asymptotic upper curvature bounds in coarse geometry we define a notion of an asymptotic upper curvature bound for gromov hyperbolic metric spaces that is invariant under rough isometries and examine the basic properties of this concept

interactive proofs and the hardness of approximating cliques the contribution of this paper is two fold first a connection is established between approximating the size of the largest clique in a graph and multi prover interactive proofs second an efficient multi prover interactive proof for np languages is constructed where the verifier uses very few random bits and communication bits last the connection between cliques and efficient multi prover interaction proofs is shown to yield hardness results on the complexity of approximating the size of the largest clique in a graph of independent interest is our proof of correctness for the multilinearity test of functions

one and a half quantum de finetti theorems when n k systems of an n partite permutation invariant state are traced out the resulting state can be approximated by a convex combination of tensor product states this is the quantum de finetti theorem in this paper we show that an upper bound on the trace distance of this approximation is given by 2 frac kd 2 n where d is the dimension of the individual system thereby improving previously known bounds our result follows from a more general approximation theorem for representations of the unitary group consider a pure state that lies in the irreducible representation u mu nu subset u mu otimes u nu of the unitary group u d for highest weights and let be the state obtained by tracing out u then is close to a convex combination of the coherent states u mu g v mu rangle where g in u d and v mu rangle is the highest weight vector in u

on the positivity of the effective action in a theory of random surfaces it is shown that the functional s eta frac 1 24 pi int left frac 1 2 left nabla eta right 2 2 eta right d mu 0 defined onc functions on the two dimensional sphere satisfies the inequalitys 0 if is subject to the constraint int e eta 1 d mu 0 0 the minimums 0 is attained at the solutions of the euler lagrange equations the proof is based on a sharper version of moser trudinger s inequality due to aubin which holds under the additional constraint int e eta xd mu 0 0 this condition can always be satisfied by exploiting the invariance ofs under the conformal transformations ofs2 the result is relevant for a recently proposed formulation of a theory of random surfaces

logarithmic potentials and quasiconformal flows on the heisenberg group abstract let h be the sub riemannian heisenberg group that h supports a rich family of quasiconformal mappings was demonstrated by koranyi and reimann using the so called flow method here we supply further evidence of the flexible nature of this family constructing quasiconformal mappings with extreme behavior on small sets more precisely we establish criteria to determine when a given logarithmic potential on h is such that there exists a quasiconformal mapping of h with jacobian comparable to e 2 so that the jacobian is zero or infinity at the same points as e 2 when is continuous and meets the criteria we show the canonical sub riemannian metric g 0 and the weighted metric g e g 0 generate bi lipschitz equivalent distance functions these results rest on an extension to the theory of quasiconformal flows on h and constructions that adapt the iterative method of bonk heinonen and saksman

sharp constants for moser trudinger inequalities on spheres in complex space ℂn the main results of this paper concern sharp constants for the moser trudinger inequalities on spheres in complex space c n we derive moser trudinger inequalities for smooth functions and holomorphic functions with different sharp constants see theorem 1 1 the sharp moser trudinger inequalities under consideration involve the complex tangential gradients for the functions and thus we have shown here such inequalities in the cr setting though there is a close connection in spirit between inequalities proven here on complex spheres and those on the heisenberg group for functions with compact support in any finite domain proven earlier by the same authors 17 derivation of the sharp constants for moser trudinger inequalities on complex spheres are more complicated and difficult to obtain than on the heisenberg group variants of moser onofri type inequalities are also given on complex spheres as applications of our sharp inequalities see theorems 1 2 and 1 3 one of the key ingredients in deriving the main theorems is a sharp representation formula for functions on the complex spheres in terms of complex tangential gradients see theorem 1 4 c 2004 wiley periodicals inc

compensated compactness and the heisenberg group jacobians of maps on the heisenberg group are shown to map suitable group sobolev spaces into the group hardy space h1 from this result and a weak convergence theorem for the hardy space h1 of the heisenberg group a compensated compactness property for these jacobians is obtained 0 introduction we investigate compensated compactness properties of jacobians of maps on the heisenberg group and we prove results analogous to those for the jacobians of maps on r let h c r be the lie group with multiplicative structure z t z t z z t t 2 im z z where z z1 zn z z 1 z n and z z n j 1 zj z j this is the n order heisenberg group it can be shown that the group operation is c on the manifold c r and hence h is a locally compact lie group let z1 zn t be coordinates on h write zj xj iyj and define the vector fields xj xj 2yj t yj yj 2xj t t t it turns out that the xj yj and t form a basis for the left invariant vector fields on h they satisfy the commutation relations xj yj 4t j 1 n and all other work partially supported by the national science foundation work partially supported by nsf grant dms 9007491 mathematics subject classification 1991 42 typeset by ams tex 1 commutators vanish the group h is equipped with a natural dilation structure r z t rz rt r 0 which is consistent with the group multiplication the associated norm z t z 4 t is homogeneous of degree 1 with respect to this group of dilations we denote by br u0 the heisenberg group ball u h u 1u0 0 f u where u 2n 2 1u if 2n 2 2n 3 the hardy space h h h is the set of all f such that f l h an alternative definition of h h can be given via the atomic decomposition see fos or cw for details given a c map f f1 fn from h into r define 0 3 jac f det l1f1 l1f2 l1fn l2f1 l2f2 l2fn lnf1 lnf2 lnfn where lj is either xj or yj we would like to show that jac f maps a product of suitable group sobolev spaces into the spaces h h for 1 2n 2 2n 3 the analogous result for the vector fields xj 1 j n on r n has been proved by p l lions and y meyer when 1 and extended by clms for 1 n n 1 note that the lower bound for in both cases is d d 1 where d is the homogeneous dimension of the group our first result is theorem 1 let n 2 and for 1 j n let 1 0 that depends only on n and on the pj s such that for every map f f1 fn on h we have 0 4 jac f h c n

optimal mass transportation in the heisenberg group in this paper we consider the problem of optimal transportation of absolutely continuous masses in the heisenberg group hn in the case when the cost function is either the square of the carnot caratheodory distance or the square of the koranyi norm in both cases we show existence and uniqueness of an optimal transport map in the former case the proof requires a delicate analysis of minimizing geodesics of the group and of the differentiability properties of the squared distance function in the latter case the proof requires some fine properties of bv functions in the heisenberg group

the zeta functional determinants on manifolds with boundary abstract in this paper we derive some geometric formulas for the quotient of the zeta functional determinants for certain elliptic boundary value problems on riemannian 3 and 4 manifolds with smooth boundary

groups quasi isometric to complex hyperbolic space we show that any finitely generated group quasi isometric to complex hyperbolic space is a finite extension of a properly discontinuous cocompact subgroup of the isometry group

sharp sobolev inequalities on the sphere and the moser trudinger inequality where de denotes normalized surface measure v is the conformal gradient and q 2n n 2 a modern folklore theorem is that by taking the infinitedimensional limit of this inequality one obtains the gross logarithmic sobolev inequality for gaussian measure which determines nelson s hypercontractive estimates for the hermite semigroup see 8 one observes using conformal invariance that the above inequality is equivalent to the sharp sobolev inequality on rn for which boundedness and extremal functions can be easily calculated using dilation invariance and geometric symmetrization the roots here go back to hardy and littlewood the advantage of casting the problem on the sphere is that the role of the constants is evident and one is led immediately to the conjecture that this inequality should hold whenever possible for example 2 q 0o if n 2 this is in fact true and will be demonstrated in section 2 a clear question at this point is what is the situation in dimension 2 two important arguments 25 26 27 dealt with this issue both motivated by geometric variational problems because q goes to infinity for dimension 2 the appropriate function space is the exponential class responding in part

the cayley transform and uniformly bounded representations let g be a simple lie group of real rank one with iwasawa decomposition kan and bruhat big cell nman then the space g man may be almost identified with n and with k m and these identifications induce the generalised cayley transform c n k m we show that c is a conformal map of carnot caratheodory manifolds and that composition with the cayley transform combined with multiplication by appropriate powers of the jacobian induces isomorphisms of sobolev spaces h n and h k m we use this to construct uniformly bounded and slowly growing representations of g

area stationary surfaces in the heisenberg group h1 we consider area stationary surfaces perhaps with a volume constraint in the heisenberg group h1 endowed with its carnot caratheodory distance by analyzing the first variation of area we characterize c2 area stationary surfaces as those with mean curvature zero or constant if a volume preserving condition is assumed and such that the characteristic curves meet orthogonally the singular curves moreover a minkowski type formula relating the area the mean curvature and the volume is obtained for volume preserving area stationary surfaces enclosing a given region as a consequence of the characterization of area stationary surfaces we refine the bernstein type theorem given in jih hsin cheng jenn fang hwang andrea malchiodi paul yang minimal surfaces in pseudohermitian geometry ann sc norm super pisa cl sci 5 4 1 2005 129 177 mr 2165405 and nicola garofalo scott pauls the bernstein problem in the heisenberg group arxiv math dg 0209065 to describe all c2 entire area stationary graphs over the xy plane in a calibration argument shows that these graphs are globally area minimizing finally by using the description of the singular set in jih hsin cheng jenn fang hwang andrea malchiodi paul yang minimal surfaces in pseudohermitian geometry ann sc norm super pisa cl sci 5 4 1 2005 129 177 mr 2165405 the characterization of area stationary surfaces and the ruling property of constant mean curvature surfaces we prove our main results where we classify area stationary surfaces in with or without a volume constraint and non empty singular set in particular we deduce the following counterpart to alexandrov uniqueness theorem in euclidean space any compact connected c2 surface in area stationary under a volume constraint must be congruent to a rotationally symmetric sphere obtained as the union of all the geodesics of the same curvature joining two points as a consequence we solve the isoperimetric problem in assuming c2 smoothness of the solutions 13 completely reducible hypersurfaces in a pencil pages 672 688 j v pereira s yuzvinsky preview pdf 324 k related articles 14 modules over motivic cohomology pages 689 727 oliver rondigs paul arne ostvaer preview pdf 303 k related articles 15 the fundamental group of period domains over finite fields pages 728 741 sascha orlik preview pdf 154 k related articles articles 1 15

estimates and extremals for zeta function determinants on four manifolds leta be a positive integral power of a natural conformally covariant differential operator on tensor spinors in a riemannian manifold suppose thata is formally self adjoint and has positive definite leading symbol for example a could be the conformal laplacian yamabe operator l or the square of the dirac operator open image in new window within the conformal class left g e 2w g 0 w in c infty m right of an einstein locally symmetric background metricgo on a compact four manifoldm we use an exponential sobolev inequality of adams to show that bounds on the functional determinant ofa and the volume ofg imply bounds on thew2 2 norm of the conformal factorw provided that a certain conformally invariant geometric constantk k m g o a is strictly less than 32 2 we show for the operatorsl and open image in new window that indeedk 32 2 except when m g o is the standard sphere or a hyperbolic space form on the sphere a centering argument allows us to obtain a bound of the same type despite the fact thatk is exactly equal to 32 2 in this case finally we use an inequality of beckner to show that in the conformal class of the standard four sphere the determinant ofl or of open image in new window is extremized exactly at the standard metric and its images under the conformal transformation groupo 5 1

h type groups and iwasawa decompositions since their introduction by a kaplan kpl some ten years ago generalised heisenberg groups also known as groups of heisenberg type or h type groups have provided a framework in which to construct interesting examples in geometry and analysis see for instance c2 kp2 kp3 kpr k2 rl r2 tl tv the iwasawa n groups associated to all the real rank one simple lie groups are h type so one has a convenient vehicle for studying these in a unified way many problems on these simple lie groups can be reduced to a problem on h type groups via the so called noncompact picture and often problems on h type groups can be solved on all the groups of the family in one fell swoop as in for example ch ck dr out of this approach to studying simple lie groups several problems arise such as why only some h type groups correspond to simple lie groups of real rank one in this paper we discuss various features of iwasawa n groups which distinguish them in the class of all h type groups we shall show that all h type groups which possess certain geometric properties clearly possessed by iwasawa n groups satisfy a lie algebraic condition implicit in the work of b kostant kt2 that we shall call the j condition we shall also use elementary clifford algebra to classify the

weak differentiability of bv functions on stratified groups we prove the almost everywhere approximate differentiability of functions with bounded variation in stratified groups and we study their approximate discontinuity set we introduce functions of bounded higher order variation and obtain a weak version of alexandroff differentiability theorem in this context we present a nontrivial class of functions with second order bounded variation arising from inf convolution formula of a suitable cost function

the geometry of the eisenstein picard modular group the eisenstein picard modular group rm pu 2 1 mathbb z omega is defined to be the subgroup of rm pu 2 1 whose entries lie in the ring mathbb z omega where omega is a cube root of unity this group acts isometrically and properly discontinuously on bf h 2 mathbb c that is on the unit ball in mathbb c 2 with the bergman metric we construct a fundamental domain for the action of rm pu 2 1 mathbb z omega on bf h 2 mathbb c which is a 4 simplex with one ideal vertex as a consequence we elicit a presentation of the group see theorem 5 9 this seems to be the simplest fundamental domain for a finite covolume subgroup of rm pu 2 1

non signaling parallel repetition using de finetti reductions in the context of multiplayer games the parallel repetition problem can be phrased as follows given a game g with optimal winning probability 1 alpha and its repeated version g n in which n games are played together in parallel can the players use strategies that are substantially better than ones in which each game is played independently this question is relevant in physics for the study of correlations and plays an important role in computer science in the context of complexity and cryptography in this paper the case of multiplayer non signaling games is considered i e the only restriction on the players is that they are not allowed to communicate during the game for complete support games games where all possible combinations of questions have non zero probability to be asked with any number of players we prove a threshold theorem stating that the probability that non signaling players win more than a fraction 1 alpha beta of the n games is exponentially small in n beta 2 for every 0 leq beta leq alpha for games with incomplete support we derive a similar statement for a slightly modified form of repetition the result is proved using a new technique based on a recent de finetti theorem which allows us to avoid central technical difficulties that arise in standard proofs of parallel repetition theorems

some examples of quasiisometries of nilpotent lie groups we construct quasiisometries of nilpotent lie groups in particular for any simply connected nilpotent lie group n we construct quasiisometries from n to itself that is not at finite distance from any map that is a composition of left translations and automorphisms we also construct bilipschitz maps of the heisenberg groups that send vertical lines to curves that are not vertical lines

unknown quantum states the quantum de finetti representation we present an elementary proof of the quantum de finetti representation theorem a quantum analog of de finetti s classical theorem on exchangeable probability assignments this contrasts with the original proof of hudson and moody z wahrschein verw geb 33 343 1976 which relies on advanced mathematics and does not share the same potential for generalization the classical de finetti theorem provides an operational definition of the concept of an unknown probability in bayesian probability theory where probabilities are taken to be degrees of belief instead of objective states of nature the quantum de finetti theorem in a closely analogous fashion deals with exchangeable density operator assignments and provides an operational definition of the concept of an unknown quantum state in quantum state tomography this result is especially important for information based interpretations of quantum mechanics where quantum states like probabilities are taken to be states of knowledge rather than

moser trudinger and beckner onofri s inequalities on the cr sphere we derive sharp moser trudinger inequalities on the cr sphere the rst type is in the adams form for powers of the sublaplacian and for general spectrally dened operators on the space of cr pluriharmonic functions we will then obtain the sharp beckner onofri inequality for crpluriharmonic functions on the sphere and as a consequence a sharp logarithmic hardy littlewood sobolev inequality in the form given by carlen and loss

adams inequalities on measure spaces abstract in 1988 adams obtained sharp moser trudinger inequalities on bounded domains of r n the main step was a sharp exponential integral inequality for convolutions with the riesz potential in this paper we extend and improve adams results to functions defined on arbitrary measure spaces with finite measure the riesz fractional integral is replaced by general integral operators whose kernels satisfy suitable and explicit growth conditions given in terms of their distribution functions natural conditions for sharpness are also given most of the known results about moser trudinger inequalities can be easily adapted to our unified scheme we give some new applications of our theorems including sharp higher order moser trudinger trace inequalities sharp adams moser trudinger inequalities for general elliptic differential operators scalar and vector valued for sums of weighted potentials and for operators in the cr setting

fundamental solution for the q laplacian and sharp moser trudinger inequality in carnot groups abstract for a general carnot group g with homogeneous dimension q we prove the existence of a fundamental solution of the q laplacian u q and a constant a q 0 such that exp a q u q is a homogeneous norm on g this implies a representation formula for smooth functions on g which is used to prove the sharp carnot group version of the celebrated moser trudinger inequality

parallel repetition in projection games and a concentration bound a two player game is played by cooperating players who are not allowed to communicate a referee asks the players questions sampled from some known distribution and decides whether they win or not based on a known predicate of the questions and the players answers the parallel repetition of the game is the game in which the referee samples n independent pairs of questions and sends the corresponding questions to the players simultaneously if the players cannot win the original game with probability better than 1 epsilon what s the best they can do in the repeated game we improve earlier results of r raz siam j comput 27 1998 pp 763 803 and t holenstein theory comput 5 2009 pp 141 172 who showed that the players cannot win all copies in the repeated game with probability better than 1 epsilon 2 omega n epsilon 2 c here c is the length of the answers in the game in the following ways i we show that the probability of winning all copies is 1 epsilon 2 omega epsilon n as long as the game is a projection game the type of game most commonly used in hardness of approximation results ii we prove a concentration bound for parallel repetition of general games showing that for any constant 0 0 there exists an alphabet size m epsilon for which it is np hard to distinguish a unique game with alphabet size m in which a 1 epsilon 2 fraction of the constraints can be satisfied from one in which a 1 epsilon f 1 epsilon fraction of the constraints can be satisfied

extremals of determinants of laplacians on etudie le determinant associe au laplacien en fonction de la metrique sur une surface donnee et en particulier ses valeurs extremes quand la metrique est bien restreinte

finite de finetti theorem for conditional probability distributions describing physical theories we work in a general framework where the state of a physical system is defined by its behavior under measurement and the global state is constrained by no signaling conditions we show that the marginals of symmetric states in such theories can be approximated by convex combinations of independent and identical conditional probability distributions generalizing the classical finite de finetti theorem of diaconis and freedman our results apply to correlations obtained from quantum states even when there is no bound on the local dimension so that known quantum de finetti theorems cannot be used

quasiconformal mappings on the heisenberg group on etudie les applications conformes et quasi conformes sur le groupe de heisenberg la metrique invariante a gauche naturelle sur cet espace est non isotrope sous des dilatations et ne peut se decrire par une metrique de riemann ceci engendre de nouveaux phenomenes

the quantum reverse shannon theorem based on one shot information theory the quantum reverse shannon theorem states that any quantum channel can be simulated by an unlimited amount of shared entanglement and an amount of classical communication equal to the channel s entanglement assisted classical capacity in this paper we provide a new proof of this theorem which has previously been proved by bennett devetak harrow shor and winter our proof has a clear structure being based on two recent information theoretic results one shot quantum state merging and the post selection technique for quantum channels

a sharp inequality of j moser for higher order derivatives in fact go nwl 1 where wnis the area of the surface of the unit n ball also he notes that if 3 exceeds go then 1 can no longer hold with some co independent of the u satisfying 2 the fact that this result holds for some 3 0 independent of u has been observed now by several authors with a variety of proofs see 24 20 23 22 10 4 1 2 13 14 for example the earliest of these arguments expands the exponential function in a power series then observing that the norm in the imbedding

a parallel repetition theorem we show that a parallel repetition of any two prover one round proof system mip 2 1 decreases the probability of error at an exponential rate no constructive bound was previously known the constant in the exponent in our analysis depends only on the original probability of error and on the total number of possible answers of the two provers the dependency on the total number of possible answers is logarithmic which was recently proved to be almost the best possible u feige and o verbitsky proc 11th annual ieee conference on computational complexity ieee computer society press los alamitos ca 1996 pp 70 76

eigenforms of the laplacian on real and complex hyperbolic spaces on decrit l espace total des solutions de l equation c ou est une p forme sur un espace x hyperbolique reel ou complexe a n dimensions et c une constante complexe

de finetti reductions for correlations when analysing quantum information processing protocols one has to deal with large entangled systems each consisting of many subsystems to make this analysis feasible it is often necessary to identify some additional structures de finetti theorems provide such a structure for the case where certain symmetries hold more precisely they relate states that are invariant under permutations of subsystems to states in which the subsystems are independent of each other this relation plays an important role in various areas e g in quantum cryptography or state tomography where permutation invariant systems are ubiquitous the known de finetti theorems usually refer to the internal quantum state of a system and depend on its dimension here we prove a different de finetti theorem where systems are modelled in terms of their statistics under measurements this is necessary for a large class of applications widely considered today such as device independent protocols where the underlying systems and the dimensions are unknown and the entire analysis is based on the observed correlations

ideal triangle groups dented tori and numerical analysis we prove the goldman parker conjecture a complex hyperbolic ideal triangle group is discretely embedded in pu 2 1 if and only if the product of its three standard generators is not elliptic we also prove that such a group is indiscrete if the product of its three standard generators is elliptic a novel feature of this paper is that it uses a rigorous computer assisted proof to deal with difficult geometric estimates

extremal metrics of zeta function determinants on 4 manifolds in conformal geometry the sobolev inequality at a critical exponent has received much attention in particular the determination of the best constants has played a crucial role in the yamabe problem in dimension two the analogous problem deals with the moser trudinger inequality on a compact riemann surface m2 there exists a constant c c m so that f e47w2 c m if f ivw12 1 and f w 0 the connection of this inequality with geometry comes through the zeta functional determinant of the laplacian as defined by ray singer for a riemannian metric g let 0 a1 a2 be the spectrum

symmetry algebra for multi contact structures given by 2n vector fields on mathbb r 2n 1 the notion of multicontact structures has been introduced in 2 as the geometry of manifolds with several distinguished mutually transversal subbundles of the tangent bundle such structures appear naturally as secondary structures in parabolic geometry more precisely let g be a semi simple lie group and p a parabolic subgroup then the nilpotent factor n in the iwasawa decomposition g kan naturally gives rise to mutually transversal subbundles

some nonabelian toy models in the large n limit schwinger dyson equations are used to study the largen limit ofu n gauge theory on several small lattices explicit solutions are found which are beyond the reach of existing steepest descent technique they show a phase transition in a three placquette model at couplingg2n 3 resembling the known transition in the one placquette model and lending support to expectations of a similar transition in the four dimensional lattice theory

phase space distribution of riemann zeros we present the partition function of a most generic u n single plaquette model in terms of representations of a unitary group extremising the partition function in a large n limit we obtain a relation between eigenvalues of unitary matrices and the number of boxes in the most dominant young tableaux distribution since the eigenvalues of unitary matrices behave like coordinates of free fermions whereas the number of boxes in a row is like conjugate momenta of the same a relation between them allows us to provide a phase space distribution for different phases of the unitary model under consideration this proves a universal feature that all the phases of a generic unitary matrix model can be described in terms of topology of free fermi phase space distribution finally using this result and analytic properties of resolvent that satisfy the dyson schwinger equation we present a phase space distribution of unfolded zeros of the riemann zeta function

singular solutions homogeneous norms and quasiconformal mappings in carnot groups in any carnot nilpotent stratified lie group g of homogeneous dimension q green s function u for the q laplace equation exists and is unique we prove that there exists a constant gamma gamma g so that n e gamma u is a homogeneous norm in g then the extremal lengths of spherical ring domains measured with respect to n can be computed and used to give estimates for the extremal lengths of ring domains measured with respect to the carnot caratheodory metric applications include regularity properties of quasiconformal mappings and a geometric characterization of bi lipschitz mappings

hardy space estimates of hausdorff operators on the heisenberg group abstract this is a continuation of a series of works about hausdorff operator on hardy spaces in the setting of the heisenberg group in this paper we obtain the boundedness from lebesgue spaces to hardy spaces for fractional hausdorff operators and their compositions with riesz transforms we also establish the boundedness for two kinds of special hausdorff operators the hausdorff poisson operator and the hausdorff gauss operator on hardy spaces

compatibility operators for degenerate elliptic equations on the ball and heisenberg group on considere des operateurs elliptiques degeneres d ordre 2 qui sont des variantes de l operateur de laplace beltrami pour la metrique de bergman

quasiregular mappings of the heisenberg group the condition 1 1 means that for all d g the coordinate functions o f f belong to the sobolev space w d of functions in l d whose distributional first order partial derivatives are also l integrable in d in the above definition f x is the formal derivative of f a t x defined by means of the partial derivatives i f x l is the operator norm of f x and ji x is the jacobian determinant of f a t x these are uniquely defined almost everywhere by 1 1 it turns out that quasiregular mappings are differentiable almost everywhere and a nonconstant quasiregular map is discrete open and sense preserving for these and other basic properties of qr maps we refer to books re by reshetnyak vu by vuorinen and also to a forthcoming book r6 by the second author the definition extends immediately to the case f m n where m and n are connected oriented riemannian n manifolds we say that a continuous map f m n is quasiregular if it has a local representation at each point of m which is quasiregular in the above sense and if there exists k 1 k oe such that

the riemann zeros and eigenvalue asymptotics comparison between formulae for the counting functions of the heights tn of the riemann zeros and of semiclassical quantum eigenvalues en suggests that the tn are eigenvalues of an unknown hermitean operator h obtained by quantizing a classical dynamical system with hamiltonian hcl many features of hcl are provided by the analogy for example the riemann dynamics should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers statistics of the tn have a similar structure to those of the semiclassical en in particular they display random matrix universality at short range and nonuniversal behaviour over longer ranges very refined features of the statistics of the tn can be computed accurately from formulae with quantum analogues the riemann siegel formula for the zeta function is described in detail its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations we speculate that the riemann dynamics is related to the trajectories generated by the classical hamiltonian hcl xp

spectrum generating operators and intertwining operators for representations induced from a maximal parabolic subgroup abstract we present a new method of calculating intertwining operators between principal series representations of semisimple lie groups g working in the compact realization we find the eigenvalues of the operators on the k types and give several examples among the advantages of our method is its applicability to bundle valued cases

a notable family of entire intrinsic minimal graphs in the heisenberg group which are not perimeter minimizing one of the main objectives of this paper is to unravel a new interesting phenomenon of the sub riemannian bernstein problem with respect to its euclidean ancestor with the purpose of also indicating a possible line of attack toward its solution we show that the global intrinsic graphs 1 2 are unstable critical points of the horizontal perimeter as a consequence of this fact the study of the stability acquires a central position in the problem itself

quasiconformal maps in metric spaces with controlled geometry this paper develops the foundations of the theory of quasiconformal maps in metric spaces that satisfy certain bounds on their mass and geometry the principal message is that such a theory is both relevant and viable the first main issue is the problem of definition which we next describe quasiconformal maps are commonly understood as homeomorphisms that distort the shape of infinitesimal balls by a uniformly bounded amount this requirement makes sense in every metric space given a homeomorphism f from a metric space x to a metric space y then for x x and r 0 set

on three dimensional cauchy riemann manifolds of the almost complex tensor of c2 multiplication by i the collection of these subspaces hp m forms a bundle hm c tm called the horizontal or contact bundle indeed strict pseudoconvexity of m implies that the plane field hpm is nondegenerate i e defines a contact structure the almost complex tensor of c2 then restricts to a bundle endomorphism j hm hm such that j2 id the bundle hm c tm together with this endomorphism j defines the cr structure of m slightly more generally let d d u m be a four dimensional compact manifold with boundary m interior d endowed with a smooth almost complex tensor which is integrable on d if d is strictly pseudoconvex i e in a neighborhood of any p e m there is a smooth strictly plurisubharmonic function u negative on d 0 on m but du 0 on m then m inherits a strictly pseudoconvex cr structure from d in this case we shall say that m bounds a strictly pseudoconvex surface this leads us to the abstract definition a strictly pseudoconvex threedimensional cr manifold is a compact manifold m without boundary dim m 3 endowed with a contact structure hm hpm p e m c tm and an endomorphism j of hm such that j2 id throughout this paper unless otherwise stated we shall be working with infinitely differentiable objects thus m hm j are assumed to be smooth in this sense furthermore a differentiable function f m c is a cr function if for any p e m the restriction df ihm hpm tf p c

sub riemannian calculus and monotonicity of the perimeter for graphical strips we consider the class of minimal surfaces given by the graphical strips mathcal s in the heisenberg group mathbb h 1 and we prove that for points p along the center of mathbb h 1 the quantity frac sigma h mathcal s cap b p r r q 1 is monotone increasing here q is the homogeneous dimension of mathbb h 1 we also prove that these minimal surfaces have maximum volume growth at infinity

conformality and q harmonicity in carnot groups we show that if f is a 1 quasiconformal map defined on an open subset of a carnot group g then composition with f preserves q harmonic functions we combine this with a regularity theorem for q harmonic functions and an algebraic regularity theorem for maps between carnot groups to show that f is smooth we give some applications to the study of rigidity

character expansion methods for matrix models of dually weighted graphs we consider generalized one matrix models in which external fields allow control over the coordination numbers on both the original and dual lattices we rederive in a simple fashion a character expansion formula for these models originally due to itzykson and di francesco and then demonstrate how to take the largen limit of this expansion the relationship to the usual matrix model resolvent is elucidated our methods give as a by product an extremely simple derivation of the migdal integral equation describing the largen limit of the itzykson zuber formula we illustrate and check our methods by analysing a number of models solvable by traditional means we then proceed to solve a new model a sum over planar graphys possessing even coordination numbers on both the original and the dual lattice we conclude by formulating the equations for the case of arbitrary sets of even self dual coupling constants this opens the way for studying the deep problems of phase transitions from random to flat lattices january 1995

the pcp theorem by gap amplification the pcp theorem arora and safra 1998 arora et al 1998 says that every language in np has a witness format that can be checked probabilistically by reading only a constant number of bits from the proof the celebrated equivalence of this theorem and inapproximability of certain optimization problems due to feige et al 1996 has placed the pcp theorem at the heart of the area of inapproximability in this work we present a new proof of the pcp theorem that draws on this equivalence we give a combinatorial proof for the np hardness of approximating a certain constraint satisfaction problem which can then be reinterpreted to yield the pcp theorem our approach is to consider the unsat value of a constraint system which is the smallest fraction of unsatisfied constraints ranging over all possible assignments for the underlying variables we describe a new combinatorial amplification transformation that doubles the unsat value of a constraint system with only a linear blowup in the size of the system the amplification step causes an increase in alphabet size that is corrected by a standard pcp composition step iterative application of these two steps yields a proof for the pcp theorem the amplification lemma relies on a new notion of graph powering that can be applied to systems of binary constraints this powering amplifies the unsat value of a constraint system provided that the underlying graph structure is an expander we also extend our amplification lemma towards construction of assignment testers alternatively pcps of proximity which are slightly stronger objects than pcps we then construct pcps and locally testable codes whose length is linear up to a polylog factor and whose correctness can be probabilistically verified by making a constant number of queries namely we prove sat pcp 1 2 1 log2 n poly log n o 1