kazhdan projections random walks and ergodic theorems in this paper we investigate generalizations of kazhdan s property t to the setting of uniformly convex banach spaces we explain the interplay between the existence of spectral gaps and that of kazhdan projections our methods employ markov operators associated to a random walk on the group for which we provide new norm estimates and convergence results this construction exhibits useful properties and flexibility and allows to reclassify the existence of kazhdan projections from a c algebraic phenomenon to a natural object associated to random walks on groups we give a number of applications of these results in particular we address several open questions we give a direct comparison of properties te and fe with lafforgue s reinforced banach property t we obtain shrinking target theorems for orbits of kazhdan groups and apply them to a question of kleinbock and margulis finally we construct non compact ghost projections for warped cones answering a question of willett and yu in this last case we conjecture that such warped cones provide new counterexamples to the coarse baum connes conjecture

a modular functor which is universal for quantum computation we show that the topological modular functor from witten chern simons theory is universal for quantum computation in the sense that a quantum circuit computation can be efficiently approximated by an intertwining action of a braid on the functor s state space a computational model based on chern simons theory at a fifth root of unity is defined and shown to be polynomially equivalent to the quantum circuit model the chief technical advance the density of the irreducible sectors of the jones representation has topological implications which will be considered elsewhere

on classification of modular tensor categories we classify all unitary modular tensor categories umtcs of rank 4 there are a total of 35 umtcs of rank 4 up to ribbon tensor equivalence since the distinction between the modular s matrix s and s has both topological and physical significance so in our convention there are a total of 70 umtcs of rank 4 in particular there are two trivial umtcs with s 1 each such umtc can be obtained from 10 non trivial prime umtcs by direct product and some symmetry operations explicit data of the 10 non trivial prime umtcs are given in sect 5 relevance of umtcs to topological quantum computation and various conjectures are given in sect 6

homogeneous asymptotic limits of haar measures of semisimple linear groups and their lattices we show that haar measures of connected semisimple groups embedded via a representation into a matrix space have a homogeneous asymptotic limit when viewed from far away and appropriately rescaled this is still true if the haar measure of the semisimple group is replaced by the haar measure of a irreducible lattice of the group and the asymptotic measure is the same in the case of an almost simple group of rank greater than 2 a remainder term is also obtained this extends and makes precise anterior results of duke rudnick and sarnak and eskin mcmullen in the case of a group variety

hom nijienhuis operators and t extensions of hom lie superalgebras abstract in this paper we study hom nijienhuis operators and t extensions of hom lie superalgebras we show that a linear map between hom lie superalgebras is a morphism if and only if its graph is a hom subalgebra and show that the infinitesimal deformation generated by a hom nijienhuis operator is trivial moreover we introduce the definition of t extensions of hom lie superalgebras and show that t extensions preserve many properties such as nilpotency solvability and decomposition in some sense in particular we discuss the equivalence of t extensions using cohomology

braid representations from unitary braided vector spaces we investigate braid group representations associated with unitary braided vector spaces focusing on a conjecture that such representations should have virtually abelian images in general and finite image provided the braiding has finite order we verify this conjecture for the two infinite families of gaussian and group type braided vector spaces as well as the generalization to quasi braided vector spaces of group type

propagation of singularities and maximal functions in the plane in this work we mainly generalize bourgain s circular maximal function to include variable coefficient averages our techniques involve a combination of bourgain s basic ideas plus microlocal analysis in particular to see the role of curvature we rely heavily on methods used in studying propogation of singularities for hyperbolic differential equations we also show that forp 2 there is local smoothing inl p for solutions to the wave equation

image of the braid groups inside the finite iwahori hecke algebras we determine the image of the braid groups inside the iwahori hecke algebras of type a when defined over a finite field in the semisimple case and for suitably large but controlable order of the defining quantum parameter

pointwise theorems for amenable groups in this paper we prove the pointwise ergodic theorem for general locally compact amenable groups along folner sequences that obey some restrictions these restrictions are mild enough so that such sequences exist for all amenable groups we also prove a generalization of the shannon mcmillan breiman theorem to all discrete amenable groups

local ergodic theorems for k spherical averages on the heisenberg group given a gelfand pair h n k where h n is the heisenberg group and k is a compact subgroup of the unitary group u n we consider the sphere and ball averages of certain k invariant measures on h n we prove local ergodic theorems for these measures when n geq 3 we also consider averages over annuli in the case of reduced heisenberg group and show that when the functions have zero mean value the maximal function associated to the annulus averages behave better than the spherical maximal function we use square function arguments which require several properties of the k spherical functions

ergodic properties of linear actions of 2 2 matrices in this paper we consider the ergodic properties of linear actions on the plane of 2 2 matrices with entries in the complex numbers quaternions or clifford numbers 0 introduction in this paper we study ergodic properties of linear actions of discrete groups of 2 2 matrices with determinant 1 consider the usual linear action on the real plane r2 of a discrete subgroup 0 in the 2 2 matrices with real entries sl 2 r given by x1 x2 7 ax1 bx2 cx1 dx2 where a b c d 0 the action by such matrices clearly preserves the usual lebesgue measure on the plane it follows from work of g hedlund 18 that if 0 is a cocompact group or more generally a lattice then this linear action is ergodic this result was extended by f ledrappier in 23 to the case of normal subgroups 0 g 0 for which 0 0 is an infinite abelian subgroup in this note we extend these ergodicity results to discrete subgroups of 2 2 matrices over other fields the simplest case relates to 2 2 matrices with complex entries in theorem 1 we show the ergodicity of the linear actions on the complex plane c2 by discrete cocompact groups 0 and by normal subgroups 0 g 0 for which 0 0 is an infinite abelian subgroup the next most obvious case deals with discrete subgroups of 2 2 matrices where the entries are quaternions where the nonabelian nature of the entries provides additional features of interest in the analysis the corresponding ergodicity result for the linear actions on the quaternionic plane h 2 is described in theorem 2 finally as we shall see there is a very rich general framework into which many of these results fit involving the study of matrices whose entries are if necessary we simply replace 0 by 0 i we can then identify 0 psl 2 r sl 2 r i i with orientation preserving isometries of h2 duke mathematical journal vol 116 no 2 c 2003 received 7 september 2001 revision received 23 january 2002 2000 mathematics subject classification primary 37c85 37c35 secondary 37a45 28d99

symplectic structures on quadratic lie algebras abstract we study quadratic lie algebras over a field k of null characteristic which admit at the same time a symplectic structure we see that if k is algebraically closed every such lie algebra may be constructed as the t extension of a nilpotent lie algebra admitting an invertible derivation and also as the double extension of another quadratic symplectic lie algebra by the one dimensional lie algebra finally we prove that every symplectic quadratic lie algebra is a special symplectic manin algebra and we give an inductive description in terms of symplectic quadratic double extensions

stationary measures and equidistribution for orbits of nonabelian semigroups on the torus let be a semigroup of d d nonsingular integer matrices and consider the action of on the torus t we assume throughout that the action is strongly irreducible there is no subtorus invariant under a finite index subsemigroup of the strong irreducibility assumption in particular implies that acts ergodically on t equipped with the lebesgue measure m therefore the orbit of lebesgue almost every x t is dense and in an appropriate sense even becomes equidistributed however when is cyclic there is a set of full hausdorff dimension of exceptional points x for which x fails to be dense when is bigger the distribution of individual orbits can be expected to be much more restrictive an important result in this direction is due to furstenberg who showed for d 1 in which case 1 an abelian semigroup never satisfies condition 1 indeed the group generated by a semigroup satisfying 1 is nonamenable assumption 2 is a technical condition which is in particular satisfied when is a zariski dense semigroup of sld z 16

aperiodic order and spherical diffraction i auto correlation of regular model sets we study uniform and non uniform model sets in arbitrary locally compact second countable lcsc groups which provide a natural generalization of uniform model sets in locally compact abelian groups as defined by meyer and used as mathematical models of quasi crystals we then define a notion of auto correlation for subsets of finite local complexitiy in arbitrary lcsc groups which generalizes hof s classical definition beyond the class of amenable groups and prov ide a formula for the auto correlation of a regular model set along the way we show that the punctured hull of an arbitrary regular model set admits a unique invariant probability measure even in the case where the punctured hull is non compact and the group is non amenable in fact this measure is also the unique stationary measure with respect to any admissible probability measure

quantum morphing and the jones polynomial we will explore the experimental observation that on the set of knots with bounded crossing number algebraically independent vassiliev invariants become correlated as noticed first by s willerton we will see this through the value distribution of the jones polynomial at roots of unit as the degree of the roots of unit is getting larger the higher order fluctuation is diminishing and a more organized shape will emerge from a rather random value distribution of the jones polynomial we call such a phenomenon quantum morphing evaluations of the jones polynomial at roots of unity play a crucial role for example in the volume conjecture

wave front sets local smoothing and bourgain s circular maximal theorem the purpose of this paper is to improve certain known regularity results for the wave equation and to give a simple proof of bourgain s circular maximal theorem 1 we use easy wave front analysis along with techniques previously used in proofs of the carleson sj6lin theorem see 3 5 7 and in the proof of sharp regularity properties of fourier integral operators 13 the circular means operators are defined by

the role of the ramanujan conjecture in analytic number theory we discuss progress towards the ramanujan conjecture for the group gln and its relation to various other topics in analytic number theory

expander graphs in pure and applied mathematics expander graphs are highly connected sparse finite graphs they play an important role in computer science as basic building blocks for network constructions error correcting codes algorithms and more in recent years they have started to play an increasing role also in pure mathematics number theory group theory geometry and more this expository article describes their constructions and various applications in pure and applied mathematics

the asymptotic distribution of lattice points in euclidean and non euclidean spaces the counting numbers for discrete subgroups of motions in euclidean and non euclidean spaces are obtained using the wave equation as the principal tool in dimensions 2 and 3 the error estimates are close to the best known

pointwise ergodic theorems for radial averages on the heisenberg group leth hn cn r denote the heisenberg group and let rdenote the normalized lebesgue measure on the sphere z 0 z r let x b m be a standard borel probability space on whichhacts measurably and ergodically by measure preserving transformations and let r denote the operator canonically associated with ronlp x we prove maximal and pointwise ergodic theorems inlp for radial averages ron the heisenberg grouphn n 1 the results are best possible for actions of the reduced heisenberg group the method of proof is to use the spectral theory of the banach algebra of radial measures on the group and decay estimates for its characters to establish maximal inequalities using spectral methods in particular littlewood paley stein square functions and analytic interpolation

diophantine approximation for negatively curved manifolds let m be a geometrically finite pinched negatively curved riemannian manifold with at least one cusp inspired by the theory of diophantine approximation of a real or complex number by rational ones we develop a theory of approximation of geodesic lines starting from a given cusp by ones returning to it we define a new invariant for m thehurwitz constant of m it measures how well all geodesic lines starting from the cusp are approximated by ones returning to it in the case of constant curvature we express the hurwitz constant in terms of lengths of closed geodesics and their depths outside the cusp neighborhood using the cut locus of the cusp we define an explicit approximation sequence for a geodesic line starting from the cusp and explore its properties we prove that the modular once punctured hyperbolic torus has the minimum hurwitz constant in its moduli space

algorithms for finding and updating minimum depth spanning trees in parallel abstract fast parallel algorithms are presented for finding and updating the minimum depth spanning trees of graphs the computational model used is a parallel random access machine that allows simultaneous access to the same memory location during both read and write operations the parallel minimum depth spanning tree algorithm is based on a partial spanning tree doubling technique which is also found to be useful in updating the minimum depth spanning tree when a new node or edge is added to the graph the parallel minimum depth spanning tree algorithm runs in o log n log log n time with o n 2 n loglog n processors the most notable results are o loglog n time parallel algorithms for updating the minimum depth spanning tree solution when a new node edge is added to an n node graph whose minimum depth spanning tree is known

the affine sieve we establish the main saturation conjecture connected with executing a brun sieve in the setting of an orbit of a group of affine linear transformations this is carried out under the condition that the zariski closure of the group is levi semisimple it is likely that this condition is also necessary for such saturation to hold 2013 american mathematical society

analogs of wiener s ergodic theorems for semisimple lie groups ii let g be a connected finite center simple lie group of real rank one and let k be a maximal compact subgroup of g let a at i t c r be a 1 parameter subgroup of hyperbolic translations such that g ka k is a cartan decomposition denote by ot the probability measure on g given by ot mk 6at mk where mk is the normalized haar measure on k let it denote the probablity measure with density vol bt 1xb g where bt g d gk k t and d denotes the canonical g invariant riemannian metric on g k let x b a be a standard borel probability space on which g acts measurably and ergodically by measure preserving transformations let 7r g iso lp x be the associated isometric representation and let 7r ut and 7r 3t denote the operators canonically associated with ct and it let n g dimr g k we prove maximal and pointwise ergodic theorems for the ball averages it on simple real rank one lie groups which are analogous to n wiener s maximal and pointwise ergodic theorems for ball averages on in the general case of ball averages on semisimple groups is completed in m n s our main result is the following

t extension of lie triple systems abstract in this paper we introduce the notion of t extension of a lie triple system then we show that t extension is compatible with nilpotency solvability and it preserves in certain sense the decomposition properties in addition we investigate the equivalence of t extensions using cohomology finally we show that every finite dimensional nilpotent metrised lie triple system over an algebraically closed field is the t extension of an appropriate quotient system

prime and almost prime integral points on principal homogeneous spaces we develop the affine sieve in the context of orbits of congruence subgroups of semisimple groups acting linearly on affine space in particular we give effective bounds for the saturation numbers for points on such orbits at which the value of a given polynomial has few prime factors in many cases these bounds are of the same quality as what is known in the classical case of a polynomial in one variable where the orbit consists of the integers when the orbit is the set of integral matrices of a fixed determinant we obtain a sharp result for the saturation number and thus establish the zariski density of matrices all of whose entries are prime numbers among the key tools used are explicit approximations to the generalized ramanujan conjectures for such groups and sharp and uniform counting of points on such orbits when ordered by various norms

counting lattice points for a locally compact second countable group g and a lattice subgroup we give an explicit quantitative solution of the lattice point counting problem in general domains in g provided that i g has finite upper local dimension and the domains satisfy a basic regularity condition ii the mean ergodic theorem for the action of g on g holds with a rate of convergence the error term we establish matches the best current result for balls in symmetric spaces of simple higher rank lie groups but holds in much greater generality a significant advantage of the ergodic theoretic approach we use is that the solution to the lattice point counting problem is uniform over families of lattice subgroups provided they admit a uniform spectral gap in particular the uniformity property holds for families of finite index subgroups satisfying a quantitative variant of property we discuss a number of applications including counting lattice points in general domains in semisimple s algebraic groups counting rational points on group varieties with respect to a height function and quantitative angular or conical equidistribution of lattice points in symmetric spaces and in affine symmetric varieties we note that the mean ergodic theorems which we establish are based on spectral methods including the spectral transfer principle and the kunze stein phenomenon we formulate and prove appropriate analogues of both of these results in the set up of adele groups and they constitute a necessary step in our proof of quantitative results in counting rational points

the two eigenvalue problem and density of jones representation of braid groups 1 the two eigenvalue problem 179 2 hecke algebra representations of braid groups 183 3 duality of jones wenzl representations 186 4 closed images of jones wenzl sectors 188 5 distribution of evaluations of jones polynomials 192 6 fibonacci representations 196

volume et entropie minimale des espaces localement symetriques m gromov introduit la notion de volume minimal d une variete riemannienne min vol m inf vol m g courbure sectionnelle de g 1 les questions naturelles sont alors i quelles varietes m verifient min vol m 0 min vol m 0 ii dans le second cas existe t il des metriques realisant le minimum lesquelles en guise de reponse a la seconde question on peut au mieux esperer que si m supporte une metrique naturelle elle realisera le volume minimal ceci conduit a la conjecture suivante posee par m gromov si m est une variete admettant une metrique hyperbolique le volume fini notee hyp alors min vol m vol m hyp le but de l article est de donner une reponse partielle a cette conjecture

injectivity sets for spherical means on the heisenberg group in this paper we prove that cylinders of the form r s r r where s r is the sphere z c n z r are injectivity sets for the spherical mean value operator on the heisenberg group h n in l p spaces we prove this result as a consequence of a uniqueness theorem for the heat equation associated to the sub laplacian a hecke bochner type identity for the weyl transform proved by d geller and spherical harmonic expansions are the main tools used

affine linear sieve expanders and sum product let mathcal o be an orbit in n of a finitely generated subgroup of gl n whose zariski closure zcl is suitably large e g isomorphic to sl2 we develop a brun combinatorial sieve for estimating the number of points on mathcal o at which a fixed integral polynomial is prime or has few prime factors and discuss applications to classical problems including pythagorean triangles and integral apollonian packings a fundamental role is played by the expansion property of the congruence graphs that we associate with mathcal o this expansion property is established when zcl sl2 using crucially sum product theorem in q for q square free

uniform pointwise bounds for matrix coefficients of unitary representations and applications to kazhdan constants let k be a local field and g the group of k rational points of a connected reductive linear algebraic group over k with k semisimple rank g 2 let k be a good maximal compact subgroup of g for a unitary representation of g a vector v in is called kfinite if the subspace spanned bykv is finite dimensional we will use the termk matrix coefficients resp k finite matrix coefficients of to refer to its matrix coefficients with respect to k invariant resp k finite unit vectors following bt we denote by g the subgroup generated by the unipotent k split subgroups of g the main goal of the present paper is to construct a class of uniform pointwise bounds for the k finite matrix coefficients of all infinite dimensional irreducible unitary representations of g or more generally of all unitary representations of g without a non zero g invariant vector let a be a maximal k split torus and a the closed positive weyl chamber of a such that the cartan decomposition g ka k holds where is a finite subset of the

resolvent and lattice points on symmetric spaces of strictly negative curvature we study the asymptotics of the lattice point counting function n x y r gamma in gamma d x gamma y for a riemannian symmetric space x obtained from a semisimple lie group of real rank one and a discontinuous group gamma of motions in x such that gamma backslash x has finite volume we show that as r rightarrow infty for each varepsilon 0 the constant 2 rho corresponds to the sum of the positive roots of the lie group associated to x and n dimx the sum in the main term runs over a system of orthonormal eigenfunctions varphi j in l 2 gamma backslash x of the laplacian such that the eigenvalues rho 2 nu j 2 are less than 4n rho 2 n 1 2

spherical means wave equations and hermite laguerre expansions abstract in this paper we study the maximal function associated to the weyl transformw r of the normalised surface measure ron the sphere z rin c n this operator is given by the expansion w r f k 0 k n 1 k n 1 phiv k r p k f where kare laguerre functions of type n 1 andpkare hermite projection operators we show that whenp 2n 2n 1 the maximal operator supr 0 w r f x is bounded onlp r n using this we study almost everywhere convergence to initial data of solutions of the wave equation associated to the hermite operator the above expansion forw r motivates the study of operators of the form s t f k 0 phiv k t p k f wherephiv kare laguerre functions of type we study various mapping properties of these operators with applications to hermite expansions and solutions of darboux type equations

a self stabilizing algorithm for minimum depth search of graphs abstract this paper presents a self stabilizing algorithm for the minimum depth search mds of a connected undirected graph on an asynchronous distributed or network model of computation the algorithm produces a minimum depth spanning tree mdst of the given graph in a distributed manner so that each node of the graph knows its depth and parent the algorithm is resilient to transient faults and does not require initialization it has been proved that the algorithm is correct and requires o n 2 time where n is the number of nodes of the graph

parallel matrix and graph algorithms matrix multiplication algorithms for cube connected and perfect shuffle computers are presented it is shown that in both these models two n times n matrices can be multiplied in o n m log m time when n 2 m 1 leqq m leqq n processing elements pes are available when only m 2 1 leqq m leqq n pes are available two n times n matrices can be multiplied in o n 2 m m n m 2 61 time it is shown that many graph problems can be solved efficiently using the matrix multiplication algorithms

diophantine equations and ergodic theorems let x be a probability measure space and t 1 t n be a family of commuting measure preserving invertible transformations on x let q m 1 m n be a homogeneous positive polynomial with integer coefficients and let s m e z n q m a denote the set of integer solutions m m 1 m n of the diophantine equation q m a we prove that under a certain nondegeneracy condition on the polynomial q m and an ergodic condition on the family of transformations t t 1 t n the images of the solution sets q x t m1 1 t m2 2 t m n n x m s become uniformly distributed on x w r t for a e x e x as that is the pointwise ergodic theorem holds when the standard averages are replaced by the ones where the exponents satisfy a diophantine equation the proof uses a variant of the hardy littlewood method of exponential sums developed by birch and davenport and techniques from harmonic analysis a key point is the corresponding maximal theorem which is a discrete analogue of a maximal theorem on r n corresponding to the level surfaces of the polynomial q x

spherical maximal operator on symmetric spaces of constant curvature we prove an endpoint weak type maximal inequality for the spherical maximal operator applied to radial funcions on symmetric spaces of constant curvature and dimension n 2 more explicitly in the lorentz space associated with the natural isometry invariant measure we show that for every radial function f mf n c n f n 1 n the proof uses only geometric arguments and volume estimates and applies uniformly in every dimension

flows on homogeneous spaces and diophantine approximation on manifolds we present a new approach to metric diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain flows on homogeneous spaces this approach yields a new proof of a conjecture of mahler originally settled by v g sprindzuk in 1964 we also prove several related hypotheses of baker and sprindzuk formulated in 1970s the core of the proof is a theorem which generalizes and sharpens earlier results on nondivergence of unipotent flows on the space of lattices

flows on homogeneous spaces and diophantine properties of matrices we generalize the notions of badly approximable resp singular systems of m linear forms in n variables and relate these generalizations to certain bounded resp divergent trajectories in the space of lattices in rm n

harmonic analysis and pointwise ergodic theorems for noncommuting transformations let fk denote the free group on k generators 1 where s n w i n and 1 denotes the word length on fk induced by s let x m m be a probability space on which fk acts ergodically by measure preserving transformations we prove a pointwise ergodic theorem for the sequence of operators ij i a a n acting on l 2 x namely a f x fx fdm almost everywhere for each f in l 2 x we also show that the sequence a2n converges to a conditional expectation operator with re spect to a a algebra which is invariant under fk the proof is based on the spectral theory of the commutative convolution subalgebra of e i fk gener ated by the elements an n 0 we then generalize the discussion to algebras arising as a gelfand pair associated with the group of automorphisms g ri r2 of a semi homogeneous tree t r r2 where r 2 r2 2 r r2 4 this content downloaded from 207 46 13 114 on thu 26 may 2016 06 27 50 utc all use subject to http about jstor org terms

logarithm laws for flows on homogeneous spaces in this paper we generalize and sharpen d sullivan s logarithm law for geodesics by specifying conditions on a sequence of subsets at t n of a homogeneous space g g a semisimple lie group an irreducible lattice and a sequence of elements ft of g under which t n ftx at is infinite for a e x g the main tool is exponential decay of correlation coefficients of smooth functions ong besides the general higher rank version of sullivan s result as a consequence we obtain a new proof of the classical khinchin groshev theorem on simultaneous diophantine approximation and settle a conjecture recently made by m skriganov