many connected components the examples are obtained by analyzing a family of positive sectional curvature metrics on homogeneous spaces constructed by aloff and wallach on which su 3 acts transitively respectively a family of positive einstein metrics constructed by wang and ziller on homogeneous spaces on which su 3 x su 2 x u i acts transitively max planck institut fur mathematik gottfried claren strasse 26 5300 bonn 3 germany and fachbereich mathematik universitat mainz 6500 mainz germany current address johannes gutenberg universitat mainz fachbereich mathematik staudinger weg 9 6500 mainz germany e mail address kreck topologie mathematik uni mainz de department of mathematics university of notre dame notre dame indiana 46556 e mail address stolz l nd edu license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

certainty equivalence control with forcing revisited abstract certainty equivalence control with forcing has been shown to be optimal for several stochastic adaptive control problems with the average cost per unit time criterion recently researchers have started looking at stochastic adaptive control problems with a view to minimizing the rate of increase of the learning loss this criterion is stronger than the average cost per unit time criterion certainty equivalence control with forcing does not usually suffice for the learning loss criterion and one has to develop fairly complicated schemes in order to achieve optimality the objective of this paper is to see how well one might be able to do with a certainty equivalence control with forcing type of scheme in particular we construct a class of such schemes whose learning loss is o log n 1 for 0 whereas optimal schemes typically have a o log n learning loss

compact differentiable transformation groups on exotic spheres 0 introduction 355 1 statement of the main results 357 2 higher dimensional subgroups of o n 1 358 3 higher dimensional transformation groups on homology spheres geometric weight systems and orbit structures 360 4 higher dimensional transformation groups on homotopy spheres and the proof of theorem b 375 5 degree of symmetry of exotic spheres revisited 381 6 large transformation groups on exotic spheres in bp 1 384

adaptive control of constrained markov chains adaptive control of finite state markov chains is discussed the optimal performance is characterized through the minimization of a long run average cost functional subject to constraints on several other such functionals under mild structural and feasibility conditions two explicit adaptive control policies are exhibited for the case where the transition probabilities are unknown the policies are optimal under the constrained optimization criterion they rely on a powerful estimation scheme which provides consistent estimators for the transition probabilities this scheme is of independent interest as it provides strong consistency under a large number of adaptive schemes and is independent of any identifiability conditions as an application an optimal adaptive policy is derived for a system of k competing queues with countable state space for which the constrained criteria arise naturally in the context of communication networks

exotic spheres with positive ricci curvature we show that a certain class of manifolds admit metrics of positive ricci curvature this class includes many exotic spheres including all homotopy spheres which represent elements of bp2n

twisting and nonnegative curvature metrics on vector bundles over the round sphere a complete noncompact manifold m with nonnegative sectional curvature is diffeomorphic to the normal bundle of a compact submanifold s called the soul of m when s is a round sphere we show that the clutching map of this bundle is restricted this is used to deduce that there are at most finitely many isomorphism types of such bundles with sectional curvature lying in a fixed interval 0 we also examine the opposite question of how the twisting of the bundle limits the type of possible nonnegative curvature metrics on the bundle it turns out that if the bundle does not admit a nowhere zero section then the normal exponential map is necessarily a diffeomorphism onto m and the ideal boundary of m consists of a single point in their paper cg cheeger and gromoll raised the question of which vector bundles over the round sphere admit complete metrics with nonnegative sectional curvature the significance of this problem is that it attempts to determine to what extent a converse to the soul theorem holds recall that this theorem states that every open i e complete noncompact manifold m with nonnegative curvature km is diffeomorphic to a vector bundle over a compact totally geodesic submanifold s called a soul a natural question then is whether all such vector bundles admit complete metrics with km 0 in ow it was shown that when the soul is a bieberbach manifold nonnegative curvature metrics force the vanishing of the euler class of the vector bundle it follows that among oriented plane bundles over the torus only the trivial one admits such a metric the above case is fairly rigid a priori however since for example any bundle with curvature 0 over such a manifold must also admit a flat metric and the corresponding question for a simply connected base remains open an answer could provide insights on the topological structure of compact manifolds with nonnegative curvature since any open manifold with nonnegative curvature can after modifying its metric be isometrically 1991 mathematics subject classification primary 53c20

s curvature of cohomogeneity one randers spaces abstract in this paper we study s curvature of cohomogeneity one randers spaces with orbit space s 1 in particular we present an explicit coordinate free formula of s curvature in terms of some quantities which is determined by the metric and the structure of the manifold as an application we give a sufficient and necessary condition for a cohomogeneity one randers space to have almost isotropy s curvature furthermore we give some simple criterions for a cohomogeneity one randers space to be a berwald space or a douglas space

pontryagin numbers and nonnegative curvature we prove that any rational linear combination of pontryagin numbers that is not a multiple of the signature is unbounded on connected closed oriented manifolds of nonnegative sectional curvature combining our result with gromov s boundedness result for the signature yields a new characterization of the l genus

collapsed 5 manifolds with pinched positive sectional curvature abstract let m be a closed 5 manifold of pinched curvature 0 sec m 1 we prove that m is homeomorphic to a spherical space form if one of the following conditions holds i the center of the fundamental group has index w a constant depending on ii 1 4 1 10 6 2 and the fundamental group is a non cyclic group of order c a constant iii the volume is less than and the fundamental group is either isomorphic to a spherical 5 space group or has an odd order and it has a center of index w a constant

finiteness theorems for nonnegatively curved vector bundles we prove several finiteness theorems for the normal bundles to souls in nonnegatively curved manifolds more generally we obtain finiteness results for open riemannian manifolds whose topology is concentrated on compact domains of bounded geometry

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

obstructions to nonnegative curvature and rational homotopy theory we establish a link between rational homotopy theory and the problem which vector bundles admit a complete riemannian metric of nonnegative sectional curvature as an application we show for a large class of simply connected nonnegatively curved manifolds that if c lies in the class and t is a torus of positive dimension then most vector bundles over c x t admit no complete nonnegatively curved metrics

geometric diffeomorphism finiteness in low dimensions and homotopy group finiteness the main results of this note consist in the following two geometric finiteness theorems for diffeomorphism types and homotopy groups of closed simply connected manifolds

tangential thickness of manifolds a notion of tangential thickness of a manifold is introduced an extensive calculation within the class of lens and fake lens spaces leads to complete classication of such manifolds with thickness 1 3 or 2k for k 1 on the other hand calculations of tangential thickness in terms of the dimension of the manifold and the rank of the fundamental group show very interesting and quite surprising correlations between these invariants

seven dimensional cohomogeneity one manifolds with nonnegative curvature we show that a certain family of cohomogeneity one manifolds does not admit an invariant metric of nonnegative sectional curvature unless it admits one with positive curvature as a consequence the classification of nonnegatively curved cohomogeneity one manifolds in dimension seven is reduced to only one further family of candidates

the characteristic polynomial and the matchings polynomial of a weighted oriented graph abstract let g be a weighted oriented graph with skew adjacency matrix s g then g is usually referred as the weighted oriented graph associated to s g denote by g the characteristic polynomial of the weighted oriented graph g which is defined as g det i n s g i 0 n a i g n i in this paper we begin by interpreting all the coefficients of the characteristic polynomial of an arbitrary real skew symmetric matrix in terms of its associated oriented weighted graph then we establish recurrences for the characteristic polynomial and deduce a formula on the matchings polynomial of an arbitrary weighted graph in addition some miscellaneous results concerning the number of perfect matchings and the determinant of the skew adjacency matrix of an unweighted oriented graph are given

cheeger manifolds and the classification of biquotients a closed manifold is called a biquotient if it is diffeomorphic to k g h for some compact lie group g with closed subgroups k and h such that k acts freely on g h every biquotient has a riemannian metric of nonnegative sectional curvature in fact almost all known manifolds of nonnegative curvature are biquotients the only known closed manifolds of nonnegative sectional curvature which were not defined in this way are those found by cheeger 6 in 1973 and grove and ziller 16 in 2000 in order to understand these constructions better this paper analyzes which of the cheeger and grove ziller manifolds are actually diffeomorphic to biquotients in the process we develop a general approach to the classification of biquotient manifolds a priori it is hard to determine whether a given manifold is a biquotient because there is no obvious upper bound on the dimension of the groups involved we give a procedure which allows us to reduce the dimension of the groups needed to describe a given manifold as a biquotient a surprising application is that gromoll meyer s example of an exotic 7 sphere which is a biquotient 15 is the only exotic sphere of any dimension which is a biquotient more generally we classify all biquotients which are simply connected rational homology spheres of any dimension theorem 6 1 the classification of biquotients which are simply connected rational homology spheres has been given independently by kapovitch and ziller 17 more generally they classify all simply connected biquotients whose rational cohomology ring is generated by one element another application of our general classification theory for biquotients is the precise determination of which cheeger manifolds the connected sums of two rank one symmetric spaces with any orientations are diffeomorphic to biquotients theorem 2 1 some of the cheeger manifolds were known to be biquotients such as cp2 cp2 the nontrivial s2 bundle over s2 but we find that many of the other cheeger manifolds are also biquotients such as cp2 cp2 on the other hand the cheeger manifold cp4 hp2 is not diffeomorphic or even homotopy equivalent to a biquotient the positive result that some cheeger manifolds such as cp2 cp2 are biquotients implies that they have nonnegatively curved riemannian metrics with various good properties that were not clear from cheeger s construction first the new metrics are real analytic further the new metrics determine a natural complex analytic structure on the whole tangent bundle since aguilar 2 showed that biquotients have this property my paper 26 has a weaker result which applies to all the cheeger manifolds finally the geodesic flow for the new metrics on the cheeger manifolds has zero topological entropy paternain 23 proved the latter property for cheeger s metric on cp2 cp2 but now we know it for some metric on a large class of the cheeger manifolds finally our general classification results imply that there are only finitely many

the energy of a graph abstract the energy e g of a simple graph g is defined to be the sum of the absolute values of the eigen values of g if g is a k regular graph on n vertices then e g k k n 1 n k b 2 and this bound is sharp it is shown that for each 0 there exist infinitely many n for each of which there exists a k regular graph g of order n with k e g b 2 two graphs with the same number of vertices are equienergetic if they have the same energy we show that for any positive integer n 3 there exist two equienergetic graphs of order 4n that are not cospectral

cohomogeneity one disk bundles with normal homogeneous collars we consider cohomogeneity one homogeneous disk bundles and adress the question when these admit a nonnegatively curved1 invariant metric with normal collar i e such that near the boundary the metric is the product of an interval and a normal homogeneous space if such a bundle is not the quotient of a trivial bundle then we show that its rank has to be in 2 3 4 6 8 moreover we give a complete classification of such bundles of rank 6 and 8 and a partial classification for rank 3

theory for automatic learning under partially observed markov dependent noise a vigorous branch of automatic learning is directed at the task of locating a global minimum of an unknown multimodal function f theta on the basis of noisy observations l theta i f theta i w theta i taken at sequentially chosen control points theta i in all preceding convergence deviations known to the authors the noise is postulated to depend on the past only through control selection here they allow the observation noise sequence to be stochastically dependent in particular a function of an unknown underlying markov decision process the observations being the stagewise losses in a sense in order to be made precise the algorithm offered is shown to attain asymptotically optimal performance and rates are assured a motivating example from queueing theory is offered and connections with classical problems of markov control theory and other disciplines are mentioned

codimension two souls and cancellation phenomena abstract for each m 0 we find an open 4 m 9 dimensional simply connected manifold with complete nonnegatively curved metrics whose souls are nondiffeomorphic homeomorphic and have codimension 2 we give a diffeomorphism classification of the pairs n soul when n is the total space of a nontrivial complex line bundle over s 7 cp 2 up to diffeomorphism there are precisely three such pairs distinguished by their nondiffeomorphic souls

diffeomorphism type of the berger space so 5 so 3 we compute the eells kuiper invariant of the berger manifold so 5 so 3 and determine that it is diffeomorphic to the total space of an s 3 bundle over s 4 this answers a question raised by k grove and w ziller

a classification of hyperpolar and cohomogeneity one actions an isometric action of a compact lie group on a riemannian manifold is called hyperpolar if there exists a closed connected submanifold that is flat in the induced metric and meets all orbits orthogonally in this article a classification of hyperpolar actions on the irreducible riemannian symmetric spaces of compact type is given since on these symmetric spaces actions of cohomogeneity one are hyperpolar i e normal geodesics are closed we obtain a classification of the homogeneous hypersurfaces in these spaces by computing the cohomogeneity for all hyperpolar actions this result implies a classification of the cohomogeneity one actions on compact strongly isotropy irreducible homogeneous spaces

obstructions to positive curvature and symmetry we discuss new obstructions to positive sectional curvature and symmetry the main result asserts that the index of the dirac operator twisted with the tangent bundle vanishes on a 2 connected manifold of dimension 8 if the manifold admits a metric of positive sectional curvature and isometric effective s1 action the proof relies on the rigidity theorem for elliptic genera and properties of totally geodesic submanifolds

curvature and symmetry of milnor spheres since milnor s discovery of exotic spheres mi one of the most intriguing problems in riemannian geometry has been whether there are exotic spheres with positive curvature it is well known that there are exotic spheres that do not even admit metrics with positive scalar curvature hi on the other hand there are many examples of exotic spheres with positive ricci curvature cf ch1 he po and na and this work recently culminated in wr where it is shown that every exotic sphere that bounds a parallelizable manifold has a metric of positive ricci curvature this includes all exotic spheres in dimension 7 so far however no example of an exotic sphere with positive sectional curvature has been found in fact until now only one example of an exotic sphere with nonnegative sectional curvature was known the so called gromoll meyer sphere gm in dimension 7 as one of our main results we prove

exotic spheres and curvature since their discovery by milnor in 1956 exotic spheres have provided a fascinating object of study for geometers in this article we survey what is known about the curvature of exotic spheres

finite isometry groups of 4 manifolds with positive sectional curvature let m be an oriented compact riemannian 4 manifold with positive sectional curvature let g be a finite subgroup of the isometry group of m we prove that if g is a finite group of order g ge 3 10 500 then i g is isomorphic to a subgroup of pu 3 if g is odd ii g contains an index at most 2 normal subgroup which is isomorphic to a subgroup of so 5 or pu 3 if g is even and m is not homeomorphic to s4 moreover m is homeomorphic to mathbb c p 2 if g is non abelian of odd order

edge addition singular values and energy of graphs and matrices the energy of a graph matrix is the sum of the absolute values of its eigenvalues we investigate the result of duplicating removing an edge to the energy of a graph we also deal with the problem that which graphs g have the property that if the edges of g are covered by some subgraphs then the energy of g does not exceed the sum of the subgraphs energies the problems are addressed in the general setting of energy of matrices which leads us to consider the singular values too among the other results it is shown that the energy of a complete multipartite graph increases if a new edge added or an old edge is deleted

nonnegative curvature on stable bundles over compact rank one symmetric spaces abstract in this note we show that every real or complex vector bundle over a compact rank one symmetric space carries after taking the whitney sum with a trivial bundle of sufficiently large rank a metric with nonnegative sectional curvature we also examine the case of complex vector bundles over other manifolds and give upper bounds for the rank of the trivial bundle that is necessary to add when the base is a sphere

nonnegative curvature and cobordism type we show that in each dimension n 4k k 2 there exist infinite sequences of closed simply connected riemannian n manifolds with nonnegative sectional curvature and mutually distinct oriented cobordism type

metrics of positive ricci curvature on quotient spaces one of the classical problems in differential geometry is the investigation of closed manifolds which admit riemannian metrics with given lower bounds for the sectional or the ricci curvature and the study of relations between the existence of such metrics and the topology and geometry of the underlying manifold despite many efforts during the past decades this problem is still far from being understood for example so far the only obstructions to the existence of a metric with positive ricci curvature come from the obstructions to the existence of metrics with positive scalar curvature li hi ro schy ta and the bonnet myers theorem which implies that the fundamental group of a closed manifold with positive ricci curvature must be finite fruitful constructions of metrics with positive ricci curvature on closed manifolds have so far been established by techniques that include deformation of metrics au eh we kahler geometry yau1 yau2 bundles and warping po na bb gpt special kinds of surgery sy wr metrical glueing gz2 and sasakian geometry bgn particularly large classes of examples of manifolds with positive ricci curvature are given by all compact homogeneous spaces with finite fundamental group na and all closed cohomogeneity one manifolds with finite fundamental group gz2 in this article we present several new classes of manifolds which admit metrics of positive or nonnegative ricci curvature the idea is to consider quotients of manifolds m g of positive or nonnegative ricci curvature by a free isometric action while taking such a quotient non decreases the sectional curvature it may well happen in general that the ricci curvature of the quotient is inferior to the one of m our first results however state that the quotient of m does admit metrics of positive ricci curvature if m belongs to one of the aforementioned classes of homogeneous spaces or spaces with a cohomogeneity one action more precisely we prove

a globally convergent stochastic approximation by combining a constrained kiefer wolfowitz search with an automatic learning algorithm it is shown that asymptotically normal convergence of an estimator to a global optimum under reasonably lenient assumptions can be attained it is enough that the objective function be smooth and locally strictly convex at its minima the central conclusion is that if theta n is the estimate produced by the method shown at the nth decision epoch then for some global minimizer theta n 1 3 theta n theta is asymptotically normally distributed this coincides with the conventional kiefer wolfowitz convergence rate to a local optimum whereas this study was motivated by needs of machine learning the basic plan would seem applicable to root finding tasks and to other types of stochastic approximation algorithms

strongly nonnegative curvature we prove that all currently known examples of manifolds with nonnegative sectional curvature satisfy a stronger condition their curvature operator can be modified with a 4 form to become positive semidefinite

conditions for nonnegative curvature on vector bundles and sphere bundles this paper addresses cheeger and gromoll s question of which vector bundles admit a complete metric of nonnegative curvature and relates their question to the issue of which sphere bundles admit a metric of positive curvature we show that any vector bundle which admits a metric of nonnegative curvature must admit a connection a tensor and a metric on the base space which together satisfy a certain differential inequality on the other hand a slight sharpening of this condition is sufficient for the associated sphere bundle to admit a metric of positive curvature our results sharpen and generalize walschap and strake s conditions under which a vector bundle admits a connection metric of nonnegative curvature

moduli spaces of nonnegative sectional curvature and non unique souls we apply various topological methods to distinguish connected components of moduli spaces of complete riemannian metrics of nonnega tive sectional curvature on open manifolds the new geometric ingredient is that souls of nearby nonnegatively curved metrics are ambiently isotopic