even minkowski valuations a new integral representation of smooth translation invariant and rotation equivariant even minkowski valuations is established explicit formulas relating previously obtained descriptions of such valuations with the new more accessible one are also derived moreover the action of alesker s hard lefschetz operators on these minkowski valuations is explored in detail

local geometry of singular real analytic surfaces let v r n be a compact real analytic surface with isolated singularities and assume its smooth part v 0 is equipped with a riemannian metric that is induced from some analytic riemannian metric on r n we prove 1 each point of v has a neighborhood which is quasi isometric naturally and almost isometrically to a union of metric cones and horns glued at their tips 2 a full asymptotic expansion for any p v of the length of v n q dist q p r as r 0 0 3 a gauss bonnet theorem saying that each singular point contributes 1 l 2 where l is the coefficient of the linear term in the expansion of 2 4 the l 2 stokes theorem selfadjointness and discreteness of the laplace beltrami operator on vo an estimate on the heat kernel and a gauss bonnet theorem for the l 2 euler characteristic as a central tool we use resolution of singularities

risk bounds in isotonic regression nonasymptotic risk bounds are provided for maximum likelihood type isotonic estimators of an unknown nondecreasing regression function with general average loss at design points these bounds are optimal up to scale constants and they imply uniform n 1 3 consistency of the ell p risk for unknown regression functions of uniformly bounded variation under mild assumptions on the joint probability distribution of the data with possibly dependent observations

hadwiger s theorem for definable functions abstract hadwiger s theorem states that e n invariant convex continuous valuations of definable sets in r n are linear combinations of intrinsic volumes we lift this result from sets to data distributions over sets specifically to definable r valued functions on r n this generalizes intrinsic volumes to dual pairs of non linear valuations on functions and provides a dual pair of hadwiger classification theorems

nonparametric instrumental variable estimation under monotonicity the ill posedness of the inverse problem of recovering a regression function in a nonparametric instrumental variable model leads to estimators that may suer from a very slow logarithmic rate of convergence in this paper we show that restricting the problem to models with monotone regression functions and monotone instruments signicantly weakens the ill posedness of the problem in stark contrast to the existing literature the presence of a monotone instrument implies boundedness of our measure of ill posedness when restricted to the space of monotone functions based on this result we derive a novel non asymptotic error bound for the constrained estimator that imposes monotonicity of the regression function for a given sample size the bound is independent of the degree of ill posedness as long as the regression function is not too steep as an implication the bound allows us to show that the constrained estimator converges at a fast polynomial rate independently of the degree of ill posedness in a large but slowly shrinking neighborhood of constant functions our simulation study demonstrates signicant nite sample performance gains from imposing monotonicity even when the regression function is rather far from being a constant we apply the constrained estimator to the problem of estimating gasoline demand functions from u s data

integral geometry of unitary area measures abstract the existence of kinematic formulas for area measures with respect to any connected closed subgroup of the orthogonal group acting transitively on the unit sphere is established in particular the kinematic operator for area measures is shown to have the structure of a co product in the case of the unitary group the algebra associated to this co product is described explicitly in terms of generators and relations as a consequence a simple algorithm that yields explicit kinematic formulas for unitary area measures is obtained

distribution free tests of stochastic monotonicity this article proposes an omnibus test for monotonicity of nonpara metric conditional distributions and its moments unlike previous proposals our method does not require smooth estimation of the derivatives of nonparametric curves and it can be implemented even when probability densities do not exist in fact we only require continuity of the marginal distributions under the null and xed alternatives distinguishing features of our approach are that critical values are pivotal under the null in nite samples and the test is invariant to any monotonic continuous transformation of the explanatory variable the test sta tistic is the sup norm of the dierence between the empirical copula function and its least concave majorant with respect to the explanatory variable coordinate the resulting test is able to detect local alternatives converging to the null at the parametric rate n 1 2 with n the sample size the article also discusses several applications and extensions of the proposal these include testing monotonicity of general conditional moments and the extension to multivariate explanatory variables the nite sample performance of the test is examined by means of a monte carlo experiment

on the geometry of classifying spaces and horizontal slices we study the local properties of the moduli space of a polarized calabi yau manifold let u be a neighborhood of the moduli space then we know the universal covering space v of u is a smooth manifold suppose d is the classifying space of a polarized calabi yau manifold with the automorphism group g then we prove that the map from v to d induced by the period map is a pluriharmonic map we also give a kahler metric on v which is called the hodge metric we prove that the ricci curvature of the hodge metric is negative away from zero we also proved the nonexistence of the kahler metric on the classifying space of a calabi yau threefold which is invariant under a cocompact lattice of g 1 introduction let x be a polarized simply connected calabi yau manifold that is x is a simply connected compact kahler manifold of dimension n with zero first chern class and is akform of x such that h 2 x z in this paper we study the local properties of the moduli space of the polarized calabi yau manifold x by definition is the parameter space of the complex structures over x for the fixed polarization is a quasi projective variety by a theorem of viehweg 17 suppose x is a calabi yau manifold let n d im h 1 x tx 0

nonparametric kernel regression subject to monotonicity constraints we suggest a method for monotonizing general kernel type estimators for example local linear estimators and nadaraya watson estimators attributes of our approach include the fact that it produces smooth estimates indeed with the same smoothness as the unconstrained estimate the method is applicable to a particularly wide range of estimator types it can be trivially modified to render an estimator strictly monotone and it can be employed after the smoothing step has been implemented therefore an experimenter may use his or her favorite kernel estimator and their favorite bandwidth selector to construct the basic nonparametric smoother and then use our technique to render it monotone in a smooth way implementation involves only an off the shelf programming routine the method is based on maximizing fidelity to the conventional empirical approach subject to monotonicity we adjust the unconstrained estimator by tilting the empirical distribution so as to make the least possible change in the sense of a distance measure subject to imposing the constraint of monotonicity

on the weil petersson volume and the first chern class of the moduli space of calabi yau manifolds in this paper we proved that the weil petersson volume of calabi yau moduli is a rational number we also proved that the integrations of the invariants of the ricci curvature of the weli petersson metric with respect to the weil petersson volume form are all rational numbers

second fundamental measure of geometric sets and local approximation of curvatures using the theory of normal cycles we associate with each geometric subset of a riemannian manifold a tensor valued curvature measure which we call its second fundamental measure this measure provides a finer description of the geometry of singular sets than the standard curvature measures moreover we deal with approximation of curvature measures we get a local quantitative estimate of the difference between curvature measures of two geometric subsets when one of them is a smooth hypersurface

generalized hodge metrics and bcov torsion on calabi yau moduli we establish an unexpected relation among the weil petersson metric the generalized hodge metrics and the bcov torsion using this relation we prove that certain kind of moduli spaces of polarized calabi yau manifolds do not admit complete subvarieties that is there is no complete family for certain class of polarized calabi yau manifolds we also give an estimate of the complex hessian of the bcov torsion using the relation after establishing a degenerate version of the schwarz lemma of yau we prove that the complex hessian of the bcov torsion is bounded by the poincare metric

absolute continuity for curvature measures of convex sets iii abstract this work is devoted to the investigation of the basic relationship between the geometric shape of a convex set and measure theoretic properties of the associated curvature and surface area measures we study geometric consequences of and conditions for absolute continuity of curvature and surface area measures with respect to d 1 dimensional hausdorff measure in euclidean space r d our main results are two transfer principles which allow one to translate properties connected with the absolute continuity of the r th curvature measure of a convex body to dual properties related to the absolute continuity of the d 1 r th surface area measure of the polar body and conversely applications are also considered

uniform approximation of near singular surfaces we consider the problem of a piecewise polynomial spline approximation of algebraic surfaces of a given degree conventional accuracy estimates in such approximations include bounds on the high order derivatives or on the surface curvature as an algebraic surface degenerates to a singular one the curvature blows up consequently the same happens with the complexity of the approximation to keep the required accuracy we need more and more patches at near singular high curvature areas this indeed happens in any conventional triangulation algorithm nevertheless using c k reparametrization theorem which originally appeared in dynamical applications we show in this paper that for such near singular families of surfaces there exist approximations of any fixed accuracy and of a uniformly bounded complexity to give an example of a situation where our construction becomes explicit we consider one of the possible models for singular and near singular surfaces in this model developed in y yomdin generic singularities of surfaces in d cheniot n dutertre c murolo d trotman a pichon eds singularity theory dedicated to j p brasselet on his 60 th birthday proceedings of the 2005 marseille singularity school and conference marseille france 2005 pp 24 1 25 2 d haviv ma thesis weizmann institute 2006 a surface appears as a part of the level surface f c with f f 1f 2 f m having a product form the advantage of this approach is that edges and corners appear in a generic and stable way as well as near edges and near corners using special model based representation developed in y yomdin generic singularities of surfaces in d cheniot n dutertre c murolo d trotman a pichon eds singularity theory dedicated to j p brasselet on his 60 th birthday proceedings of the 2005 marseille singularity school and conference marseille france 2005 pp 24 1 25 2 d haviv ma thesis weizmann institute 2006 we produce explicit formulae for the uniform c 2 reparametrization of some of these near singularities

sharp well posedness and ill posedness results for a quadratic non linear schrodinger equation abstract we establish that the quadratic non linear schrodinger equation iu t u xx u 2 where u r r c is locally well posed in h s r when s 1 and ill posed when s 1 previous work in c kenig g ponce l vega quadratic forms for the 1 d semilinear schrodinger equation trans amer math soc 346 1996 3323 3353 had established local well posedness for s 3 4 the local well posedness is achieved by an iteration using a modification of the standard x s b spaces the ill posedness uses an abstract and general argument relying on the high to low frequency cascade present in the non linearity and a computation of the first non linear iterate

symmetric differentials and the fundamental group helene esnault asked whether a smooth complex projective variety x with infinite fundamental group must have a nonzero symmetric differential meaning that h x s x 6 0 for some i 0 this was prompted by the second author s work 23 in fact severi had wondered in 1949 about possible relations between symmetric differentials and the fundamental group 34 p 36 we know from hodge theory that the cotangent bundle x has a nonzero section if and only if the abelianization of 1x is infinite the geometric meaning of other symmetric differentials is more mysterious and it is intriguing that they may have such a direct relation to the fundamental group in this paper we prove the following result on esnault s question in the slightly broader setting of compact kahler manifolds

valuations with crofton formula and finsler geometry valuations admitting a smooth crofton formula are studied using geometric measure theory and rumin s cohomology of contact manifolds the main technical result is a current representation of a valuation with a smooth crofton formula a geometric interpretation of alesker s product is given for such valuations as a first application in finsler geometry a short proof of the theorem of gelfand smirnov that crofton densities are projective is derived the holmes thompson volumes in a projective finsler space are studied it is shown that they induce in a natural way valuations and that the alesker product of the k dimensional and the l dimensional holmes thompson valuation is the k l dimensional holmes thompson valuation

normal cycles and curvature measures of sets with d c boundary abstract we show that for every compact domain in a euclidean space with d c delta convex boundary there exists a unique legendrian cycle such that the associated curvature measures fulfill a local version of the gauss bonnet formula this was known in dimensions two and three and was open in higher dimensions in fact we show this property for a larger class of sets including also lower dimensional sets we also describe the local index function of the legendrian cycles and we show that the associated curvature measures fulfill the crofton formula

analytic torsion for calabi yau threefolds after bershadsky cecotti ooguri vafa we introduce an invariant of calabi yau threefolds which we call the bcov invariant and which we obtain using analytic torsion we give an explicit formula for the bcov invariant as a function on the compactified moduli space when it is isomorphic to a projective line as a corollary we prove the formula for the bcov invariant of quintic mirror threefolds conjectured by bershadsky cecotti ooguri vafa

stably embedded surfaces of bounded integral curvature we study the effect of simultaneous bounds on the local l1 norms of the second fundamental form and of the gauss curvature on the geometry of surfaces embedded in a riemannian manifold m such bounds are natural since together with an area bound they amount to a local bound on the area of the manifold of unit normals to living in the sphere bundle of m the main technical point is that given a sequence of surfaces i with uniform local bounds of this type converging in the hausdorff metric topology to a surface 0 the length space structures of the i converge to that of 0 i e distances do not collapse it follows that under suitable local topological conditions the surface 0 is a manifold of bounded integral curvature mbc in the sense of alexandrov under all smooth complete changes of the metric on m in particular the boundary of a convex set with nonempty interior in any smooth complete m3 is mbc our method is to construct local coordinates for smoothly embedded surfaces satisfying w1 bv bounds based solely on curvature integrals in the process we demonstrate a regularity property for cut loci of surfaces

a steiner formula in the lp brunn minkowski theory abstract we prove an analogue of the classical steiner formula for the l p affine surface area of a minkowski outer parallel body for any real parameter p we show that the classical steiner formula and the steiner formula of lutwak s dual brunn minkowski theory are special cases of this new steiner formula this new steiner formula and its localized versions lead to new curvature measures that have not appeared before in the literature they have the intrinsic volumes of the classical brunn minkowski theory as well as the dual quermassintegrals of the dual brunn minkowski theory as special cases properties of these new quantities are investigated a connection to information theory among them a steiner formula for the s th mixed l p affine surface area of a minkowski outer parallel body for any real parameters p and s is also given

asymptotics of degenerations of mixed hodge structures abstract we construct a hermitian metric on the classifying spaces of graded polarized mixed hodge structures and prove analogs of the strong distance estimate 6 between an admissible period map and the approximating nilpotent orbit we also consider the asymptotic behavior of the biextension metric introduced by hain 12 analogs of the norm estimates of 19 and the asymptotics of the naive limit hodge filtration considered in 21

tournaments flags and harmonic maps twistor space constructions have shown that under quite general hypotheses harmonic maps of riemann surfaces into riemannian manifolds can be represented by holomorphic curves with respect to almost complex structures which are often non integrable see for example 19 17 5 for maps into complex grassmannians the almost complex structures are invariant ones on flag manifolds and can be identified with tournaments i e digraphs in which any two nodes are joined by exactly one oriented edge elementary properties of tournaments are summarised in sect i the basic references are moon 14 and reid and beineke 18 emphasis is given to certain types of tournament which are interpreted in sect 2 in terms of the homogeneous structure of complex flag manifolds holomorphic curves in such flag manifolds are described in sect 3 by collections of subbundles of a trivial bundle over a riemann surface whose second fundamental forms define a tournament this approach relies on techniques of burstall and wood 6 and gives rise naturally to harmonic maps into grassmannians when the riemann surface has genus zero a result from algebraic geometry namely the birkhoff grothendieck decomposition theorem for holomorphic vector bundles fits into the scheme and allows one to reconstruct a tournament from a harmonic map into a grassmannian in sect 4 we use such techniques to show that harmonic maps of tl2p 1 into a complex grassmannian can be built up successively from holomorphic data this last result extends results of burstall and wood 6 and is closely related to recent work of uhlenbeck 21 and wolfson 23

valuations on manifolds and rumin cohomology smooth valuations on manifolds are studied by establishing a link with the rumin de rham complex of the co sphere bundle several operations on differential forms induce operations on smooth valuations signature operator rumin laplace operator euler verdier involution and derivation operator as an application alesker s hard lefschetz theorem for even translation invariant valuations on a finite dimensional euclidean space is generalized to all translation invariant valuations the proof uses kahler identities the rumin de rham complex and spectral geometry

absolute continuity for curvature measures of convex sets ii this work is devoted to the investigation of the basic relationship between the geometric shape of a convex set and measure theoretic properties of the associated curvature and surface area measures we study geometric consequences of and conditions for absolute continuity of curvature and surface area measures with respect to d 1 dimensional hausdorff measure in euclidean space r our main results are two transfer principles which allow one to translate properties connected with the absolute continuity of the rth curvature measure of a convex body to dual properties related to the absolute continuity of the d 1 r th surface area measure of the polar body and conversely applications are also considered 2002

monotone nonparametric regression in monotone regression procedures one utilizes only the monotonicity of the regression function in nonparametric regression one utilizes only the assumed smoothness the analytic and asymptotic properties of the estimator are superior in the latter case however monotonicity is not guaranteed we study a hybrid procedure that produces monotone estimators with properties similar to those of nonparametric regression estimators

inference in nonparametric instrumental variables with partial identification this paper develops methods for hypothesis testing in a nonparametric instrumental variables setting within a partial identification framework we construct and derive the asymptotic distribution of a test statistic for the hypothesis that at least one element of the identified set satisfies a conjectured restriction the same test statistic can be employed under identification in which case the hypothesis is whether the true model satisfies the posited property an almost sure consistent bootstrap procedure is provided for obtaining critical values possible applications include testing for semiparametric specifications as well as building confidence regions for certain functionals on the identified set as an illustration we obtain confidence intervals for the level and slope of brazilian fuel engel curves a monte carlo study examines finite sample performance

anti concentration and honest adaptive confidence bands modern construction of uniform condence e and nickl 2010 this condition requires the existence of a limit distribution of an extreme value type for the supremum of a studentized empirical process equivalently for the supremum of a gaussian process with the same covariance function as that of the studentized empirical process the principal contribution of this paper is to remove the need for this classical condition we show that a considerably weaker sucient condi tion is derived from an anti concentration property of the supremum of the approximating gaussian process and we derive an inequality lead ing to such a property for separable gaussian processes we refer to the new condition as a generalized sbr condition our new result shows that the supremum does not concentrate too fast around any value we then apply this result to derive a gaussian multiplier bootstrap procedure for constructing honest condence bands for nonparametric density estimators this result can be applied in other nonparametric problems as well an essential advantage of our approach is that it ap plies generically even in those cases where the limit distribution of the supremum of the studentized empirical process does not exist or is un known this is of particular importance in problems where resolution levels or other tuning parameters have been chosen in a data driven fash ion which is needed for adaptive constructions of the condence bands furthermore our approach is asymptotically honest at a polynomial rate namely the error in coverage level converges to zero at a fast polynomial speed with respect to the sample size in sharp contrast the approach based on extreme value theory is asymptotically honest only at a logarithmic rate the error converges to zero at a slow loga rithmic speed finally of independent interest is our introduction of a new practical version of lepski s method which computes the optimal non conservative resolution levels via a gaussian multiplier bootstrap method

the steiner formula for minkowski valuations a steiner type formula for continuous translation invariant minkowski valuations is established in combination with a recent result on the symmetry of rigid motion invariant homogeneous bivaluations this new steiner type formula is used to obtain a family of brunn minkowski type inequalities for rigid motion intertwining minkowski valuations

gauss bonnet chern theorem on moduli space in this paper we prove a gauss bonnet chern type theorem in full generality for the chern weil forms of hodge bundles that is the chern weil forms compute the corresponding chern classes this settles a long standing problem second we apply the result to calabi yau moduli and proved the corresponding gauss bonnet chern type theorem in the setting of weil petersson geometry as an application of our results in string theory we prove that the number of flux vacua of type ii string compactified on a calabi yau manifold is finite and their number is bounded by an intrinsic geometric quantity

theory of valuations on manifolds iii multiplicative structure in the general case this is the third part of a series of articles where the theory of valuations on manifolds is constructed in the second part of this series the notion of a smooth valuation on a manifold was introduced the goal of this article is to put a canonical multiplicative structure on the space of smooth valuations on general manifolds thus extending some of the affine constructions from the first author s 2004 paper and from the first part of this series

hermitian integral geometry we give in explicit form the principal kinematic formula for the action of the ane unitary group on c n together with a straightforward algebraic method for computing the full array of unitary kinematic formulas expressed in terms of certain convex valuations introduced essentially by h tasaki we introduce also several other canonical bases for the algebra of unitary invariant valuations explore their interrelations and characterize in these terms the cones of positive and monotone elements

theory of valuations on manifolds ii abstract this article is the second part in the series of articles where we are developing theory of valuations on manifolds roughly speaking valuations could be thought as finitely additive measures on a class of nice subsets of a manifold which satisfy some additional assumptions the goal of this article is to introduce a notion of a smooth valuation on an arbitrary smooth manifold and establish some of the basic properties of it

kinematic formulas for sets defined by differences of convex functions abstract the class wdc m consists of all subsets of a smooth manifold m that may be expressed in local coordinates as sufficiently regular sublevel sets of dc differences of convex functions if m is riemannian and g is a group of isometries acting transitively on the sphere bundle sm we define the invariant curvature measures of compact wdc subsets of m and show that pairs of such subsets are subject to the array of kinematic formulas known to apply to smoother sets restricting to the case m g r d s o d this extends and subsumes federer s theory of sets with positive reach in an essential way the key technical point is equivalent to a sharpening of a classical theorem of ewald larman and rogers characterizing the dimension of the set of directions of line segments lying in the boundary of a given convex body

the monotone smoothing of scatterplots we consider the problem of summarizing a scatterplot with a smooth monotone curve a solution that combines local averaging and isotonic regression is proposed and we demonstrate its use with two examples in the second example the procedure is applied in a regression setting to some data from box and cox 1964 and it is shown how this new procedure generalizes box and cox s well known family of transformations in the same example the bootstrap is applied to obtain a measure of the variability of the procedure

stiefel whitney classes and the conormal cycle of a singular variety a geometric construction of sullivan s stiefel whitney homology classes of a real analytic variety x is given by means of the conormal cycle of an embedding of x in a smooth variety we prove that the stiefel whitney classes define additive natural transformations from certain constructible functions to homology we also show that for a complex analytic variety these classes are the mod 2 reductions of the chern macpherson classes we present a new definition of the stiefel whitney homology classes of a possibly singular real analytic variety x the original definition due to sullivan s involves a triangulation of x its geometric meaning is unclear our definition uses the conormal cycle of an embedding of x in a smooth variety the conormal cycle of a subanalytic subset x of an analytic manifold m was defined by the first author f4 using geometric measure theory the conormal cycle is an integral current representing up to sign kashiwara s characteristic cycle of the sheaf dmrx f4 4 7 our construction of the stiefel whitney classes is based on the fundamental observation that the conormal cycle of a real analytic variety is antipodally symmetric mod 2 we use this observation to give a definition of stiefel whitney classes which is parallel to the first author s definition of the chern schwartz macpherson classes of a complex analytic variety f5 this definition of chern classes is related to earlier work of brylinski dubson kashiwara bdk and sabbah sa in fact we show that the stiefel whitney homology classes of a complex analytic variety are the mod 2 reductions of the chern classes we prove that our stiefel whitney classes satisfy axioms similar to the deligne grothendieck axioms for chern classes and we prove a specialization formula for the stiefel whitney classes of a family of varieties we show that the stiefel whitney classes of an affine real analytic variety x are represented by the polar cycles of x introduced for simplicial spaces by banchoff b and mccrory mc and we give a new proof of the combinatorial formula for stiefel whitney classes of manifolds our central result is a specialization formula for the conormal cycle theorem 3 7 we use this formula to prove the basic results of kashiwara and schapira s calculus of subanalytically constructible functions sc ks 9 7 as well as the pushforward and specialization formulas for stiefel whitney classes received by the editors october 2 1995 1991 mathematics subject classification primary 14p25 57r20 secondary 14p15 49q15

valuation theory of indefinite orthogonal groups abstract let so p q denote the identity connected component of the real orthogonal group with signature p q we give a complete description of the spaces of continuous and generalized translation and so p q invariant valuations generalizing hadwiger s classification of euclidean isometry invariant valuations as a result of independent interest we identify within the space of translation invariant valuations the class of klain schneider continuous valuations which strictly contains all continuous translation invariant valuations the operations of pull back and push forward by a linear map extend naturally to this class