sharp systolic inequalities for reeb flows on the three sphere the systolic ratio of a contact form alpha on the three sphere is the quantity begin aligned rho mathrm sys alpha frac t min alpha 2 mathrm vol s 3 alpha wedge d alpha end aligned where t min alpha is the minimal period of closed reeb orbits on s 3 alpha a zoll contact form is a contact form such that all the orbits of the corresponding reeb flow are closed and have the same period our first main result is that rho mathrm sys le 1 in a neighbourhood of the space of zoll contact forms on s 3 with equality holding precisely at zoll contact forms this implies a particular case of a conjecture of viterbo a local middle dimensional non squeezing theorem and a sharp systolic inequality for finsler metrics on the two sphere which are close to zoll ones our second main result is that rho mathrm sys is unbounded from above on the space of tight contact forms on s 3

large gaps between consecutive prime numbers let g x denote the size of the largest gap between consecutive primes below x answering a question of erd os we show that g x f x logx log logx log log log logx log log logx 2 where f x is a function tending to innity with x our proof combines existing arguments with a random construction covering a set of primes by arithmetic progressions as such we rely on recent work on the existence and distribution of long arithmetic progressions consisting entirely of primes

single and multiple recurrence along non polynomial sequences abstract we establish new recurrence and multiple recurrence results for a rather large family f of non polynomial functions which contains tempered functions and non polynomial functions from a hardy field with polynomial growth in particular we show that somewhat surprisingly and in the contrast to the multiple recurrence along polynomials the sets of return times along functions from f are thick i e contain arbitrarily long intervals a major component of our paper is a new result about equidistribution of sparse sequences on nilmanifolds whose proof borrows ideas from the work of green and tao 26 among other things we show that for any f f any invertible probability measure preserving system x b t any a b with a 0 and any e 0 the sets of returns n n a t f n a 2 a e n n a t f n a t f n 1 a t f n k a 0 are thick our recurrence theorems imply via furstenberg s correspondence principle some new combinatorial results for example we show that given a set e n with positive upper density for every k n there are a n n such that a a f n a f n k e when f n n c with c 0 non integer this result provides a positive answer to a question posed by frantzikinakis 16 problem 23

a systolic inequality for geodesic flows on the two sphere for a riemannian metric g on the two sphere let ell min g be the length of the shortest closed geodesic and ell max g be the length of the longest simple closed geodesic we prove that if the curvature of g is positive and sufficiently pinched then the sharp systolic inequalities begin aligned ell mathrm min g 2 le pi mathrm area s 2 g le ell max g 2 end aligned hold and each of these two inequalities is an equality if and only if the metric g is zoll the first inequality answers positively a conjecture of babenko and balacheff the proof combines arguments from riemannian and symplectic geometry

c1 weierstrass for compact sets in hilbert space abstract the c 1 weierstrass approximation theorem is proved for any compact subset x of a hilbert space h the same theorem is also proved for whitney 1 jets on x when x satisfies the following further condition there exist finite dimensional linear subspaces h 1 h 2 h such that n 1 h n is dense in span x and n x x h n for each n 1 here n h h n is the orthogonal projection it is also shown that when x is compact convex with span x h and satisfies the above condition then c 1 x is complete if and only if the c 1 whitney extension theorem holds for x finally for compact subsets of h an extension of the c 1 weierstrass approximation theorem is proved for c 1 maps h h with compact derivatives

the mobius function and distal flows we prove that the mobius function is linearly disjoint from an analytic skew product on the 2 torus these flows are distal and can be irregular in the sense that their ergodic averages need not exist for all points the previous cases for which such disjointness has been proved are all regular we also establish the linear disjointness of the mobius function from various distal homogeneous flows

self similar groups and the zig zag and replacement products of graphs abstract every finitely generated self similar group naturally produces an infinite sequence of finite d regular graphs n we construct self similar groups whose graphs n can be represented as an iterated zig zag product and graph powering n 1 n k z k 1 we also construct self similar groups whose graphs n can be represented as an iterated replacement product and graph powering n 1 n k r k 1 this gives simple explicit examples of self similar groups whose graphs n form an expanding family and examples of automaton groups whose graphs n have linear diameters diam n o n and bounded girth

nonconventional ergodic averages and nilmanifolds we study the l2 convergence of two types of ergodic averages the first is the average of a product of functions evaluated at return times along arithmetic progressions such as the expressions appearing in furstenberg fs proof of szemer ledi fs theorem the second average is taken along cubes whose sizes tend to for each average we show that it is sufficient to prove the convergence for special systems the characteristic factors we build these factors in a general way independent of the type of the average to each of these factors we associate a natural group of transformations and give them the structure of a nilmanifold from the second convergence result we derive a combinatorial interpretation for the arithmetic structure inside a set of integers of positive upper density

a multidimensional szemeredi theorem for hardy sequences of different growth abstract we prove a variant of the multidimensional polynomial szemeredi theorem of bergelson and leibman where one replaces polynomial sequences with other sparse sequences defined by functions that belong to some hardy field and satisfy certain growth conditions we do this by studying the limiting behavior of the corresponding multiple ergodic averages and obtaining a simple limit formula a consequence of this formula in topological dynamics shows denseness of certain orbits when the iterates are restricted to suitably chosen sparse subsequences another consequence is that every syndetic set of integers contains certain nonshift invariant patterns and every finite coloring of n with each color class a syndetic set contains certain polychromatic patterns results very particular to our non polynomial setup

volumes of balls in large riemannian manifolds we prove two lower bounds for the volumes of balls in a riemannian manifold if m n g is a complete riemannian manifold with lling radius at least r then it contains a ball of radius r and volume at least n r n if m n hyp is a closed hyperbolic manifold and if g is another metric on m with volume no greater than n vol m hyp then the universal cover of m g contains a unit ball with volume greater than the volume of a unit ball in hyperbolic n space

free abelian covers short loops stable length and systolic inequalities we explore the geometry of the abel jacobi map f from a closed orientable riemannian manifold x to its jacobi torus h 1 x mathbb r h 1 x mathbb z mathbb r applying m gromov s filling inequality to the typical fiber of f we prove an interpolating inequality for two flavors of shortest length invariants of loops the procedure works provided the lift of the fiber is non trivial in the homology of the maximal free abelian cover overline x classified by f we show that the finite dimensionality of the rational homology of overline x is a sufficient condition for the homological non triviality of the fiber when applied to nilmanifolds our fiberwise inequality typically gives stronger information than the filling inequality for x itself in dimension 3 we present a sufficient non vanishing condition in terms of massey products this condition holds for certain manifolds that do not fiber over their jacobi torus such as 0 framed surgeries on suitable links our systolic inequality applies to surface bundles over the circle provided the algebraic monodromy has 1 dimensional coinvariants even though the massey product invariant vanishes for some of these bundles

an inverse theorem for the gowers u s 1 n norm we prove the inverse conjecture for the gowers u s 1 n norm for all s 1 this is new for s 4 more precisely we establish that if f n 1 1 is a function with f u s 1 n d then there is a bounded complexity s step nilsequence f g n g that correlates with f where the bounds on the complexity and correlation depend only on s and d from previous results this conjecture implies the hardy littlewood prime tuples conjecture for any linear system of finite complexity

the quantitative behaviour of polynomial orbits on nilmanifolds a theorem of leibman asserts that a polynomial orbit g n g n z on a nilmanifold g g is always equidistributed in a union of closed sub nilmanifolds of g g in this paper we give a quantitative version of leibman s result describing the uniform distribution properties of a finite polynomial orbit g n g n n in a nilmanifold more specifically we show that there is a factorisation g eg where e n is smooth n g n z is periodic and rational and g n g n p is uniformly distributed up to a specified error d inside some subnilmanifold g g of g g for all sufficiently dense arithmetic progressions p n our bounds are uniform in n and are polynomial in the error tolerance d in a companion paper we shall use this theorem to establish the mobius and nilsequences conjecture from an earlier paper of ours

discrete time controlled markov processes with average cost criterion a survey this work is a survey of the average cost control problem for discrete time markov processes the authors have attempted to put together a comprehensive account of the considerable research on this problem over the past three decades the exposition ranges from finite to borel state and action spaces and includes a variety of methodologies to find and characterize optimal policies the authors have included a brief historical perspective of the research efforts in this area and have compiled a substantial yet not exhaustive bibliography the authors have also identified several important questions that are still open to investigation

isometries rigidity and universal covers the goal of this paper is to describe all closed aspherical riemannian manifolds m whose universal covers m have have a nontrivial amount of symmetry by this we mean that isom m is not discrete by the well known theorem of myers steenrod ms this condition is equivalent to isom m 1 m also note that if any cover of m has a nondiscrete isometry group then so does its universal cover m our description of such m is given in theorem 1 2 below the proof of this theorem uses methods from lie theory harmonic maps large scale geometry and the homological theory of transformation groups the condition that m have nondiscrete isometry group appears in a wide variety of problems in geometry since theorem 1 2 provides a taxonomy of such m it can be used to reduce many general problems to verifications of specific examples actually it is not always theorem 1 2 which is applied directly but the main subresults from its proof after explaining in 1 1 the statement of theorem 1 2 we give in 1 2 a number of such applications these range from new characterizations of locally symmetric manifolds to the classification of contractible manifolds covering both compact and finite volume manifolds to a new proof of the nadel frankel theorem in complex geometry

polynomial sequences in groups abstract given a groupgwith lower central seriesg g1 g2 g3 we say that a sequenceg z gispolynomialif for anykthere isdsuch that the sequence obtained fromgby applying the difference operatordg n g n 1g n 1 dtimes takes its values ingk we introduce the notion ofthe degree of a polynomial sequenceand we prove that polynomial sequences of degrees not exceeding a given one form a group as an application we obtain the following extension of the hall petresco theorem theorem letg g1 g2 g3 be the lower central series of a group g let x gk y gland letp qbe polynomials z z of degrees k and l respectively then there is a sequencez0 g zi gifori n such thatfor all n n

linear equations in primes consider a system of nonconstant affine linear forms 1 t d no two of which are linearly dependent let n be a large integer and let k n n d be convex a generalisation of a famous and difficult open conjecture of hardy and littlewood predicts an asymptotic as n for the number of integer points n d k for which the integers 1 n t n are simultaneously prime this implies many other well known conjectures such as the twin prime conjecture and the weak goldbach conjecture it also allows one to count the number of solutions in a convex range to any simultaneous linear system of equations in which all unknowns are required to be prime in this paper we conditionally verify this asymptotic under the assumption that no two of the affine linear forms 1 t are affinely related this excludes the important binary cases such as the twin prime or goldbach conjectures but does allow one to count nondegenerate configurations such as arithmetic progressions our result assumes two families of conjectures which we term the inverse gowers norm conjecture gi s and the mobius and nilsequences conjecture mn s where s 1 2 is the complexity of the system and measures the extent to which the forms i depend on each other the case s 0 is somewhat degenerate and follows from the prime number theorem in aps roughly speaking the inverse gowers norm conjecture gi s asserts the gowers u s 1 norm of a function f n 1 1 is large if and only if f correlates with an s step nilsequence while the mobius and nilsequences conjecture mn s asserts that the mobius function is strongly asymptotically orthogonal to s step nilsequences of a fixed complexity these conjectures have long been known to be true for s 1 essentially by work of hardy littlewood and vinogradov and were established for s 2 in two papers of the authors thus our results in the case of complexity s 2 are unconditional in particular we can obtain the expected asymptotics for the number of 4 term progressions p 1 p 2 p 3 p 4 n of primes and more generally for any nondegenerate problem involving two linear equations in four prime unknowns

effective equidistribution for closed orbits of semisimple groups on homogeneous spaces we prove effective equidistribution with polynomial rate for large closed orbits of semisimple groups on homogeneous spaces under certain technical restrictions notably the acting group should have finite centralizer in the ambient group the proofs make extensive use of spectral gaps and also of a closing lemma for such actions

local boundary rigidity of a compact riemannian manifold with curvature bounded above this paper considers the boundary rigidity problem for a compact convex riemannian manifold m g with boundary o9m whose curvature satisfies a general upper bound condition this includes all nonpositively curved manifolds and all sufficiently small convex domains on any given riemannian manifold it is shown that in the space of metrics g on m there is a c3 neighborhood of g such that g is the unique metric with the given boundary distance function i e the function that assigns to any pair of boundary points their distance as measured in m more precisely given any metric g in this neighborhood with the same boundary distance function there is diffeomorphism o which is the identity on o9m such that g g there is also a sharp volume comparison result for metrics in this neighborhood in terms of the boundary distance function 1 statement of the result the general boundary rigidity problem reads to which extent is a riemannian metric on a compact manifold with boundary determined from the distances between boundary points more precisely it can be formulated as follows let m g be a compact riemannian manifold with boundary om let g be another riemannian metric on m we say that g and g have the same boundary distance function if d1 x y d9 x y for arbitrary boundary points x y e om where d1 resp d9 represents distance in m with respect to g resp g it is easy to give examples of pairs of metrics with the same boundary distance function indeed if m m is an arbitrary diffeomorphism of m onto itself which is the identity on the boundary then the metrics g and g g have the same boundary distance function here g og is the pull back of g under i e for arbitrary vectors 1 e txm we have r1 71 x i where p txm t x m is the differential of at x and resp is the inner product with respect to the metric g resp g we say that a compact riemannian manifold is boundary rigid if this is the only type of nonuniqueness more precisely m g is boundary rigid if for every received by the editors august 10 1998 2000 mathematics subject classification primary 53c65 53c22 53c20 secondary 58c25 58c40 the first and third authors were supported by crdf grant rm2 143 the first author was also supported by nsf grants mds 9505175 and mds 9626232 the third author was also supported by intas rfbr grant 95 0763 2000 american matlhematical society 3937 this content downloaded from 207 46 13 167 on thu 21 jul 2016 04 20 00 utc all use subject to http about jstor org terms 3938 c b croke n s dairbekov and v a sharafnjtdinov riemannian metric g on m1with the same boundary distance function as g there is a diffeomorphism o m m which is the identity on the boundary and for which g 0 of in ward and outward vectors here v is the unit outward norn al to om1 given x c qmvi we denote by k x the maximum of the sectional curvatures of all two planes oc t1 m such that c a for m g a cdrm we define the following invariant

long gaps between primes let pn denote the n th prime we prove that max pn 1 x pn 1 pn log x log log x log log log log x log log log x for sufficiently large x improving upon recent bounds of the first three and fifth authors and of the fourth author our main new ingredient is a generalization of a hypergraph covering theorem of pippenger and spencer proven using the rodl nibble method

on the quantitative distribution of polynomial nilsequences erratum this is an erratum to the paper the quantitative behaviour of polynomial orbits on nilmanifolds by the authors published as ann of math 2 175 2012 no 2 465 540 the proof of theorem 8 6 of that paper which claims a distribution result for multiparameter polynomial sequences on nilmanifolds was incorrect we provide two xes for this issue here first we deduce the equal sides case n1 nt n of this result from the 1 parameter results in the paper this is the same basic mode of argument we attempted originally though the details are dierent the equal sides case is the only one required in applications such as the proof of the inverse conjectures for the gowers norms due to the authors and ziegler to remove the equal sides condition one must rerun the entire argument of our paper in the context of multiparameter polynomial sequences g z t g rather than 1 parameter sequences g z g as is currently done a more detailed sketch of how this may be done is available online

the length of a shortest geodesic loop at a point in this paper we prove that given a point p m n where m n is a closed riemannian manifold of dimension n the length of a shortest geodesic loop lp m n at this point is bounded above by 2nd where d is the diameter of m n moreover we show that on a closed simply connected riemannian manifold m n with a nontrivial second homotopy group there either exist at least three geodesic loops of length less than or equal to 2d at each point of m n or the length of a shortest closed geodesic on m n is bounded from above by 4d

flowers on riemannian manifolds in this paper we will present two upper bounds for the length of a smallest flower shaped geodesic net in terms of the volume and the diameter of a manifold minimal geodesic nets are critical points of the length functional on the space of graphs immersed into a riemannian manifold let mn be a closed riemannian manifold of dimension n we prove that there exists a minimal geodesic net that consists of one vertex and at most 2n 1 geodesic loops based at that vertex of total length 2n d where d is the diameter of mn we also show that there exists a minimal geodesic net that consists of one vertex and at most 3 n 1 2 loops of total length leq2 n 1 2 3 n 1 3 fill rad m n leq2 n 1 frac 5 2 3 n 1 3 n 1 n n vol m n frac 1 n where fill rad mn denotes the filling radius and vol mn denotes the volume of mn

logarithmic growth of systole of arithmetic riemann surfaces along congruence subgroups we apply a study of orders in quaternion algebras to the differential geometry of riemann surfaces the least length of a closed geodesic on a hyperbolic surface is called its systole and denoted sys 1 p buser and p sarnak constructed riemann surfaces x whose systole behaves logarithmically in the genus g x the fuchsian groups in their examples are principal congruence subgroups of a fixed arithmetic group with rational trace field we generalize their construction to principal congruence subgroups of arbitrary arithmetic surfaces the key tool is a new trace estimate valid for an arbitrary ideal in a quaternion algebra we obtain a particularly sharp bound for a principal congruence tower of hurwitz surfaces pch namely the 4 3bound sys 1 xpch 4 log g xpch similar results are obtained for the systole of hyperbolic 3 manifolds relative to their simplicial volume

multiple recurrence and nilsequences aiming at a simultaneous extension of khintchine s and furstenberg s recurrence theorems we address the question if for a measure preserving system x mathcal x mu t and a set a in mathcal x of positive measure the set of integers n such that mu a cap t n a cap t 2n a cap ldots cap t kn a mu a k 1 epsilon is syndetic the size of this set surprisingly enough depends on the length k 1 of the arithmetic progression under consideration in an ergodic system for k 2 and k 3 this set is syndetic while for k 4 it is not the main tool is a decomposition result for the multicorrelation sequence int f x f t n x f t 2n x ldots f t kn x d mu x where k and n are positive integers and f is a bounded measurable function we also derive combinatorial consequences of these results for example showing that for a set of integers e with upper banach density d e 0 and for all e 0 the set big n in mathbb z colon d big e cap e n cap e 2n cap e 3n big d e 4 epsilon big is syndetic

fourier uniformity of bounded multiplicative functions in short intervals on average let lambda denote the liouville function we show that as x rightarrow infty begin aligned int x 2x sup alpha left sum x 0 fixed but arbitrarily small previously this was only known for theta 5 8 for smaller values of theta this is the first non trivial case of local fourier uniformity on average at this scale we also obtain the analogous statement for non pretentious 1 bounded multiplicative functions we illustrate the strength of the result by obtaining cancellations in the sum of lambda n lambda n h lambda n 2h over the ranges h x theta and n x and where lambda is the von mangoldt function

stability for the lens rigidity problem let g be a riemannian metric for mathbb r d d geqq 3 which differs from the euclidean metric only in a smooth and strictly convex bounded domain m the lens rigidity problem is concerned with recovering the metric g inside m from the corresponding lens relation on the boundary partial m in this paper the stability of the lens rigidity problem is investigated for metrics which are a priori close to a given non trapping metric satisfying the strong fold regular condition a metric g is called strong fold regular if for each point x in m there exists a set of geodesics passing through x whose conormal bundle covers t x m moreover these geodesics contain either no conjugate points or only fold conjugate points with a non degeneracy condition examples of strong fold regular metrics are constructed our main result gives the first stability result for the lens rigidity problem in the case of anisotropic metrics which allow conjugate points the approach is based on the study of the linearized inverse problem of recovering a metric from its induced geodesic flow which is a weighted geodesic x ray transform problem for symmetric 2 tensor fields a key ingredient is to show that the kernel of the x ray transform on symmetric solenoidal 2 tensor fields is of finite dimension it remains open whether the kernel space is trivial or not

volume of minimal hypersurfaces in manifolds with nonnegative ricci curvature we establish a min max estimate on the volume width of a closed riemannian manifold with nonnegative ricci curvature more precisely we show that every closed riemannian manifold with nonnegative ricci curvature admits a pl morse function whose level set volume is bounded in terms of the volume of the manifold as a consequence of this sweep out estimate there exists an embedded closed possibly singular minimal hypersurface whose volume is bounded in terms of the volume of the manifold

bounding the distance of quantum surface codes homological quantum codes also called topological codes are low density parity check error correcting codes that come from surfaces and higher dimension manifolds homological codes from surfaces i e surface codes have also been suggested as a possible way to construct stable quantum memory and fault tolerant computation it has been conjectured that all homological codes have a square root bound on there distance and therefore cannot produce good codes this claim has been disputed in dimension four using the geometric property of systolic freedom we will show in this paper that the conjecture holds in dimension two due to the negation of systolic freedom i e systolic rigidity

adapted solution of a backward stochastic differential equation abstract let wt t 0 1 be a standard k dimensional weiner process defined on a probability space f p and let ft denote its natural filtration given a f1 measurable d dimensional random vector x we look for an adapted pair of processes x t y t t 0 1 with values in rd and rd k respectively which solves an equation of the form x t t 1 f s x s y s ds t 1 g s x s y s dw s x a linearized version of that equation appears in stochastic control theory as the equation satisfied by the adjoint process we also generalize our results to the following equation x t t 1 f s x s y s ds t 1 g s x s y s dw s x under rather restrictive assumptions on g

short geodesic loops on complete riemannian manifolds with a finite volume in this paper we will show that on any complete noncompact riemannian manifold with a finite volume there exist uncountably many geodesic loops of arbitrarily small length

geodesic loops and periodic geodesics on a riemannian manifold diffeomorphic to s 3 let m n be a closed 2 connected riemannian manifold such that 3 m n 0 in this paper we prove that either there exists a periodic geodesic on m n of length 6d where d is the diameter of m n or at each point p m n there exists a geodesic loop of length 2d

global rigidity of higher rank anosov actions on tori and nilmanifolds an anosov diffeomorphism f on a torus t is affine if f lifts to an affine map on r by a classical result of franks and manning any anosov diffeomorphism g on t is topologically conjugate to an affine anosov diffeomorphism more precisely there is a homeomorphism t t such that f g 1 is an affine anosov diffeomorphism we call the franks manning conjugacy the linear part of f is the map induced by g on h1 t anosov diffeomorphisms are rarely c conjugate to affine ones for example one can perturb a linear anosov diffeomorphism locally around a fixed point p to change the conjugacy class of the derivative at p the resulting diffeomorphism will still be anosov but cannot be c conjugate to its linearization the situation is radically different for z actions with many anosov diffeomorphisms in other words anosov diffeomorphisms rarely commute with other anosov diffeomorphisms it follows easily from the result for a single anosov diffeomorphism that an anosov z action on t is topologically conjugate to a z action by affine anosov diffeomorphisms we call this action the linearization of and denote it by again for any a z the linear part of a is the map induced by a on h1 t the logarithms of the moduli of the eigenvalues of these linear parts define additive maps i z r which extend to linear functionals on r a weyl chamber of is a connected component of r i ker i theorem 1 1 let be a c action of z k 2 on a torus t and let be its linearization suppose that there is a z subgroup of z such that a is ergodic for every nonzero a z further assume that there is an anosov element for in each weyl chamber of then is c conjugate to furthermore for a linear z action on t having a z subgroup acting by ergodic elements is equivalent to several other properties in particular to being genuinely higher rank 37 a linear z action is called genuinely higher rank if for all finite index subgroups z of z no quotient of the z action factors through a finite extension of z hence we obtain the following corollary

fundamental groups of manifolds with positive isotropic curvature a central theme in riemannian geometry is understanding the relationships between the curvature and the topology of a riemannian manifold positive isotropic curvature pic is a natural and much studied curvature condition which includes manifolds with pointwise quarter pinched sectional curvatures and manifolds with positive curvature operator by the results of micallef and moore there is only one topological type of compact simply connected manifold with pic namely any such manifold must be homeomorphic to the sphere on the other hand there is a large class of nonsimply connected manifolds with pic an important open problem has been to understand the fundamental groups of manifolds with pic in this paper we prove a new result in this direction we show that the fundamental group of a compact manifold m n with pic n 5 does not contain a subgroup isomorphic to the techniques used involve minimal surfaces

stability of bsdes with random terminal time and homogenization of semilinear elliptic pdes in this paper we extend the probabilistic method for homogenization of semi linear parabolic pdes developed by buckdahn hu and peng to the case of elliptic pdes first we give a stability result for bsdes with random terminal time which are related to elliptic pdes as shown in peng stochastics stochastics rep 37 1991 61 74 in the one dimensional case we also partially relax the monotonicity assumption on the coefficient then we use these stability results for bsdes with random terminal time to study homogenization of systems of semilinear elliptic pdes

the mobius function is strongly orthogonal to nilsequences we show that the mobius function n is strongly asymptotically or thogonal to any polynomial nilsequence f g n n2n here g is a sim ply connected nilpotent lie group with a discrete and cocompact subgroup so g is a nilmanifold g z g is a polynomial sequence and f g r is a lipschitz function more precisely we show that j 1 n p n n 1 n f g n j f g a log a n for all a 0 in particu lar this implies the mobius and nilsequence conjecture mn s from our earlier paper for every positive integer s this is one of two major in gredients in our programme to establish a large number of cases of the generalised hardy littlewood conjecture which predicts how often a collec tion 1 t z d z of linear forms all take prime values the proof is a relatively quick application of the results in our recent companion paper we give some applications of our main theorem we show for exam ple that the mobius function is uncorrelated with any bracket polynomial such as n p 3bn p 2c we also obtain a result about the distribution of nilse quences a n x n2n as n ranges only over the primes

multifractal behavior of polynomial fourier series abstract we study the spectrum of singularities of a family of fourier series with polynomial frequencies in particular we prove that they are multifractal functions the case of degree two was treated by s jaffard in 1996 higher degrees require completely different ideas essentially because harmonic analysis techniques poisson summation are useless to study the oscillation at most of the points we introduce a new approach involving special diophantine approximations with prime power denominators and fine analytic and arithmetic aspects of the estimation of exponential sums to control the holder exponent in thin cantor like sets

an effective ratner equidistribution result for operatorname sl 2 mathbb r ltimes mathbb r 2 let g sl 2 r proportional to r 2 be the affine special linear group of the plane and set gamma sl 2 z proportional to z 2 we prove a polynomially effective asymptotic equidistribution result for the orbits of a 1 dimensional nonhorospherical unipotent flow on gamma g

upper bounds on the length of the shortest closed geodesic on simply connected manifolds the subject of this paper is upper bounds on the length of the shortest closed geodesic on simply connected manifolds with non trivial second homology group we will give three estimates the first estimate will explicitly depend on volume and the upper bound for the sectional curvature the second estimate will depend on diameter a positive lower bound for the volume and on the possibly negative lower bound on sectional curvature the third estimate will depend on diameter on a possibly negative lower bound for the sectional curvature and on a lower bound for the simply connectedness radius the technique that we develop in order to obtain the last result will also enable us to estimate the homotopy distance between any two closed curves on compact simply connected manifolds of sectional curvature bounded from below and diameter bounded from above more precisely let c be a constant such that any metric ball of radius leq c is simply connected there exists a homotopy connecting any two closed curves such that the length of the trajectory of the points during this homotopy has an upper bound in terms of the lower bound of the curvature the upper bound of diameter and c

ergodic bsdes and optimal ergodic control in banach spaces in this paper we introduce a new kind of backward stochastic differential equations called ergodic bsdes which arise naturally in the study of optimal ergodic control we study the existence uniqueness and regularity of solution to ergodic bsdes then we apply these results to the optimal ergodic control of a banach valued stochastic state equation we also establish the link between the ergodic bsdes and the associated hamilton jacobi bellman equation applications are given to the optimal ergodic control of stochastic partial differential equations

ergodic bsdes driven by markov chains we consider ergodic backward stochastic differential equations bsdes in a setting where noise is generated by a countable state uniformly ergodic markov chain we show that for lipschitz drivers such that a comparison theorem holds these equations admit unique solutions to obtain this result we show by coupling and splitting techniques that uniform ergodicity estimates of markov chains are robust to perturbations of the rate matrix and that these perturbations correspond in a natural way to ergodic bsdes we then consider applications of this theory to markov decision problems with a risk averse average reward criterion

mixing properties of commuting nilmanifold automorphisms we study mixing properties of commutative groups of automorphisms acting on compact nilmanifolds assuming that every non trivial element acts ergodically we prove that such actions are mixing of all orders we further show exponential 2 mixing and 3 mixing as an application we prove smooth cocycle rigidity for higher rank abelian groups of nilmanifold automorphisms

volume growth of some uniformly contractible riemannian manifolds it is shown that if a uniformly contractible riemannian n manifold m g is k quasi isometric to an n dimensional normed space v n cdot k 1 then liminf r rightarrow infty frac vol g ball r r n omega n geq frac 1 k 2n where n is the volume of the unit euclidean ball in particular if m is uniformly contractible and d gh m d g v n cdot infty then m has at least euclidean volume growth this corollary covers an earlier result by burago and ivanov our results are motivated by a volume growth theorem contained in gromov s book gromov in progress in mathematics vol 152 birkhauser boston 1999 p 256 we give a detailed proof of this theorem using the same argument we also derive a generalization of the theorem which is pointed out by gromov

drift and monotonicity conditions for continuous time controlled markov chains with an average criterion we give conditions for the existence of average optimal policies for continuous time controlled markov chains with a denumerable state space and borel action sets the transition rates are allowed to be unbounded and the reward cost rates may have neither upper nor lower bounds in the spirit of the drift and monotonicity conditions for continuous time markov processes we propose a new set of conditions on the controlled process primitive data under which the existence of optimal deterministic stationary policies in the class of randomized markov policies is proved using the extended generator approach instead of kolmogorov s forward equation used in the previous literature and under which the convergence of a policy iteration method is also shown moreover we use a controlled queueing system to show that all of our conditions are satisfied whereas those in the previous literature fail to hold

metric and isoperimetric problems in symplectic geometry introduction 411 1 some basic results in symplectic topology 413 2 capacity and symplectic reduction 414 3 volume estimates for lagrange submanifolds 416 3 1 the case of r 417 3 2 deformations of the zero section in cotangent bundles 419 3 3 generalization to the case of cp 422 4 an application to billiards 423 5 geometry of convex sets and periodic orbits 425 6 compensated compactness and closure of the symplectic group 426 appendix a a summary of symplectic geometry through generating functions 427 appendix b john s ellipsoid 428 references 430

systoles des surfaces plates singulieres de genre deux the extremal metrics for the isosystolic problem on surfaces of genus two are studied in this paper we show that contrary to e calabi s conjecture in genus three no flat metric with conical singularities is extremal for this problem in genus two

isosystolic inequalities for optical hypersurfaces abstract we explore a natural generalization of systolic geometry to finsler metrics and optical hypersurfaces with special emphasis on its relation to the mahler conjecture and the geometry of numbers in particular we show that if an optical hypersurface of contact type in the cotangent bundle of the 2 dimensional torus encloses a volume v then it carries a periodic characteristic whose action is at most v 3 this result is deduced from an interesting dual version of minkowski s lattice point theorem if the origin is the unique integer point in the interior of a planar convex body the area of its dual body is at least 3 2

the local limit theorem on nilpotent lie groups a local limit theorem is proven on connected simply connected nilpotent lie groups for a class of generating measures satisfying a moment condition and a condition on the characteristic function of the abelianization the result extends an earlier local limit theorem of alexopoulos which treated absolutely continuous measures with a continuous density of compact support and also extends local limit theorems of breuillard and diaconis hough which treated general measures on the heisenberg group

orbit of the diagonal in the power of a nilmanifold let x be a nilmanifold that is a compact homogeneous space of a nilpotent lie group g and let a 2 g we study the closure of the orbit of the diagonal of x r under the action a p1 n

length of geodesics on a two dimensional sphere let m be an arbitrary riemannian manifold diffeomorphic to s 2 let x y be two arbitrary points of m we prove that for every k 1 2 3 ldots there exist k distinct geodesics between x and y of length less than or equal to 4k 2 2k 1 d where d denotes the diameter of m to prove this result we demonstrate that for every riemannian metric on s 2 there are two not mutually exclusive possibilities either every two points can be connected by many short geodesics of index 0 or the resulting riemannian sphere can be swept out by short meridians

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

convergence and the length spectrum abstract the author defines and analyzes the 1 k length spectra l 1 k m whose union over all k n is the classical length spectrum these new length spectra are shown to converge in the sense that lim k k 1 k m i l 1 k m 0 as m i m in the gromov hausdorff sense energy methods are introduced to estimate the shortest element of l 1 k as well as a concept called the minimizing index which may be used to estimate the length of the shortest closed geodesic of a simply connected manifold in any dimension a number of gap theorems are proven including one for manifolds m n with ricci n 1 and volume close to vol s n many results in this paper hold on compact length spaces in addition to riemannian manifolds

filtration consistent nonlinear expectations and related g expectations from a general definition of nonlinear expectations viewed as operators preserving monotonicity and constants we derive under rather general assumptions the notions of conditional nonlinear expectation and nonlinear martingale we prove that any such nonlinear martingale can be represented as the solution of a backward stochastic equation and in particular admits continuous paths in other words it is a g martingale

counting and equidistribution in heisenberg groups we strongly develop the relationship between complex hyperbolic geometry and arithmetic counting or equidistribution applications that arises from the action of arithmetic groups on complex hyperbolic spaces especially in dimension 2 we prove a mertens formula for the integer points over a quadratic imaginary number fields k in the light cone of hermitian forms as well as an equidistribution theorem of the set of rational points over k in heisenberg groups we give a counting formula for the cubic points over k in the complex projective plane whose galois conjugates are orthogonal and isotropic for a given hermitian form over k and a counting and equidistribution result for arithmetic chains in the heisenberg group when their cygan diameter tends to 0

boundary rigidity with partial data we study the boundary rigidity problem with partial data consisting of determining locally the riemannian metric of a riemannian manifold with boundary from the distance function measured at pairs of points near a fixed point on the boundary we show that one can recover uniquely and in a stable way a conformal factor near a strictly convex point where we have the information in particular this implies that we can determine locally the isotropic sound speed of a medium by measuring the travel times of waves joining points close to a convex point on the boundary the local results lead to a global lens rigidity uniqueness and stability result assuming that the manifold is foliated by strictly convex hypersurfaces

homological and homotopical dehn functions are different the homological and homotopical dehn functions are different ways of measuring the difficulty of filling a closed curve inside a group or a space the homological dehn function measures fillings of cycles by chains whereas the homotopical dehn function measures fillings of curves by disks because the two definitions involve different sorts of boundaries and fillings there is no a priori relationship between the two functions however before this work there were no known examples of finitely presented groups for which the two functions differ this paper gives such examples constructed by amalgamating a free by cyclic group with several bestvina brady groups

on the spread of positively curved alexandrov spaces it was proved by f wilhelm that gromov s filling radius of closed positively curved manifolds with a uniform lower bound on sectional curvature attains the maximum with the round sphere recently the author proved that this is also the case for closed finite dimensional alexandrov spaces with a positive lower curvature bound these were proved as a corollary of a comparison theorem for the invariant called spread of those spaces in this paper we extend the latter result to infinite dimensional alexandrov spaces

lengths of three simple periodic geodesics on a riemannian 2 sphere let m be a riemannian 2 sphere a classical theorem of lyusternik and shnirelman asserts the existence of three distinct simple non trivial periodic geodesics on m in this paper we prove that there exist three simple periodic geodesics with lengths that do not exceed 20d where d is the diameter of m we also present an upper bound that depends only on the area and diameter for the lengths of the three simple periodic geodesics with positive indices that appear as minimax critical values in the classical proofs of the lyusternik shnirelman theorem finally we present better bounds for these three lengths for thin spheres when the area a is much less than d 2 where the bounds for the lengths of the first two simple periodic geodesics are asymptotically optimal when a over d 2 longrightarrow 0