competition interfaces and second class particles the one dimensional nearest neighbor totally asymmetric simple exclusion process can be constructed in the same space as a last passage percolation model in z 2 we show that the trajectory of a second class particle in the exclusion process can be linearly mapped into the competition interface between two growing clusters in the last passage percolation model using technology built up for geodesics in percolation we show that the competition interface converges almost surely to an asymptotic random direction as a consequence we get a new proof for the strong law of large numbers for the second class particle in the rarefaction fan and describe the distribution of the asymptotic angle of the competition interface

on ol structure of nuclear quasidiagonal c algebras we continue the study of ol structure of nuclear c algebras initiated by junge ozawa and ruan in particular we prove if ol a 1

projective fraisse limits and the pseudo arc the aim of the present work is to develop a dualization of the fraisse limit construction from model theory and to indicate its surprising connections with the pseudo arc as corollaries of general results on the dual fraisse limits we obtain mioduszewski s theorem on surjective universality of the pseudo arc among chainable continua and a theorem on projective homogeneity of the pseudo arc which generalizes a result of lewis and smith on density of homeomorphisms of the pseudo arc among surjective continuous maps from the pseudo arc to itself we also get a new characterization of the pseudo arc via the projective homogeneity property

circular free spectrahedra abstract this paper considers matrix convex sets invariant under several types of rotations it is known that matrix convex sets that are free semialgebraic are solution sets of linear matrix inequalities lmis they are called free spectrahedra we classify all free spectrahedra that are circular that is closed under multiplication by e i t up to unitary equivalence the coefficients of a minimal lmi defining a circular free spectrahedron have a common block decomposition in which the only nonzero blocks are on the superdiagonal a matrix convex set is called free circular if it is closed under left multiplication by unitary matrices as a consequence of a hahn banach separation theorem for free circular matrix convex sets we show the coefficients of a minimal lmi defining a free circular free spectrahedron have up to unitary equivalence a block decomposition as above with only two blocks this paper also gives a classification of those noncommutative polynomials invariant under conjugating each coordinate by a different unitary matrix up to unitary equivalence such a polynomial must be a direct sum of univariate polynomials

the dynamics of pseudographs in convex hamiltonian systems we study the evolution under convex hamiltonian flows on cotangent bundles of compact manifolds of certain distinguished subsets of the phase space these subsets are generalizations of lagrangian graphs we call them pseudographs they emerge in a natural way from fathi s weak kam theory by this method we find various orbits which connect prescribed regions of the phase space our study is inspired by works of john mather as an application we obtain the existence of diffusion in a large class of a priori unstable systems and provide a solution to the large gap problem we hope that our method will have applications to more examples

transfer of energy to high frequencies in the cubic defocusing nonlinear schrodinger equation we consider the cubic defocusing nonlinear schrodinger equa tion on the two dimensional torus we exhibit smooth solutions for which the support of the conserved energy moves to higher fourier modes this behavior is quantified by the growth of higher sobolev norms given any 1 k 1 s 1 we construct smooth initial data u0 withu0 h s k at some time t this growth occurs despite the hamiltonian s bound onu t h 1 and despite the conservation of the quantityu t l2

arnold diffusion in arbitrary degrees of freedom and normally hyperbolic invariant cylinders we prove a form of arnold diffusion in the a priori stable case let h 0 p epsilon h 1 theta p t quad theta in mathbb t n p in b n t in mathbb t mathbb r mathbb t be a nearly integrable system of arbitrary degrees of freedom n geqslant 2 with a strictly convex h0 we show that for a generic epsilon h 1 there exists an orbit theta p satisfying p t p 0 l h 1 0 where l h 1 is independent of epsilon the diffusion orbit travels along a codimension 1 resonance and the only obstruction to our construction is a finite set of additional resonances

existence of exponentially small separatrix splittings and homoclinic connections between whiskered tori in weakly hyperbolic near integrable hamiltonian systems abstract in this paper we study a generalization of arnold s original example in which he discussed the existence of a mechanism for instability caused by the splitting of the homoclinic manifolds of the weakly hyperbolic tori that has subsequently been referred to as arnold diffusion in case when the number of degrees of freedom n 3 namely we consider a widely studied model of a pendulum weakly coupled with n 1 rotors with the degeneracies in the unperturbed hamiltonian corresponding to different timescales existing in the problem using an alloy of the iterative and direct methods developed within the last years we give exponentially small upper bounds for the splitting measure of transversality for the case of an even analytic perturbation thus improving the estimate of gallavotti 1994 which he calls quasiflat and generalizing the analogous recent estimate of delshams et al 1996 for the rapidly quasiperiodically forced pendulum to a much larger class of hamiltonian systems in particular the exponentially small upper bound for the transversality measure of the splitting applies when the hamiltonian has extra degeneracies namely when the frequencies on a torus become near resonant in fact we show that in such a case the quantity in question becomes smaller which is the incarnation of the general fact that resonant regions in the action space are more stable in the sense that they have larger nekhoroshev exponent nekhoroshev exponent nevertheless we emphasize that getting uniform estimates for an arbitrary n 3 is very hard unless one makes some additional assumptions on the approximation properties of the frequency vector although recent developments show that the first order of canonical perturbation theory given by melnikov integrals generally cannot be accepted as the leading order answer for the splitting distance for the case of more than two degrees of freedom including the rapidly quasiperiodically forced pendulum problem we suggest an analytic perturbation the majority of whose fourier components are strictly nondashzero for which melnikov integrals can be vindicated as the leading order approximation for the components of the splitting distance in different directions if the frequencies on the invariant tori satisfy certain arithmetic conditions this allows us to bound the splitting distance from below furthermore having such a perturbation for the case of three degrees of freedom we use a simple number theoretical argument to find the asymptotics of the fourier series with exponentially small coefficients involved this enables us to compute the numerous homoclinic orbits for the whiskered tori of asymptotically full measure and by proving the domineering contribution of the first order of perturbation theory for the transversality measure to suggest a leading order answer for this quantity which implies the existence of an infinite number of heteroclinic connections between tori with close diophantine frequencies we elucidate the numerous arithmetic issues that obstruct getting a compact leading order approximation for the splitting size most of which can be overcome in the case of three degrees of freedom as our example shows these obstacles can be also possibly avoided in the same fashion for an arbitrary n 3 if one treats the case when the frequencies of the rotors are near a resonance of multiplicity n 3 or n 2

the super operator system structures and their applications in quantum entanglement theory an operator system s with unit e can be viewed as an archimedean order unit space s s e using this archimedean order unit space for a fixed k n we construct a super k minimal operator system omink s and a super k maximal operator system omaxk s which are the general versions of the minimal operator system omin s and the maximal operator system omax s introduced recently such that for k 1 we obtain the equality respectively we develop some of the key properties of these super operator systems and make some progress on characterizing when an operator system s is completely boundedly isomorphic to either omink s or to omaxk s then we apply these concepts to the study of k partially entanglement breaking maps we prove that for matrix algebras a linear map is completely positive from omink mn to omaxk mm for some fixed k min n m if and only if it is a k partially entanglement breaking map

banach spaces which are m ideals in their biduals we investigate banach spaces x such that x is an m ideal in x subspaces quotients and co sums of spaces which are m ideals in their biduals are again of this type a nonreflexive space x which is an m ideal in x contains a copy of co recently lima has shown that if k x is an m ideal in lf x then x is an m ideal in x here we show that if xis reflexive and k x is an m ideal in l x then k x is isometric to l x i e k x is an m ideal in its bidual moreover for real such spaces we show that k x contains a proper m ideal if and only if x or x contains a proper m ideal in 3 we start to study the properties of spaces which are m ideals in their biduals we show that subspaces and quotients of spaces which are m ideals in their biduals have the same property the proof of this result is based upon the fact that x is an m ideal in x if and only if the natural projection from x onto x is an l projection using local reflexivity we show that if x is an m ideal in x and x is nonreflexive then x contains almost isometric copies of co from this it follows that subspaces and quotients of such papers which are isomorphic to dual spaces are reflexive m structure of spaces which are m ideals in their biduals are studied in 4 here we show that if x is an m ideal in x without proper m summands and j o is an m ideal in x then x c j in 5 we treat the relation between spaces which are m ideals in their biduals and spaces of compact operators in 12 lima proved that if k x is an m ideal in lf x then x is an m ideal in x and x has the rnp here we show that if x is reflexive and k x is an m ideal in l x then k x is isometric to l x i e k x is an m ideal in its bidual moreover if k x is an m ideal in its bidual then x is reflexive we prove that for real spaces k x y contains a proper m summand

duality between coalescence times and exit points in last passage percolation models in this article we prove a duality relation between coalescence times and exit points in last passage percolation models with exponential weights as a consequence we get lower bounds for coalescence times with scaling exponent 3 23 2 and we relate its distribution with variational problems involving the brownian motion process and the airy22 process the proof relies on the relation between busemann functions and the burke property for stationary versions of the last passage percolation model with boundary

consequences of universality among toeplitz operators the invariant subspace problem for hilbert spaces is a long standing question and the use of universal operators in the sense of rota has been one tool for studying the problem the best known universal operators have been adjoints of analytic toeplitz operators or unitarily equivalent to them we present many examples of toeplitz operators whose adjoints are universal operators and exhibit some of their common properties some ways in which the invariant subspaces of these universal operators interact with operators in their commutants are given special attention is given to the closed subalgebra not always the zero algebra of compact operators in their commutants finally three questions connecting shift invariant subspaces and invariant subspaces of analytic toeplitz operators are raised positive answers for both of the first two imply the existence of non trivial invariant subspaces for every bounded operator on separable hilbert spaces of dimension two or more

amalgamation and ramsey properties of lp spaces abstract we study the dynamics of the group of isometries of l p spaces in particular we study the canonical actions of these groups on the space of isometric embeddings of finite dimensional subspaces of l p 0 1 into itself and we show that for every real number 1 p with p 4 6 8 they are e transitive provided that is small enough we achieve this by extending the classical equimeasurability principle of plotkin and rudin we define the central notion of a fraisse banach space which underlies these results and of which the known separable examples are the spaces l p 0 1 p 4 6 8 and the gurarij space we also give a proof of the ramsey property of the classes l p n n p 2 viewing it as a multidimensional borsuk ulam statement we relate this to an arithmetic version of the dual ramsey theorem of graham and rothschild as well as to the notion of a spreading vector of matousek and rodl finally we give a version of the kechris pestov todorcevic correspondence that links the dynamics of the group of isometries of an approximately ultrahomogeneous space x with a ramsey property of the collection of finite dimensional subspaces of x

operator space structure and amenability for figa talamanca herz algebras column and row operator spaces which we denote by col and row respectively over arbitrary banach spaces were introduced by the first named author for hilbert spaces these definitions coincide with the usual ones given a locally compact group g and p p 1 with 1p 1p 1 we use the operator space structure on cb col lp g to equip the figa talamanca herz algebra ap g with an operator space structure turning it into a quantized banach algebra moreover we show that for p q 2 or 2 q p and amenable g the canonical inclusion aq g ap g is completely bounded with cb norm at most kg2 where kg is grothendieck s constant as an application we show that g is amenable if and only if ap g is operator amenable for all and equivalently for one p 1 this extends a theorem by ruan

turbulence amalgamation and generic automorphisms of homogeneous structures we study topological properties of conjugacy classes in polish groups with emphasis on automorphism groups of homogeneous countable structures we first consider the existence of dense conjugacy classes the topological rokhlin property we then characterize when an automorphism group admits a comeager conjugacy class answering a question of truss and apply this to show that the homeomorphism group of the cantor space has a comeager conjugacy class answering a question of akin hurley and kennedy finally we study polish groups that admit comeager conjugacy classes in any dimension in which case the groups are said to admit ample generics we show that polish groups with ample generics have the small index property generalizing results of hodges hodkinson lascar and shelah and arbitrary homomorphisms from such groups into separable groups are automatically continuous moreover in the case of oligomorphic permutation groups they have uncountable cofinality and the bergman property these results in particular apply to automorphism groups of many stable 0 categorical structures and of the random graph in this connection we also show that the infinite symmetric group s has a unique non trivial separable group topology for several interesting groups we also establish serre s properties fh and fa

uniqueness universality and homogeneity of the noncommutative gurarij space we realize the noncommutative gurarij space ng defined by oikhberg as the fraisse limit of the class of finite dimensional 1 exact operator spaces as a consequence we deduce that the noncommutative gurarij space is unique up to completely isometric isomorphism homogeneous and universal among separable 1 exact operator spaces we also prove that ng is the unique separable nuclear operator space with the property that the canonical triple morphism from the universal tro to the triple envelope is an isomorphism we deduce from this fact that ng does not embed completely isometrically into an exact c algebra and it is not completely isometrically isomorphic to a c algebra or to a tro we also provide a canonical construction of ng which shows that the group of surjective complete isometries of ng is universal among polish groups analog results are proved in the commutative setting and more generally for m n spaces in particular we provide a new characterization and canonical construction of the gurarij banach space

coalescence of geodesics in exactly solvable models of last passage percolation coalescence of semi infinite geodesics remains a central question in planar first passage percolation in this paper we study finer properties of the coalescence structure of finite and semi infinite geodesics for exactly solvable models of last passage percolation consider directed last passage percolation on z2 with independent and identically distributed exponential weights on the vertices fix two points v1 0 0 and v2 0 k2 3 for some k 0 and consider the maximal paths 1 and 2 starting at v1 and v2 respectively to the point n n for n k our object of study is the point of coalescence i e the point v 1 2 with smallest v 1 we establish that the distance to coalescence v 1 scales as k by showing the upper tail bound p v 1 rk r c for some c 0 we also consider the problem of coalescence for semi infinite geodesics for the almost surely unique semi infinite geodesics in the direction 1 1 starting from v3 k2 3 k2 3 and v4 k2 3 k2 3 we establish the optimal tail estimate p v 1 rk r 2 3 for the point of coalescence v this answers a question left open by pimentel ann probab 44 5 3187 3206 2016 who proved the corresponding lower bound

some remarks on c convexity abstract recently the study of completely positive maps has become important to the results of brown douglas and fillmore on ext a c algebra attempts to solve questions related to ext have often turned into questions about the matrix algebras mn in this paper we wish to discuss a notion of c convexity related to completely positive linear maps to state some facts about c convexity and to ask some questions about c convexity to a large degree the tone of this paper is expository

shape fluctuations and random matrices we study a certain random growth model in two dimensions closely related to the one dimensional totally asymmetric exclusion process the results show that the shape fluctuations appropriately scaled converges in distribution to the tracy widom largest eigenvalue distribution for the gaussian unitary ensemble gue

on the distribution of the length of the longest increasing subsequence of random permutations let sn be the group of permutations of 1 2 n if 7r e sn we say that 7 i1 7f ik is an increasing subsequence in 7r if il i2 ik and 7r ii 7r i2 7r ik let 1n r be the length of the longest increasing subsequence for example if n 5 and 7r is the permutation 5 1 3 2 4 in one line notation thus 7r 1 5 7r 2 1 then the longest increasing subsequences are 1 2 4 and 1 3 4 and n 3 equip sn with uniform distribution

geodesics in two dimensional first passage percolation we consider standard first passage percolation on z 2 geodesics are nearest neighbor paths in z 2 each of whose segments is time minimizing we prove part of the conjecture that doubly infinite geodesics do not exist our main tool is a result of independent interest about the coalescing of semi infinite geodesics

the kirchberg wassermann operator system is unique in 1998 kirchberg and wassermann constructed a separable nuclear operator system with the property that the canonical unital homomorphism from the maximal c algebra onto the c envelope is injective we show that such an operator system is unique and completely order isomorphic to the operator system associated with the noncommutative poulsen simplex

busemann functions and infinite geodesics in two dimensional first passage percolation we study first passage percolation on mathbb z 2 where the edge weights are given by a translation ergodic distribution addressing questions related to existence and coalescence of infinite geodesics some of these were studied in the late 1990s by c newman and collaborators under strong assumptions on the limiting shape and weight distribution in this paper we develop a framework for working with distributional limits of busemann functions and use it to prove forms of newman s results under minimal assumptions for instance we show a form of coalescence of long finite geodesics in any deterministic direction we also introduce a purely directional condition which replaces newman s global curvature condition and whose assumption we show implies the existence of directional geodesics without this condition we prove existence of infinite geodesics which are directed in sectors last we analyze distributional limits of geodesic graphs proving almost sure coalescence and nonexistence of infinite backward paths this result relates to the conjecture of nonexistence of bigeodesics

homoclinic orbits for second order hamiltonian systems possessing superquadratic potentials received by the editors november 27 1990 1991 mathematics subject classification primary 34c99 58e99 58f99 the first author was supported by ministero p i gruppo 40 calcolo delle variazioni the second author s research was sponsored in part by the u s army research office under contract daal03 87 k 0043 the national science foundation under grant mcs 8110556 and the office of naval research under grant n00014 88 ko134 any reproduction for the purpose of the united states govemment is permitted

compressions of compact tuples abstract we study the matrix range of a tuple of compact operators on a hilbert space and examine the notions of minimal nonsingular and fully compressed tuples in this pursuit we refine previous results by characterizing nonsingular compact tuples in terms of matrix extreme points of the matrix range further we find that a compact tuple a is fully compressed if and only if it is multiplicity free and the shilov ideal is trivial which occurs if and only if a is minimal and nonsingular fully compressed compact tuples are therefore uniquely determined up to unitary equivalence by their matrix ranges we also produce a proof of this fact which does not depend on the concept of nonsingularity

moderate deviation and exit time estimates for stationary last passage percolation we consider planar stationary exponential last passage percolation in the positive quadrant with boundary weights for rho in 0 1 and points v n 1 rho 2 n rho 2 n going to infinity along the characteristic direction we establish right tail estimates with the optimal exponent for the exit time of the geodesic along with optimal exponent estimates for the upper tail moderate deviations for the passage time for the case rho frac 1 2 in the stationary model we establish the lower bound estimate with the optimal exponent for the lower tail of the passage time our arguments are based on moderate deviation estimates for point to point and point to line exponential last passage percolation which are obtained via random matrix estimates

quotients exactness and nuclearity in the operator system category abstract we continue our study of tensor products in the operator system category we define operator system quotients and exactness in this setting and refine the notion of nuclearity by studying operator systems that preserve various pairs of tensor products one of our main goals is to relate these refinements of nuclearity to the kirchberg conjecture in particular we prove that the kirchberg conjecture is equivalent to the statement that every operator system that is min er nuclear is also el c nuclear we show that operator system quotients are not always equal to the corresponding operator space quotients and then study exactness of various operator system tensor products for the operator system quotient we prove that an operator system is exact for the min tensor product if and only if it is min el nuclear we give many characterizations of operator systems that are min er nuclear el c nuclear min el nuclear and el max nuclear these characterizations involve operator system analogues of various properties from the theory of c algebras and operator spaces including the wep and llp

brownian gibbs property for airy line ensembles consider a collection of n brownian bridges b i n n to mathbb r b i n b i n 0 1 i n conditioned not to intersect the edge scaling limit of this system is obtained by taking a weak limit as n of the collection of curves scaled so that the point 0 21 2 n is fixed and space is squeezed horizontally by a factor of n 2 3 and vertically by n 1 3 if a parabola is added to each of the curves of this scaling limit an x translation invariant process sometimes called the multi line airy process is obtained we prove the existence of a version of this process which we call the airy line ensemble in which the curves are almost surely everywhere continuous and non intersecting this process naturally arises in the study of growth processes and random matrix ensembles as do related processes with wanderers and outliers we formulate our results to treat these relatives as well note that the law of the finite collection of brownian bridges above has the property called the brownian gibbs property of being invariant under the following action select an index 1 k n and erase b k on a fixed time interval a b n n then replace this erased curve with a new curve on a b according to the law of a brownian bridge between the two existing endpoints a b k a and b b k b conditioned to intersect neither the curve above nor the one below we show that this property is preserved under the edge scaling limit and thus establish that the airy line ensemble has the brownian gibbs property an immediate consequence of the brownian gibbs property is a confirmation of the prediction of m prahofer and h spohn that each line of the airy line ensemble is locally absolutely continuous with respect to brownian motion we also obtain a proof of the long standing conjecture of k johansson that the top line of the airy line ensemble minus a parabola attains its maximum at a unique point this establishes the asymptotic law of the transversal fluctuation of last passage percolation with geometric weights our probabilistic approach complements the perspective of exactly solvable systems which is often taken in studying the multi line airy process and readily yields several other interesting properties of this process

on ℒ structures of nuclear c algebras we study the local operator space structure of nuclear c algebras it is shown that a c algebra is nuclear if and only if it is an space for some and actually for every 6 the constant provides an interesting invariant for nuclear c algebras indeed if is a nuclear c algebra then we have 1 6 and if is a unital nuclear c algebra with we show that must be stably finite we also investigate the connection between the rigid 1 structure and the rigid complete order 1 structure on c algebras where the latter structure has been studied by blackadar and kirchberg in their characterization of strong nf c algebras another main result of this paper is to show that these two local structrues are actually equivalent on unital nuclear c algebras we obtain this by showing that if a unital nuclear c algebra is a rigid 1 space then it is inner quasi diagonal and thus is a strong nf algebra it is also shown that if a unital nuclear c algebra is an 1 space then it is quasi diagonal and thus is an nf algebra

on subspaces of non commutative lp spaces abstract we study some structural aspects of the subspaces of the non commutative haagerup l p spaces associated with a general non necessarily semi finite von neumann algebra if a subspace x of l p contains uniformly the spaces l p n n 1 it contains an almost isometric almost 1 complemented copy of l p if x contains uniformly the finite dimensional schatten classes s p n it contains their l p direct sum too we obtain a version of the classical kadec pelczynski dichotomy theorem for l p spaces p 2 we also give operator space versions of these results the proofs are based on previous structural results on the ultrapowers of l p together with a careful analysis of the elements of an ultrapower l p u which are disjoint from the subspace l p these techniques permit to recover a recent result of n randrianantoanina concerning a subsequence splitting lemma for the general non commutative l p spaces various notions of p equiintegrability are studied one of which is equivalent to randrianantoanina s one and some results obtained by haagerup rosenthal and sukochev for l p spaces based on finite von neumann algebras concerning subspaces of l p containing l p are extended to the general case

matrix convex sets without absolute extreme points abstract this article shows the existence of a class of closed bounded matrix convex sets which do not have absolute extreme points the sets we consider are noncommutative sets k x formed by taking matrix convex combinations of a single tuple x in the case that x is a tuple of compact operators with no nontrivial finite dimensional reducing subspaces k x is a closed bounded matrix convex set with no absolute extreme points a central goal in the theory of matrix convexity is to find a natural notion of an extreme point in the dimension free setting which is minimal with respect to spanning matrix extreme points are the strongest type of extreme point known to span matrix convex sets however they are not necessarily the smallest set which does so absolute extreme points a more restricted type of extreme points that are closely related to arveson s boundary enjoy a strong notion of minimality should they span this result shows that matrix convex sets may fail to be spanned by their absolute extreme points

dynamical entropy in banach spaces we introduce a version of voiculescu brown approximation entropy for isometric automorphisms of banach spaces and develop within this framework the connection between dynamics and the local theory of banach spaces as discovered by glasner and weiss our fundamental result concerning this contractive approximation entropy or ca entropy characterizes the occurrence of positive values both geometrically and topologically this leads to various applications for example we obtain a geometric description of the topological pinsker factor and show that a c algebra is type i if and only if every multiplier inner automorphism has zero ca entropy we also examine the behaviour of ca entropy under various product constructions and determine its value in many examples including isometric automorphisms of l p for 1 p and noncommutative tensor product shifts

total minimality of the unitary groups dierolf and schwanengel 7 showed that if x is an infinite diescrete space then the group f x of all bijections f x x provided with the topology of pointwise convergence is a totally minimal topological group that was the first example of a totally minimal group which is not precompact other examples of such groups can be found in 8 see also 21 w 2 infinite groups which does not admit non discrete hausdorff group topologies were constructed first by shelah 22 assuming ch and then by hesse 14 and a ol ghanski who noted that a quotient of the adian s group a m n has the property in question cf 1 w note that all the examples mentioned above are examples of non abelian groups prodanov 19 established the totally minimal abelian groups are precompact and recently prodanov and the author 20 proved that all minimal abelian groups are precompact the main purpose of the present paper is to show that the unitary group of every real or complex hilbert space provided with the strong operator topology is a totally minimal topological group this result gives an affirmative answer to a question posed by i prodanov in sects 2 and 3 we study the equivariant with respect to the action of the unitary group compactifications of the unit sphere s of an infinite dimensional hilbert space it is shown in sect 2 that the unit ball endowed with the weak topology is the greatest equivariant compactification of s this fact is used in sect 3 to describe all equivariant compactification of s the main theorem is proved in sect 4 the proof uses the scheme of the proof of 7 1 a generalization of which is discussed in proposition 4 6

geometry of arnold diffusion the goal of this paper is to present to nonspecialists what is perhaps the simplest possible geometrical picture explaining the mechanism of arnold diffusion we choose to speak of a specific model that of geometric rays in a periodic optical medium this model is equivalent to that of a particle in a periodic potential in mathbb r n with energy prescribed and to the geodesic flow in a riemannian metric on mathbb r n

a general mechanism of diffusion in hamiltonian systems qualitative results we present a general mechanism to establish the existence of diffusing orbits in a large class of nearly integrable hamiltonian systems our approach relies on the scattering map outer dynamics and on the recurrence property of the inner dynamics restricted to a normally hyperbolic invariant manifold we apply topological methods to find trajectories that follow these two dynamics this method differs in several crucial aspects from earlier works there are virtually no assumptions on the inner dynamics as the method does not use at all the invariant objects for the inner dynamics e g primary and secondary tori lower dimensional hyperbolic tori and their stable unstable manifolds aubry mather sets the method applies when the unperturbed hamiltonian is not necessarily convex and of arbitrary degrees of freedom in addition this mechanism is easy to verify analytically or numerically in concrete examples as well as to establish diffusion in generic systems we include several applications such as bridging large gaps in a priori unstable models in any dimension and establishing diffusion in cases when the inner dynamics in a non twist map

weakly almost periodic functions model theoretic stability and minimality of topological groups we investigate the automorphism groups of aleph 0 categorical structures and prove that they are exactly the roelcke precompact polish groups we show that the theory of a structure is stable if and only if every roelcke uniformly continuous function on the automorphism group is weakly almost periodic analysing the semigroup structure on the weakly almost periodic compactification we show that continuous surjective homomorphisms from automorphism groups of stable aleph 0 categorical structures to hausdorff topological groups are open we also produce some new wap trivial groups and calculate the wap compactification in a number of examples

quasi banach spaces of almost universal disposition we show that for each p 0 1 p 0 1 there exists a separable p banach space gpgp of almost universal disposition that is having the following extension property for each e 0e 0 and each isometric embedding g x yg x y where y is a finite dimensional p banach space and x is a subspace of gpgp there is an e isometry f y gpf y gp such that x f g x x f g x for all x xx x such a space is unique up to isometries does contain an isometric copy of each separable p banach space and has the remarkable property of being locally injective amongst p banach spaces we also present a nonseparable generalization which is of universal disposition for separable spaces and separably injective no separably injective p banach space was previously known for p 1p 1

pure matrix states on operator systems abstract an operator system is a complex matricially ordered vector space that is completely order isomorphic to a unital selfadjoint subspace of a unital c algebra a matrix state on an operator system v is a unital completely positive linear map of v into a full matrix algebra pure matrix states are studied and a new and somewhat simplified proof of a krein milman type theorem of webster and winkler is given if v is 3 dimensional then the matrix state space of v is matrix affinely homeomorphic to the matricial range of some hilbert space operator with the aid of this representation pure matrix states on 3 dimensional operator systems are examined and in some cases completely determined

speed of arnold diffusion for analytic hamiltonian systems for a convex real analytic e close to integrable hamiltonian system with n 5 degrees of freedom we construct an orbit exhibiting arnold diffusion with the diffusion time bounded by exp c epsilon frac 1 2 n 2 this upper bound of the diffusion time almost matches the lower bound of order exp epsilon frac 1 2 n 1 predicted by the nekhoroshev type stability results our method is based on the variational approach of bessi and mather and includes a new construction on the space of frequencies

colp spaces the local structure of non commutative lp spaces abstract developing the theory of col p spaces a variation of the non commutative analogue of l p spaces we provide new tools to investigate the local structure of non commutative l p spaces under mild assumptions on the underlying von neumann algebras non commutative l p spaces with grothendieck s approximation property behave locally like the space of matrices equipped with the p norm of the sequences of their singular values as applications we obtain a basis for non commutative l p spaces associated with hyperfinite von neumann algebras with separable predual von neumann algebras generated by free groups and obtain a basis for separable nuclear c algebras

operator space and operator system analogs of kirchberg s nuclear embedding theorem abstract the gurarij operator space ng introduced by oikhberg is the unique separable 1 exact operator space that is approximately injective in the category of 1 exact operator spaces and completely isometric linear maps we prove that a separable operator space x is nuclear if and only if there exist a linear complete isometry x ng and a completely contractive projection from ng onto the range of this can be seen as the operator space analog of kirchberg s nuclear embedding theorem with similar methods we also establish the natural operator system analog of kirchberg s nuclear embedding theorem involving the gurarij operator system gs

matrix convexity operator analogues of the bipolar and hahn banach theorems abstract several basic results of convexity theory are generalized to the quantized matrix convex sets of wittstock these include the bipolar theorem a gauge version of the hahn banach theorem and the existence theorem for support functionals

c convexity of norm unit balls abstract we characterize the norms on b h whose unit balls are c convex we call such norms m norms and investigate their dual norms named l norms we show that the class of l norms greater than a given norm enjoys a minimum element and the class of m norms less than a given norm enjoys a maximum element these minimum and maximum elements will be determined in some cases finally we give a constructive result to obtain m norms and l norms on b h

fraisse limits in functional analysis we provide a unified approach to fraisse limits in functional analysis including the gurarij space the poulsen simplex and their noncommutative analogs we obtain in this general framework many known and new results about the gurarij space and the poulsen simplex and at the same time establish their noncommutative analogs particularly we construct noncommutative analogs of universal operators in the sense of rota

perturbations of completely positive maps and strong nf algebras let mn b h be an injective completely positive contraction with 1 mn mncb 1 we show that if either c mn contains no non trivial compact operators or mn approximately contains a rank 1 projection then there is a complete order embedding mn b h w ith cb we also give examples showing that such a perturbation does not exist in general as an application we show that every c algebra a with ol a 1 and a finite separating family of primitive ideals is a strong nf algebra providing a partial answer to a question of junge ozawa and ruan

geodesics and the competition interface for the corner growth model we study the directed last passage percolation model on the planar integer lattice with nearest neighbor steps and general i i d weights on the vertices outside the class of exactly solvable models in georgiou et al probab theory relat fields 2016 doi 10 1007 s00440 016 0729 x we constructed stationary cocycles and busemann functions for this model using these objects we prove new results on the competition interface on existence uniqueness and coalescence of directional semi infinite geodesics and on nonexistence of doubly infinite geodesics

delocalization of polymers in lower tail large deviation directed last passage percolation models on the plane where one studies the weight as well as the geometry of optimizing paths called polymers in a field of i i d weights are paradigm examples of models in kpz universality class in this article we consider the large deviation regime i e when the polymer has a much smaller lower tail or larger upper tail weight than typical precise asymptotics of large deviation probabilities have been obtained in a handful of the so called exactly solvable scenarios including the exponential johansson in commun math phys 209 2 437 476 2000 and poissonian deuschel and zeitouni in comb probab comput 8 03 247 263 1999 seppalainen in probab theory relat fields 112 2 221 244 1998 cases how the geometry of the optimizing paths change under such a large deviation event was considered in deuschel and zeitouni 1999 where it was shown that the paths from 0 0 to n n say remain concentrated around the straight line joining the end points in the upper tail large deviation regime but the corresponding question in the lower tail was left open we establish a contrasting behaviour in the lower tail large deviation regime showing that conditioned on the latter in both the models the optimizing paths are not concentrated around any deterministic curve our argument does not use any ingredient from integrable probability and hence can be extended to other planar last passage percolation models under fairly mild conditions and also to other non integrable settings such as last passage percolation in higher dimensions

stationary cocycles and busemann functions for the corner growth model we study the directed last passage percolation model on the planar square lattice with nearest neighbor steps and general i i d weights on the vertices outside of the class of exactly solvable models stationary cocycles are constructed for this percolation model from queueing fixed points these cocycles serve as boundary conditions for stationary last passage percolation solve variational formulas that characterize limit shapes and yield existence of busemann functions in directions where the shape has some regularity in a sequel to this paper the cocycles are used to prove results about semi infinite geodesics and the competition interface

local properties of ternary rings of operators and their linking c algebras abstract we show that some local properties such as nuclearity exactness and local reflexivity of ternary rings of operators tros are closely related to the local properties of their linking c algebras we also show some equivalent conditions for nuclear tros and show that haagerup s decomposition property for completely bounded maps and pisier s norm can be naturally generalized to tros

dynamics near an isolated m semi static homoclinic orbit abstract we show that there are infinitely many periodic orbits in any neighborhood of an isolated m semi static orbit homoclinic to an aubry set for time periodic positive lagrangian systems

an approximation theorem for nuclear operator systems abstract we prove that an operator system s is nuclear in the category of operator systems if and only if there exist nets of unital completely positive maps s m n and m n s such that converges to id s in the point norm topology our proof is independent of the choi effros kirchberg characterization of nuclear c algebras and yields this characterization as a corollary we give an explicit example of a nuclear operator system that is not completely order isomorphic to a unital c algebra

operator spaces with few completely bounded maps we construct several examples of hilbertian operator spaces with few completely bounded maps in particular we give an example of a separable 1 hilbertian operator space x 0 such that whenever x is an infinite dimensional quotient of x 0 x is a subspace of x and t x rightarrow x is a completely bounded map then t i x s where s is compact hilbert schmidt and s 2 16 s cb s 2 moreover every infinite dimensional quotient of a subspace of x 0 fails the operator approximation property we also show that every banach space can be equipped with an operator space structure without the operator approximation property

geometric properties of the scattering map of a normally hyperbolic invariant manifold given a normally hyperbolic invariant manifold for a map f whose sta ble and unstable invariant manifolds intersect transversally we consider its associated scattering map that is the map that given an asymptotic orbit in the past gives the asymptotic orbit in the future we show that when f and are symplectic resp exact symplectic then the scattering map is symplectic resp exact symplectic furthermore we show that in the exact symplectic case there are extremely easy formulas for the primitive function which have a variational interpretation as dierence of actions we use this geometric information to obtain ecien t perturbative calculations of the scattering map using deformation theory this perturbation theory generalizes and extends several results already obtained using the melnikov method analogous results are true for hamiltonian o ws the proofs are obtained by geometrically natural methods and do not involve the use of particular coordinate systems hence the results can be used to obtain intersection properties of objects of any type we also reexamine the calculation of the scattering map in a geodesic o w perturbed by a quasi periodic potential we show that the geometric theory reproduces the results obtained in dls06b using methods of fast slow systems moreover the geometric theory allows to compute perturbatively the dependence on the slow variables which does not seem to be accessible to the previous methods

orbits of unbounded energy in quasi periodic perturbations of geodesic flows abstract we show that certain mechanical systems including a geodesic flow in any dimension plus a quasi periodic perturbation by a potential have orbits of unbounded energy the assumptions we make in the case of geodesic flows are a the metric and the external perturbation are smooth enough b the geodesic flow has a hyperbolic periodic orbit such that its stable and unstable manifolds have a tranverse homoclinic intersection c the frequency of the external perturbation is diophantine d the external potential satisfies a generic condition depending on the periodic orbit considered in b the assumptions on the metric are c 2 open and are known to be dense on many manifolds the assumptions on the potential fail only in infinite codimension spaces of potentials the proof is based on geometric considerations of invariant manifolds and their intersections the main tools include the scattering map of normally hyperbolic invariant manifolds as well as standard perturbation theories averaging kam and melnikov techniques we do not need to assume that the metric is riemannian and we obtain results for finsler or lorentz metrics indeed there is a formulation for hamiltonian systems satisfying scaling hypotheses we do not need to make assumptions on the global topology of the manifold nor on its dimension

subspaces of c algebras abstract basic properties of matricially normed spaces are considered and a simple matrix norm characterization of the subspaces of c algebras is given the latter result is used to study the haagerup tensor products and quotients of such subspaces

facial characterizations of complex lindenstrauss spaces we characterize complex banach spaces a whose banach dual spaces are l 1 spaces in terms of l ideals generated by certain extremal subsets of the closed unit ball k of a our treatment covers the case of spaces a containing constant functions and also spaces not containing constants separable spaces are characterized in terms of w compact sets of extreme points of k whereas the nonseparable spaces necessitate usage of the w closed faces of k our results represent natural extensions of known characterizations of choquet simplexes we obtain also a characterization of complex lindenstrauss spaces in terms of boundary annihilating measures and this leads to a characterization of the closed subalgebras of c c x which are complex lindenstrauss spaces

eberlein oligomorphic groups we study the fourier stieltjes algebra of roelcke precompact non archimedean polish groups and give a model theoretic description of the hilbert compactification of these groups we characterize the family of such groups whose fourier stieltjes algebra is dense in the algebra of weakly almost periodic functions those are exactly the automorphism groups of 0 stable 0 categorical structures this analysis is then extended to all semitopological semigroup compactifica tions s of such a group s is hilbert representable if and only if it is an inverse semigroup we also show that every factor of the hilbert compactification is hilbert representable

transversal fluctuations for increasing subsequences on the plane consider a realization of a poisson process in 2 with intensity 1 and take a maximal up right path from the origin to n n consisting of line segments between the points where maximal means that it contains as many points as possible the number of points in such a path has fluctuations of order n where 1 3 bdj here we show that typical deviations of a maximal path from the diagonal x y is of order n with 2 3 this is consistent with the scaling identity 2 1 which is believed to hold in many random growth models

moderate deviations for longest increasing subsequences the upper tail we derive the upper tail moderate deviations for the length of a longest increasing subsequence in a random permutation this concerns the regime between the upper tail large deviation regime and the central limit regime our proof uses a formula to describe the relevant probabilities in terms of the solution of the rank 2 riemann hilbert problem rhp this formula was invented by baik deift and johansson 3 to find the central limit asymptotics of the same quantities in contrast to the work of these authors who apply a third order nonstandard steepest descent approximation at an inflection point of the transition matrix elements of the rhp our approach is based on a more classical second order gaussian saddle point approximation at the stationary points of the transition function matrix elements 2001 john wiley sons inc

operator positivstellensatze for noncommutative polynomials positive on matrix convex sets abstract this article studies algebraic certificates of positivity for noncommutative nc operator valued polynomials on matrix convex sets such as the solution set d l called a free hilbert spectrahedron of the linear operator inequality loi l x a 0 i j 1 g a j x j 0 where a j are self adjoint linear operators on a separable hilbert space x j matrices and i is an identity matrix if a j are matrices then l x 0 is called a linear matrix inequality lmi and d l a free spectrahedron for monic lmis i e a 0 i and nc matrix valued polynomials the certificates of positivity were established by helton klep and mccullough in a series of articles with the use of the theory of complete positivity from operator algebras and classical separation arguments from real algebraic geometry since the full strength of the theory of complete positivity is not restricted to finite dimensions but works well also in the infinite dimensional setting we use it to tackle our problems first we extend the characterization of the inclusion d l 1 d l 2 from monic lmis to monic lois l 1 and l 2 as a corollary one immediately obtains the description of a polar dual of a free hilbert spectrahedron d l and its projection called a free hilbert spectrahedrop further on using this characterization in a separation argument we obtain a certificate for multivariate matrix valued nc polynomials f positive semidefinite on a free hilbert spectrahedron defined by a monic loi replacing the separation argument by an operator fejer riesz theorem enables us to extend this certificate in the univariate case to operator valued polynomials f finally focusing on the algebraic description of the equality d l 1 d l 2 we remove the assumption of boundedness from the description in the lmis case by an extended analysis however the description does not extend to lois case by counterexamples