hard lefschetz theorem for valuations complex integral geometry and unitarily invariant valuations we obtain new general results on the structure of the space of translation invariant continuous valuations on convex sets a version of the hard lefschetz theorem using these and our previous results we obtain explicit characterization of unitarily invariant translation invariant continuous valuations it implies new integral geometric formulas for real submanifolds in hermitian spaces generalizing the classical kinematic formulas in euclidean

mass concentration and aging in the parabolic anderson model with doubly exponential tails we study the non negative solution u u x t to the cauchy problem for the parabolic equation partial t u vardelta u xi u on mathbb z d times 0 infty with initial data u x 0 mathsf 1 0 x here vardelta is the discrete laplacian on mathbb z d and xi xi z z in mathbb z d is an i i d random field with doubly exponential upper tails we prove that for large t and with large probability most of the total mass u t sum x u x t of the solution resides in a bounded neighborhood of a site z t that achieves an optimal compromise between the local dirichlet eigenvalue of the anderson hamiltonian vardelta xi and the distance to the origin the processes t mapsto z t and t mapsto tfrac 1 t log u t are shown to converge in distribution under suitable scaling of space and time aging results for z t as well as for the solution to the parabolic problem are also established the proof uses the characterization of eigenvalue order statistics for vardelta xi in large sets recently proved by the first two authors

delocalising the parabolic anderson model through partial duplication of the potential the parabolic anderson model on mathbb z d with i i d potential is known to completely localise if the distribution of the potential is sufficiently heavy tailed at infinity in this paper we investigate a modification of the model in which the potential is partially duplicated in a symmetric way across a plane through the origin in the case of potential distribution with polynomial tail decay we exhibit a surprising phase transition in the model as the decay exponent varies for large values of the exponent the model completely localises as in the i i d case by contrast for small values of the exponent we show that the model may delocalise more precisely we show that there is an event of non negligible probability on which the solution has non negligible mass on two sites

inequalities between projection functions of convex bodies we obtain injectivity results for certain integral transforms on grassmann manifolds these are used to solve the intermediate cases of a problem of shephard and to give other applications to projection functions of convex bodies of revolution

long time tails in the parabolic anderson model with bounded potential we consider the parabolic anderson problem t u u u on 0 z d with random i i d potential z z zd and the initial condition u 0 1 our main assumption is that esssup 0 0 depending on the thickness of the distribution prob 0 close to its essential supremum we identify both the asymptotics of the moments of u t 0 and the almost sure asymptotics of u t 0 as t in terms of variational problems as a by product we establish lifshitz tails for the random schrodinger operator at the bottom of its spectrum in our class of distributions the lifshitz exponent ranges from d 2 to the power law is typically accompanied by lower order corrections

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

generalized intersection bodies abstract we study the structures of two types of generalizations of intersection bodies and the problem of whether they are in fact equivalent intersection bodies were introduced by lutwak and played a key role in the solution of the busemann petty problem a natural geometric generalization of this problem considered by zhang led him to introduce one type of generalized intersection bodies a second type was introduced by koldobsky who studied a different analytic generalization of this problem koldobsky also studied the connection between these two types of bodies and noted that an equivalence between these two notions would completely settle the unresolved cases in the generalized busemann petty problem we show that these classes share many identical structural properties proving the same results using integral geometry techniques for zhang s class and fourier transform techniques for koldobsky s class using a functional analytic approach we give several surprising equivalent formulations for the equivalence problem which reveal a deep connection to several fundamental problems in the integral geometry of the grassmann manifold

range characterization of the cosine transform on higher grassmannians abstract we characterize the range of the cosine transform on real grassmannians in terms of the decomposition under the action of the special orthogonal group so n we also give a geometric interpretation of this image in terms of valuations in addition we discuss the non archimedean analogues

sections of convex bodies the generalized busemann petty problem asks if the volume of i dimensional central section of a centrally symmetric convex body in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i is smaller than that of another such body is the volume of the body also smaller it is proved that the answer is negative if 2 i n the case of a 2 dimensional section remains open the proof uses techniques in functional analysis and radon transforms on grassmannians it also requires the notion of an i intersection body which generalizes the notion of an intersection body inequalities among the volumes of projection bodies polar projection bodies and their central sections are proved they are related to the maximal slice problem

the universality classes in the parabolic anderson model we discuss the long time behaviour of the parabolic anderson model the cauchy problem for the heat equation with random potential on mathbb z d we consider general i i d potentials and show that exactly four qualitatively different types of intermittent behaviour can occur these four universality classes depend on the upper tail of the potential distribution 1 tails at that are thicker than the double exponential tails 2 double exponential tails at studied by gartner and molchanov 3 a new class called almost bounded potentials and 4 potentials bounded from above studied by biskup and konig the new class 3 which contains both unbounded and bounded potentials is studied in both the annealed and the quenched setting we show that intermittency occurs on unboundedly increasing islands whose diameter is slowly varying in time the characteristic variational formulas describing the optimal profiles of the potential and of the solution are solved explicitly by parabolas respectively gaussian densities our analysis of class 3 relies on two large deviation results for the local times of continuous time simple random walk one of these results is proved by brydges and the first two authors in bhk04 and is also used here to correct a proof in bk01

a uniqueness result for mean section bodies abstract for 1 k d the k th mean section body m k k of a convex body k in r d is the minkowski sum of all its sections by k dimensional flats we will show that for any fixed 1 k d the body k is uniquely determined by the body m k k assuming dim k d k 2 this result was previously known only for centrally symmetric bodies or in full generality only for the case k 2

convolutions transforms and convex bodies the paper studies convex bodies and star bodies in bbb r n by using radon transforms on grassmann manifolds p cosine transforms on the unit sphere and convolutions on the rotation group of bbb r n it presents dual mixed volume characterizations of i intersection bodies and l p balls which are related to certain volume inequalities for cross sections of convex bodies it considers approximations of special convex bodies by analytic bodies and various finite sums of ellipsoids which preserve special geometric properties convolution techniques are used to derive formulas for mixed volumes mixed surface measures and p cosine transforms they are also used to prove characterizations of geometric functionals such as surface area and dual quermassintegrals

some concepts of negative dependence abstract the theory of positive dependence notions cannot yield useful results for some widely used distributions such as the multinomial dirichlet and the multivariate hypergeometric some conditions of negative dependence that are satisfied by these distributions and which have practical meaning are introduced preservation results for some of these concepts are derived useful inequalities for some widely used distributions are obtained results of mallows 1969 that apply to the multinomial distributions are extended to more distributions examples are listed author

convolutions and multiplier transformations of convex bodies rotation intertwining maps from the set of convex bodies in into itself that are continuous linear operators with respect to minkowski and blaschke addition are investigated the main focus is on blaschke minkowski homomorphisms we show that such maps are represented by a spherical convolution operator an application of this representation is a complete classification of all even blaschke minkowski homomorphisms which shows that these maps behave in many respects similar to the well known projection body operator among further applications is the following result if an even blaschke minkowski homomorphism maps a convex body to a polytope then it is a constant multiple of the projection body operator

hitting times and periodicity in random dynamics we prove quenched laws of hitting time statistics for random subshifts of finite type in particular we prove a dichotomy between the law for periodic and for non periodic points we show that this applies to random gibbs measures

statistical properties of dynamical systems with some hyperbolicity this paper is about the ergodic theory of attractors and conservative dynamical systems with hyperbolic properties on large parts though not necessarily all of their phase spaces the main results are for discrete time systems to put this work into context recall that for axiom a attractors the picture has been fairly complete since the 1970 s see s1 b r2 since then much progress has been made on two fronts there is a general nonuniform theory that deals with properties common to all diffeomorphisms with nonzero lyapunov exponents o p1 ka ly and there are detailed analyses of specific kinds of dynamical systems including for example billiards 1 dimensional and henon type maps s2 bsc hk j bc2 by1 statistical properties such as exponential decay of correlations are not enjoyed by all diffeomorphisms with nonzero lyapunov exponents the goal of this paper is a systematic understanding of these and other properties for a class of dynamical systems larger than axiom a this class will not be defined explicitly but it includes some of the much studied examples by looking at regular returns to sets with good hyperbolic properties one could give systems in this class a simple dynamical representation conditions for the existence of natural invariant measures exponential mixing and central limit theorems are given in terms of the return times these conditions can be checked in concrete situations giving a unified way of proving a number of results some new and some old among the new results are the exponential decay of correlations for a class of scattering billiards and for a positive measure set of henon type maps

isomorphic properties of intersection bodies we study isomorphic properties of two generalizations of intersection bodies the class ikn of k intersection bodies in rn and the class bpkn of generalized k intersection bodies in rn in particular we show that all convex bodies can be in a certain sense approximated by intersection bodies namely if k is any symmetric convex body in rn and 1 k n 1 then the outer volume ratio distance from k to the class bpkn can be estimated by o v r k bpkn inf c k 1n c bpkn k c cnklogenk where c 0 is an absolute constant next we prove that if k is a symmetric convex body in rn 1 k n 1 and its k intersection body ik k exists and is convex then dbm ik k b2n c k where c k is a constant depending only on k dbm is the banach mazur distance and b2n is the unit euclidean ball in rn this generalizes a well known result of hensley and borell we conclude the paper with volumetric estimates for k intersection bodies

diophantine properties of iets and general systems quantitative proximality and connectivity three properties of dynamical systems recurrence connectivity and proximality are quantified by introducing and studying the gauges measurable functions corresponding to each of these properties the properties of the proximality gauge are related to the results in the active field of shrinking targets the emphasis in the present paper is on the iets interval exchange transformations mathcal i t mathcal i 0 1 in particular we prove that if an iet t is ergodic relative to the lebesgue measure then the equality a1 liminf n to infty n bigl t n x y bigr 0 holds for a a x y in mathcal i 2 the ergodicity assumption is essential the result does not extend to all minimal iets also the factor n in a1 is optimal e g it cannot be replaced by n ln ln lnn on the other hand for lebesgue almost all 3 iets mathcal i t we prove that for all 0 a2 liminf n to infty n epsilon bigl t n x t n y bigr infty quad text for lebesgue a a x y in mathcal i 2 this should be contrasted with the equality lim inf n t n x t n y 0 for a a x y in mathcal i 2 which holds since mathcal i 2 t times t is ergodic because generic 3 iets mathcal i t are weakly mixing we introduce the notion of entropy of an iet which is related to obtaining estimates of type a2 we also prove that no 3 iet is strongly topologically mixing

geometric characterization of intermittency in the parabolic anderson model we consider the parabolic anderson problem tu u x u on r z d with localized initial condition u 0 x 0 x and random i i d potential under the assumption that the distribution of 0 has a double exponential or slightly heavier tail we prove the following geometric characterisation of intermittency with probability one as t 1 the overwhelming contribution to the total mass p x u t x comes from a slowly increasing number of islands which are located far from each other these islands are local regions of those high exceedances of the eld in a box of side length 2t log 2 t for which the local principal dirichlet eigenvalue of the random operator is close to the top of the spectrum in the box we also prove that the shape of in these regions is non random and that u t is close to the corresponding positive eigenfunction this is the geometric picture suggested by localization theory for the anderson hamiltonian

correlation structure of intermittency in the parabolic anderson model consider the cauchy problem u x t t u x t x d t 0 with initial condition u x 0 1 and with the anderson hamiltonian here is the discrete laplacian 0 is a diffusion constant and x x d is an i i d random field taking values in gartner and molchanov 1990 have shown that if the law of 0 is nondegenerate then the solution u is asymptotically intermittent in the present paper we study the structure of the intermittent peaks for the special case where the law of 0 is in the vicinity of the double exponential prob 0 s exp e s s here 0 is a parameter that can be thought of as measuring the degree of disorder in the field our main result is that for fixed x y d and t the correlation coefficient of u x t and u y t converges to w 2 l2 z d w x z w y z in this expression while w d is given by w v d with v the unique centered ground state i e the solution in l2 with minimal l 2 norm of the 1 dimensional nonlinear equation v 2 v log v 0 the uniqueness of the ground state is actually proved only for large but is conjectured to hold for any 0 empty it turns out that if the right tail of the law of 0 is thicker or thinner than the double exponential then the correlation coefficient of u x t and u y t converges to x y resp the constant function 1 thus the double exponential family is the critical class exhibiting a nondegenerate correlation structure

generalized intersection bodies are not equivalent abstract in a koldobsky a functional analytic approach to intersection bodies geom funct anal 10 2000 1507 1526 a koldobsky asked whether two types of generalizations of the notion of an intersection body are in fact equivalent the structures of these two types of generalized intersection bodies have been studied by the author in e milman generalized intersection bodies j funct anal 240 2 2006 530 567 providing substantial evidence for a positive answer to this question the purpose of this note is to construct a counter example which provides a surprising negative answer to this question in a strong sense this implies the existence of non trivial non negative functions in the range of the spherical radon transform and the existence of non trivial spaces which embed in l p for certain negative values of p

intersection bodies and valuations all gl n covariant star body valued valuations on convex polytopes are completely classified it is shown that there is a unique nontrivial such valuation this valuation turns out to be the so called intersection operator an operator that played a critical role in the solution of the busemann petty problem

parabolic problems for the anderson model the objective of this paper is a mathematically rigorous investigation of intermittency and related questions intensively studied in different areas of physics in particular in hydrodynamics on a qualitative level intermittent random fields are distinguished by the appearance of sparsely distributed sharp peaks which give the main contribution to the formation of the statistical moments the paper deals with the cauchy problem t u t x hu t x u 0 x t0 x 0 t x d for the anderson hamiltonianh x x d where is a generally unbounded spatially homogeneous random potential this first part is devoted to some basic problems using percolation arguments a complete answer to the question of existence and uniqueness for the cauchy problem in the class of all nonnegative solutions is given in the case of i i d random variables necessary and sufficient conditions for intermittency of the fieldsu t ast are found in spectral terms ofh rough asymptotic formulas for the statistical moments and the almost sure behavior ofu t x ast are also derived

poincare recurrence for observations a high dimensional dynamical system is often studied by experimentalists through the measurement of a relatively low number of different quantities called an observation following this idea and in the continuity of boshernitzan s work for a measure preserving system we study poincare recurrence for the observation the link between the return time for the observation and the hausdorff dimension of the image of the invariant measure is considered we prove that when the decay of correlations is super polynomial the recurrence rates for the observations and the pointwise dimensions relative to the push forward are equal

processes of flats induced by higher dimensional processes stationary processes of k flats in mathbb e d can be thought of as point processes on the grassmannian mathcal l kd of k dimensional subspaces of mathbb e d if such a process is sampled by a d k j dimensional space f it induces a process of j flats in f in this work we will investigate the possibility of determining the original k process from knowledge of the intensity measures of the induced j processes we will see that this is impossible precisely when 1 k d 1 and j 0 2 r 2 1 where r is the rank of the manifold mathcal l kd we will show how the problem is equivalent to the study of the kernel of various integral transforms these will then be investigated using harmonic analysis on grassmannian manifolds

a positive answer to the busemann petty problem in three dimensions we prove that in e3 the busemann petty problem concerning cen tral sections of centrally symmetric convex bodies has a positive answer together with other results this settles the problem in each dimension

contravariant minkowski valuations a complete classification of all continuous gl n contravariant minkowski valuations is established as an application we present a family of sharp isoperimetric inequalities for such valuations which generalize the classical petty projection inequality

parabolic problems for the anderson model i intermittency and related topics the objective of this paper is a mathematically rigorous investigation of intermittency and related questions intensively studied in different areas of physics in particular in hydrodynamics on a qualitative level intermittent random fields are distinguished by the appearance of sparsely distributed sharp peaks which give the main contribution to the formation of the statistical moments the paper deals with the cauchy problem t u t x hu t x u 0 x t 0 x 0 t x d for the anderson hamiltonianh x x d where is a generally unbounded spatially homogeneous random potential this first part is devoted to some basic problems using percolation arguments a complete answer to the question of existence and uniqueness for the cauchy problem in the class of all nonnegative solutions is given in the case of i i d random variables necessary and sufficient conditions for intermittency of the fieldsu t ast are found in spectral terms ofh rough asymptotic formulas for the statistical moments and the almost sure behavior ofu t x ast are also derived

localization for random walks among random obstacles in a single euclidean ball place an obstacle with probability 1 p independently at each vertex of mathbb z d and run a simple random walk before hitting one of the obstacles for d ge 2 and p strictly above the critical threshold for site percolation we condition on the environment where the origin is contained in an infinite connected component free of obstacles and we show that for environments with probability tending to one as n rightarrow infty there exists a unique discrete euclidean ball of volume d log 1 p n asymptotically such that the following holds conditioned on survival up to time n we have that at any time t in epsilon n n n for some epsilon n rightarrow n rightarrow infty 0 with probability tending to one the simple random walk is in this ball this work relies on and substantially improves a previous result of the authors on localization in a region of volume poly logarithmic in n for the same problem

radon transforms on higher grassmannians on definit une transformee de radon r a partir des fonctions cr k n les grassmanniens des k plans projectifs dans cp n aux fonctions sur gr l n si f c gr k n et l gr l n alors rf l est l integrale de f h sur tous les k plans h qui sont dans l si r t est la transformee duale on montre sous de bonnes hypotheses sur k et l que r t r est inversible par un polynome dans les operateurs de casimir de u n 1 le groupe des isometries de cp n

projection functions area measures and the alesker fourier transform abstract dual to koldobsky s notion of j intersection bodies the class of j projection bodies is introduced generalizing minkowski s classical notion of projection bodies of convex bodies a fourier analytic characterization of j projection bodies in terms of their area measures of order j is obtained in turn this yields an equivalent characterization of j projection bodies involving alesker s fourier type transform on translation invariant smooth spherical valuations as applications of these results several basic properties of j projection bodies are established and new non trivial examples are constructed

crofton measures and minkowski valuations a description of continuous rigid motion compatible minkowski valuations is established as an application we present a brunn minkowski type inequality for intrinsic volumes of these valuations

hausdorff dimension of measures via poincare recurrence we study the quantitative behavior of poincare recurrence in particular for an equilibrium measure on a locally maximal hyperbolic set of a c1 diffeomorphism f we show that the recurrence rate to each point coincides almost everywhere with the hausdorff dimension d of the measure that is inf k 0 fkx b x r r d this result is a non trivial generalization of work of boshernitzan concerning the quantitative behavior of recurrence and is a dimensional version of work of ornstein and weiss for the entropy we stress that our approach uses different techniques furthermore our results motivate the introduction of a new method to compute the hausdorff dimension of measures

even valuations on convex bodies the notion of eveil valuation is introduced as a natural generalization of volume on cornpact convex subsets of euclidean space a recent characterization theorem for volume leads in turn to a connection between even valuations on compact convex sets and continuous functions on grassmannians this connection can be described in part using generating distributions for symmetric compact convex sets we also explore some consequences of these characterization results in convex and integral geometry recent interest in volume as a valuation on compact conlvex sets stems from hilbert s third problem which is actually an ancient problem recast in modern terms hilbert asked if two polytopes p and q can be each cut into a finite number of pieces p pn and q qm with each pi congruent to qi by a rigid euclidean motion provided that p and q have the same volume 18 this question was answered in the negative by max dehn 2 4 29 who found a functionlal on polytopes that is invariant under dissections over rigid motions while varying in value among polytopes of equal volume in other words the dehn invariant is a simple rigid motion invariant valuation on polytopes that is not equal to volume under any normalization dehn s solution left open the question of exactly what conditions oil p and q imply equidissectability over the group of rigid motions although this problem was solved by hadwiger in the case where only translations and no rotations nor reflections are permitted see 2 16 17 26 29 in the course of studying this and related problems hadwiger discovered a characterization of euclidean volume as a continuous rigid motion invariant simple valuation on compact convex sets that is a continuous rigid motion invariant valuation that vanishes on convex sets of less than full dimension this result led in turn to a complete characterization of all continuous rigid motion inivariant valuations on compact convex sets in 1r as consisting of a real n 4 dimensional vector space spanned by the intrinsic volumes or quermassintegrals 16 also 20 21 31 since many standard functionals and integral operators can be interpreted as invariant valuations such as intrinsic volumes mean projections crofton and kinematic formulas what came to be known as hadwiger s chtracterization theorem proved to be a valuable tool for generating quick and effortless proofs of many formulas and equations in integral geometry unfortunately hadwiger s original proof was long and difficult 161 while seeking a shorter proof of hadwiger s volume characterization the author discovered received by the editors june 24 1996 and in revised form september 29 1997 1991 mathematics subject classification primary 52a22 52a38 52a39 52b45 research supported in part by nsf grants dms 9022140 to msri and dms 9626688 to the author 1999 american mathematical society 71 this content downloaded from 207 46 13 52 on mon 24 oct 2016 04 15 57 utc all use subject to http about jstor org terms

extremal theory for spectrum of random discrete schrodinger operator iii localization properties in this paper we study the asymptotic localization properties with high probability of the kth eigenfunction associated with the kth largest eigenvalue k 1 fixed of the multidimensional anderson hamiltonian in torus v increasing to the whole of lattice denote by z k v v the site at which the kth largest value of potential is attained it is well known that if the tails of potential distribution are heavier than the double exponential function and satisfies additional regularity and continuity conditions at infinity then the kth eigenfunction is asymptotically delta function at the site z k v localization centre for some random k v k 1 we study the asymptotic behavior of the index v k by distinguishing between three cases of the tails of potential distribution i for the heavy tails including gaussian v k is asymptotically bounded ii for the light tails but heavier than the double exponential the index v k unboundedly increases like v o 1 iii finally for the double exponential tails with high disorder the index v k behaves like a power of v for weibull s and fractional double exponential types distributions associated with the case ii we obtain the first order expansion formulas for log v k

projection bodies in complex vector spaces the space of minkowski valuations on an m dimensional complex vector space which are continuous translation invariant and contravariant under the complex special linear group is explicitly described each valuation with these properties is shown to satisfy geometric inequalities of the brunn minkowski aleksandrov fenchel and minkowski type

lutwak petty projection inequalities for minkowski valuations and their duals abstract lutwak s volume inequalities for polar projection bodies of all orders are generalized to polarizations of minkowski valuations generated by even zonal measures on the euclidean unit sphere this is based on analogues of mixed projection bodies for such minkowski valuations and a generalization of the notion of centroid bodies a new integral representation is used to single out lutwak s inequalities as the strongest among these families of inequalities which in turn are related to a conjecture on affine quermassintegrals in the dual setting a generalization of volume inequalities for intersection bodies of all orders by leng and lu is proved these results are related to grinberg s inequalities for dual affine quermassintegrals

an analytic solution to the busemann petty problem on sections of convex bodies we derive a formula connecting the derivatives of parallel section functions of an origin symmetric star body in r n with the fourier transform of powers of the radial function of the body a parallel section function or ni 1 dimensional x ray gives the ni 1 dimensional volumes of all hyperplane sections of the body orthogonal to a given direction this formula provides a new characterization of intersection bodies in r n and leads to a unifled analytic solution to the busemann petty problem suppose that k and l are two origin symmetric convex bodies inr n such that the ni 1 dimensional volume of each central hyperplane section of k is smaller than the volume of the corresponding section of l is the n dimensional volume of k smaller than the volume of l in conjunction with earlier established connections between the busemann petty problem intersection bodies and positive deflnite distributions our formula shows that the answer to the problem depends on the behavior of the ni 2 nd derivative of the parallel section functions the a rmative answer to the busemann petty problem for n 4 and the negative answer for n 5 now follow from the fact that convexity controls the second derivatives but does not control the derivatives of higher orders

intersection bodies and lp spaces abstract we prove that the convex intersection bodies are isomorphically equivalent to unit balls of subspaces of l q for each q 0 1 this is done by extending to negative values of p the factorization theorem of maurey and nikishin which states that for any 0 p q 1 every banach subspace of l p is isomorphic to a subspace of l q

intersection bodies and dual mixed volumes the setting for this article is euclidean n space ky n 2 let x denote the set of convex bodies compact convex subsets with nonempty interiors in r and let x denote the subset of x that contains the centered centrally symmetric with respect to the origin bodies for ue s let e denote the hyperplane through the origin that is orthogonal to u and let ki e denote the image of the orthogonal projection of the body ke x onto e we shall use v to denote i dimensional lebesgue measure for v and v we shall usually write v and u shephard 38 poses the question if k l e xe and

projection bodies and valuations let be the projection operator which maps every polytope to its projection body it is well known that maps the set of polytopes pn in rn into pn that it is a valuation and that for every p pn p is affinely associated to p it is shown that these properties characterize the projection operator this proves a conjecture by lutwak

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

a two cities theorem for the parabolic anderson model the parabolic anderson problem is the cauchy problem for the heat equation partial derivative t u t z delta u t z xi z u t z on 0 infinity x z d with random potential xi z z is an element of z d we consider independent and identically distributed potentials such that the distribution function of z converges polynomially at infinity if u is initially localized in the origin that is if u 0 z 1 0 z we show that as time goes to infinity the solution is completely localized in two points almost surely and in one point with high probability we also identify the asymptotic behavior of the concentration sites in terms of a weak limit theorem

a fourier transform approach to christoffel s problem we use fourier transform techniques to provide a new approach to berg s solution of the christoffel problem this leads to an explicit description of berg s spherical kernel and to new regularity properties of the associated integral transform

extremal theory for spectrum of random discrete schrodinger operator ii distributions with heavy tails we study the asymptotic structure of the first k largest eigenvalues k v and the corresponding eigenfunctions k v of a finite volume anderson model discrete schrodinger operator mathcal h v kappa delta v xi cdot on the multidimensional lattice torus v increasing to the whole of lattice provided the distribution function f of i i d potential satisfies condition log 1 f t o t3 and some additional regularity conditions as t for z v denote by 0 z the principal eigenvalue of the single peak hamiltonian v z z in l2 v and let lambda 0 k v be the kth largest value of the sample 0 in v we first show that the eigenvalues k v are asymptotically close to lambda 0 k v we then prove extremal type limit theorems i e poisson statistics for the normalized eigenvalues k v bv av where the normalizing constants av 0 and bv are chosen the same as in the corresponding limit theorems for lambda 0 k v the eigenfunction k v is shown to be asymptotically completely localized as v at the sites zk v v defined by lambda 0 z k v lambda 0 k v proofs are based on the finite rank in particular rank one perturbation arguments for discrete schrodinger operator when potential peaks are sparse

quantitative recurrence results letx be a probability measure spacex x endowed with a compatible metricd so that x d has a countable base it is well known that ift x x is measure preserving then almost all pointsx x are recurrent i e lim begin array 20 c inf n geqq 1 end array d x t n x 0 we show that under the additional assumption that the hausdorff measureh x ofx is finite for some 0 this result can be strengthened lim begin array 20 c inf n geqq 1 end array left n 1 alpha d x t n x right infty for almost all pointsx x a number of applications are considered

brownian confinement and pinning in a poissonian potential ii we study the behavior of ad dimensional brownian motion in a soft repulsive poissonian potential over long time intervals 0 t we introduce certaint and configuration dependent scales which grow almost linearly witht for typical configurations with probability tending to 1 ast goes to the size of displacements of the process is bounded above by these scales confinement effect the proof involves calculations beyond leading order to this end we use a coarse grained picture of the environment method of enlargement of obstacles and of the path a backbone of excursions between clearings and forest parts in the environment these coarse grained pictures are also used in the sequel 11 to the present article when proving the pinning effect