hamiltonian stationary cones and self similar solutions in higher dimension in lw we construct examples of two dimensional hamiltonian stationary self shrinkers and self expanders for lagrangian mean curvature flows which are asymptotic to the union of two schoen wolfson cones these self shrinkers and self expanders can be glued together to yield solutions of the brakke flow a weak formulation of the mean curvature flow moreover there is no mass loss along the brakke flow in this paper we generalize these results to higher dimension we construct new higher dimensional hamiltonian stationary cones of different topology as generalizations of the schoen wolfson cones hamiltonian stationary self shrinkers and self expanders that are asymptotic to these hamiltonian stationary cones are also constructed they can also be glued together to produce eternal solutions of the brakke flow without mass loss finally we show the same conclusion holds for those lagrangian self similar examples recently found by joyce tsui and the first author in jlt

generic mean curvature flow i generic singularities it has long been conjectured that starting at a generic smooth closed embedded surface in r 3 the mean curvature ow remains smooth until it arrives at a singularity in a neighborhood of which the ow looks like concentric spheres or cylinders that is the only singularities of a generic ow are spherical or cylindrical we will address this conjecture here and in a sequel the higher dimensional case will be addressed elsewhere the key to showing this conjecture is to show that shrinking spheres cylinders and planes are the only stable self shrinkers under the mean curvature ow we prove this here in all dimensions an easy consequence of this is that every singularity other than spheres and cylinders can be perturbed away

extremal energies of integral circulant graphs via multiplicativity abstract the energy of a graph is the sum of the moduli of the eigenvalues of its adjacency matrix integral circulant graphs can be characterised by their order n and a set d of positive divisors of n in such a way that they have vertex set z n z and edge set a b a b z n z gcd a b n d among integral circulant graphs of fixed prime power order p s those having minimal energy e min p s or maximal energy e max p s respectively are known we study the energy of integral circulant graphs of arbitrary order n with so called multiplicative divisor sets this leads to good bounds for e min n and e max n as well as conjectures concerning the true value of e min n

expander graphs and their applications a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists since expander graphs the protagonists of our story come up in numerous and often surprising contexts in both fields but perhaps we should start with a few words about graphs in general they are of course one of the prime objects of study in discrete mathematics however graphs are among the most ubiquitous models of both natural and human made structures in the natural and social sciences they model relations among species societies companies etc in computer science they represent networks of communication data organization computational devices as well as the flow of computation and more in mathematics cayley graphs are useful in group theory graphs carry a natural metric and are therefore useful in geometry and though they are just one dimensional complexes they are useful in certain parts of topology e g knot theory in statistical physics graphs can represent local connections between interacting parts of a system as well as the dynamics of a physical process on such systems the study of these models calls then for the comprehension of the significant structural properties of the relevant graphs but are there nontrivial structural properties which are universally important expansion of a graph requires that it is simultaneously sparse and highly connected expander graphs were first defined by bassalygo and pinsker and their existence first proved by pinsker in the early 70s the property of being an expander seems significant in many of these mathematical computational and physical contexts it is not surprising that expanders are useful in the design and analysis of communication networks what is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness in mathematics we will encounter e g their role in the study of metric embeddings and in particular in work around the baum connes conjecture expansion is closely related to the convergence rates of markov chains and so they play a key role in the study of monte carlo algorithms in statistical mechanics and in a host of practical computational applications the list of such interesting and fruitful connections goes on and on with so many applications we will not even

are bitvectors optimal we study the it static membership problem given a set s of at most n keys drawn from a universe u of size m store it so that queries of the form is u in s can be answered by making few accesses to the memory we study schemes for this problem that use space close to the information theoretic lower bound of omega n log frac m n bits and yet answer queries by reading a small number of bits of the memory we show that for epsilon 0 there is a scheme that stores o frac n epsilon 2 log m bits and answers membership queries using a randomized algorithm that reads just one bit of memory and errs with probability at most epsilon we consider schemes that make no error for queries in s but are allowed to err with probability at most epsilon for queries not in s we show that there exist such schemes that store o frac n epsilon 2 log m bits and answer queries using just one bitprobe if multiple probes are allowed then the number of bits stored can be reduced to o n 1 delta log m for any delta 0 the schemes mentioned above are based on probabilistic constructions of set systems with small intersections we show lower bounds that come close to our upper bounds for a large range of n and epsilon schemes that answer queries with just one bitprobe and error probability epsilon must use omega frac n epsilon log 1 epsilon log m bits of storage if the error is restricted to queries not in s then the scheme must use omega frac n 2 epsilon 2 log n epsilon log m bits of storage we also consider deterministic schemes for the static membership problem and show tradeoffs between space and the number of probes

probabilistic checking of proofs a new characterization of np we give a new characterization of np the class np contains exactly those languages l for which membership proofs a proof that an input x is in l can be verified probabilistically in polynomial time using logarithmic number of random bits and by reading sublogarithmic number of bits from the proof we discuss implications of this characterization specifically we show that approximating clique and independent set even in a very weak sense is np hard

infinite non simple c algebras absorbing the cuntz algebra o abstract the first named author has given a classification of all separable nuclear c algebras a that absorb the cuntz algebra o we say that a absorbs o if a is isomorphic to a o motivated by this classification we investigate here if one can give an intrinsic characterization of c algebras that absorb o this investigation leads us to three different notions of pure infiniteness of a c algebra all given in terms of local algebraic conditions on the c algebra the strongest of the three properties strongly purely infinite is shown to be equivalent to absorbing o for separable nuclear c algebras that either are stable or have an approximate unit consisting of projections in a previous paper 2000 amer j math 122 637 666 we studied an intermediate and perhaps more natural condition purely infinite that extends a well known property for simple c algebras the weakest condition of the three weakly purely infinite is shown to be equivalent to the absence of quasitraces in an ultrapower of the c algebra the three conditions may be equivalent for all c algebras and we prove this to be the case for c algebras that are either simple of real rank zero or approximately divisible

entropy waves the zig zag graph product and new constant degree expanders the main contribution of this work is a new type of graph product which we call the zig zag product taking a product of a large graph with a small graph the resulting graph inherits roughly its size from the large one its degree from the small one and its expansion properties from both iteration yields simple explicit constructions of constant degree expanders of arbitrary size starting from one constant size expander crucial to our intuition and simple analysis of the properties of this graph product is the view of expanders as functions which act as entropy wave propagators they transform probability distributions in which entropy is concentrated in one area to distributions where that concentration is dissipated in these terms the graph product affords the constructive interference of two such waves

finite simple groups as expanders abstract we prove that there exist k and 0 e such that every non abelian finite simple group g which is not a suzuki group has a set of k generators for which the cayley graph cay g s is an e expander expander graphs ramanujan complexes

singular traces and compact operators abstract we give a necessary and sufficient condition on a positive compact operatortfor the existence of a singular trace i e a trace vanishing on the finite rank operators which takes a finite non zero value ont this generalizes previous results by dixmier and varga we also give an explicit description of these traces and associated ergodic states onl n using tools of non standard analysis in an essential way

on the kernel of integral circulant graphs abstract given a positive integer n every integral circulant graph on n vertices is isomorphic to some graph icg n d having vertex set z n z and edge set a b a b z n z gcd a b n d for a uniquely determined set d of positive divisors of n by virtue of its adjacency matrix one defines the spectrum of a graph g and naturally can ask to which degree the eigenvalues of g determine the graph itself with respect to integral circulant graphs little is known about this question which is related to a conjecture of so in this note we examine the role of the eigenvalue 0 and clarify the interrelation between the dimension of the kernel of icg n d and the graph itself for all prime powers n p k and for all positive integers n in case d is a multiplicative divisor set

the spectra of infinite hypertrees a model of regular infinite hypertrees is developed to mimic for hypergraphs what infinite trees do for graphs two notions of spectra or first eigenvalue are then examined for the infinite tree obtaining a precise value for the first notion and obtaining some estimates for the second the results indicate agreement of the first eigenvalue of the infinite hypertree with the second eigenvalue of a random hypergraph of the same degree to within logarithmic factors at least for the first notion of first eigenvalue author s abstract

universal properties of group actions on locally compact spaces abstract we study universal properties of locally compact g spaces for countable infinite groups g in particular we consider open invariant subsets of the g space g and their minimal closed invariant subspaces these are locally compact free g spaces and the latter are also minimal we examine the properties of these g spaces with emphasis on their universal properties as an example of our results we use combinatorial methods to show that each countable infinite group admits a free minimal action on the locally compact non compact cantor set

on the structure of simple c algebras tensored with a uhf algebra ii let d be a simple unital c algebra let b be a uhf algebra and put a b d it is proved that if p q a are projections and p q for all quasitraces on a then p q in the sense of murray and von neumann a more general result involving positive operators in a is also proved if d has finitely many extremal quasi traces and the projections in d k separate these then it is proved that a has real rank zero finally it is proved that provided d is stably finite then each positive state on k0 d lifts to a quasi trace

extending states on preordered semigroups and the existence of quasitraces on c algebras abstract it is shown that every state on a subsemigroup of a scaled preordered semigroup extends to the whole semigroup extending a theorem of goodearl and handelman as an application every state on k0 a for a unital c algebra a comes from a quasitrace

difference equations isoperimetric inequality and transience of certain random walks the difference laplacian on a square lattice in rn has been stud ied by many authors in this paper an analogous difference operator is studied for an arbitrary graph it is shown that many properties of the laplacian in the continuous setting e g the maximum principle the harnack inequality and cheeger s bound for the lowest eigenvalue hold for this difference oper ator the difference laplacian governs the random walk on a graph just as the laplace operator governs the brownian motion as an application of the theory of the difference laplacian it is shown that the random walk on a class of graphs is transient the random walks we consider are defined as follows let k be a connected graph i e a one dimensional simplicial complex for a vertex x e k let m x denote the number of edges emanating from x the probability that a particle moves from x to another vertex y e k is l m x if x and y are connected by an edge and it is zero otherwise as observed by courant friedrichs and lewy cfl for the case of a square lattice in the plane this random walk is intimately related to the difference analog of the laplacian

flow by mean curvature of convex surfaces into spheres the motion of surfaces by their mean curvature has been studied by brakke 1 from the viewpoint of geometric measure theory other authors investigated the corresponding nonparametric problem 2 5 9 a reason for this interest is that evolutionary surfaces of prescribed mean curvature model the behavior of grain boundaries in annealing pure metal in this paper we take a more classical point of view consider a compact uniformly convex w dimensional surface m mo without boundary which is smoothly imbedded in r let mo be represented locally by a diffeomorphism

an index theorem on open manifolds ii on decrit et on applique un nouveau theoreme d indice pour des operateurs elliptiques sur certaines varietes non compactes on calcule un indice a valeur reelle pour des operateurs de type dirac sur des varietes non compactes de geometrie bornee

self similar solutions and translating solitons for lagrangian mean curvature flow we construct many self similar and translating solitons for lagrangian mean curvature flow including self expanders and translating solitons with arbitrarily small oscillation on the lagrangian angle our translating solitons play the same role as cigar solitons in ricci flow and are important in studying the regularity of lagrangian mean curvature flow given two transverse lagrangian planes wn in cn with sum of characteristic angles less than a we show there exists a lagrangian self expander asymptotic to this pair of planes the maslov class of these self expanders is zero thus they can serve as local models for surgeries on lagrangian mean curvature flow families of self shrinkers and self expanders with different topologies are also constructed this paper generalizes the work of anciaux i joyce 12 lawlor 15 and lee and wang 17 18

how to compute the volume in high dimension in some areas of theoretical computer science we feel that randomized algorithms are better and in some others we can prove that they are more efficient than the deterministic ones

self shrinkers for the mean curvature flow in arbitrary codimension in this paper we generalize colding minicozzi s recent results about codimension 1 self shrinkers for the mean curvature flow to higher codimension in particular we prove that the sphere bf s n sqrt 2n is the only complete embedded connected f stable self shrinker in mathbf r n k with mathbf h ne 0 polynomial volume growth flat normal bundle and bounded geometry we also discuss some properties of symplectic self shrinkers proving that any complete symplectic self shrinker in mathbf r 4 with polynomial volume growth and bounded second fundamental form is a plane as a corollary we show that there is no finite time type i singularity for symplectic mean curvature flow which has been proved by chen li using different method we also study lagrangian self shrinkers and prove that for lagrangian mean curvature flow the blow up limit of the singularity may be not f stable

non simple purely infinite c algebras a c algebra a is defined to be purely infiniteif there are no characters on a a nd if for every pair of positive elements a b in a such that b lies in the closed two sided ideal generated by a there exists a sequence rn in a such that rn arn b this definition agrees with the usual definition by j cuntz when a is simple it is shown that the property of being purely infinite is preserved under extensions morita equivalence inductive limits and it passes to quotients and to hereditary sub c algebras it is shown that a is purely infinite for every c algebra a purely infinite c algebras admit no traces and conversely an approximately divisible exact c algebra is purely infinite if it admits no nonzero trace

integral circulant ramanujan graphs via multiplicativity and ultrafriable integers abstract any integral circulant graph icg n d is characterised by its order n and a set d of positive divisors of n in such a way that it has vertex set z n z and edge set a b a b z n z gcd a b n d such graphs are regular and a connected regular graph g is called ramanujan if the second largest modulus of the eigenvalues of the adjacency matrix of g is at most 2 1 in 2010 droll described all ramanujan unitary cayley graphs i e graphs of type x n icg n 1 having the ramanujan property recently le and the author classified all ramanujan graphs icg p s d for prime powers p s and arbitrary divisor sets d we greatly extend the established results to graphs icg n d with arbitrary n and multiplicative divisor set d i we derive a criterion in terms of euler s totient function for icg n d to be ramanujan ii we prove that for all even integers n 2 and a positive proportion of the odd integers n namely those having a dominating prime power factor there exists a multiplicative divisor set d such that icg n d is ramanujan iii we show that the set of odd n for which no ramanujan graph icg n d with multiplicative divisor set d exists viz ultrafriable integers has positive density as well the proofs of ii and iii use methods from analytic number theory

the cone of lower semicontinuous traces on a c algebra the cone of lower semicontinuous traces is studied with a view to its use as an invariant its properties include compactness hausdorffness and continuity with respect to inductive limits a suitable notion of dual cone is given the cone of lower semicontinuous 2 quasitraces on a non exact c algebra is considered as well these results are applied to the study of the cuntz semigroup it is shown that if a c algebra absorbs the jiang su algebra then the subsemigroup of its cuntz semigroup consisting of the purely non compact elements is isomorphic to the dual cone of the cone of lower semicontinuous 2 quasitraces this yields a computation of the cuntz semigroup for the following two classes of c algebras c algebras that absorb the jiang su algebra and have no non zero simple subquotients and simple c algebras that absorb the jiang su algebra 1 introduction the most standard invariants in the classification of nu clear c algebras are their k groups and their traces the traces at least for simple c algebras are assumed to be bounded in the unital case and lower semicontinuous and densely finite in the non unital case if one has in mind the classification of non simple c algebras it is clear that these two kinds of traces will not suffice and a broader class should be considered in this paper we study the properties of the cone of all lower semicontinuous traces on a c algebra with the purpose of applying our results to questions in the classification of non simple c algebras one sign that this is natural is that this cone in particular determines the lattice of closed two sided ideals an important invariant in its own right see 3 4 we shall also consider lower semicontinuous 2 quasitraces since these appear naturally as functionals on the cuntz semigroup of the algebra if the