adding handles to the helicoid there exist two new embedded minimal surfaces asymptotic to the helicoid one is periodic with quotient by orientation preserving translations of genus one the other is nonperiodic of genus one

dominated and pointwise ergodic theorems with weighted averages for bounded lamperti representations of amenable groups abstract group representations by bounded lamperti operators in the spaces l 1 form a wide class of representations including representations by bounded positive operators and when 2 representations by isometric operators the dominated and the pointwise ergodic theorems det and pet for cesaro averages for the bounded lamperti representations of amenable compact locally compact groups in l 1 were proved by a tempelman in proc amer math soc 143 2015 4989 5004 by using a completely different functional analytical method developed by a shulman in his phd thesis in 1988 we generalize this result to weighted averages of such representations and discuss various conditions on the weights under which the det and the pet hold we conclude with applications of the general results to the bounded lamperti representations of groups of polynomial growth and of the groups r m and z m

time regularity for random walks on locally compact groups let g be a compactly generated locally compact group and let t be the operator of convolution with a probability measure on g our main results give sufficient conditions on for the operator t to be analytic in l p g 1 p where analyticity means that one has an estimate of form vert i t t n vert leq cn 1 for all n 1 2 in l p operator norm counterexamples show that analyticity may not hold if some of the conditions are not satisfied

positivity of the universal pairing in 3 dimensions associated to a closed oriented surface s is the complex vector space with basis the set of all compact oriented 3 manifolds which it bounds gluing along s defines a hermitian pairing on this space with values in the complex vector space with basis all closed oriented 3 manifolds the main result in this paper is that this pairing is positive i e that the result of pairing a nonzero vector with itself is nonzero this has bearing on the question of what kinds of topological information can be extracted in principle from unitary 2 1 dimensional tqfts the proof involves the construction of a suitable complexity function c on all closed 3 manifolds satisfying a gluing axiom which we call the topological cauchy schwarz inequality namely that c ab max c aa c bb for all a b which bound s with equality if and only if a b the complexity function c involves input from many aspects of 3 manifold topology and in the process of establishing its key properties we obtain a number of results of independent interest for example we show that when two finite volume hyperbolic 3 manifolds are glued along an incompressible acylindrical surface the resulting hyperbolic 3 manifold has minimal volume only when the gluing can be done along a totally geodesic surface this generalizes a similar theorem for closed hyperbolic 3 manifolds due to agol storm thurston

thermodynamic properties of fully connected q ising neural networks the thermodynamic and retrieval properties of fully connected q ising networks are studied in the replica symmetric mean field approximation in particular capacity gain parameter and capacity temperature phase diagrams are derived for q 3 4 and q infinity and different distributions of the stored patterns furthermore the optimal gain function is determined in order to obtain the best performance where appropriate the results are compared with the diluted and layered versions of these models

content addressability and learning in neural networks the content addressability of patterns stored in ising spin neural network models with symmetric interactions is studied numerical results from simulations on the icl distributed array processor dap involving systems with up to 2048 neurons are presented behaviour consistent with finite size scaling characteristic of a first order phase transition is shown to be exhibited by the basins of attraction of the stored patterns both in the case of the hopfield model and for systems using a local iterative learning algorithm designed to optimise the basins of attraction estimates are obtained for the critical minimum overlaps which an input pattern must have with a stored pattern in order to successfully retrieve it

the structure of singly periodic minimal surfaces the only properly embedded minimal surface with more than one topological end and a cont inuous g roup of symmetries is the catenoid considering periodic minimal surfaces with more than one end a surface in n 3 is periodic if it is invariant under an infinite discrete g roup of isometrics that act freely on 1r3 the only classical examples are those discovered by riemann the riemann examples possess a quite special set of properties 1 they have an infinite number of flat annular ends 1 2 they are invariant under a nontrivial t ranslat ion t 3 the surfaces t h a v e genus 1 and 2 ends 4 the surfaces t have total curvature equal to 8 zr mot iva ted by the riemann examples we established the existence of an infinite family of proper ly embedded periodic minimal surfaces d k each with an infinite number of ends 1 these surfaces which are discussed below see sect 1 2 on rigidity also have an infinite number of flat annular ends and are invariant under a t ranslat ion t the quot ient of these surfaces by t is a surface of genus 2 k 1 with two ends and total curvature 8 r e k 1 k l we have been able to twis t each of these new surfaces that is we construct surfaces with the same properties that are invariant under screw motions t r where r is a rota t ion a round an axis in the direction of t see sect 1 3 below together with the catenoid and the examples of riemann these new surfaces comprise all the known properly embedded minimal surfaces with more than one topological end whose symmetry g roup contains an infinite discrete subgroup these examples share m a n y of the properties of the riemann examples a major goal of this paper is to show that this is not accidental

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

doubly power bounded operators on lp 2 p 1 abstract we show the existence of a doubly power bounded t on l p 1 p p 2 such that t is spectral of scalar type hence polynomially bounded t is not similar to a lamperti operator hence is not similar to an isometry none of the powers of t is similar to a lamperti operator none of the powers is similar to a positive operator and for some f l p the averages 1 n k 1 n t k f or the averages along the primes or the squares fail to be a e convergent

on the overlap dynamics of multi state neural networks with a finite number of patterns neural networks with multi state neurons are studied in the case of low loading for symmetric couplings satisfying a certain positivity condition a lyapunov function is shown to exist in the space of overlaps between the instantaneous microscopic state of the system and the learned patterns furthermore an algorithm is derived for zero temperature to determine all the fixed points as an illustration the three state model is worked out explicitly for hebbian couplings for finite temperature the time evolution of the overlap is studied for couplings which need not be symmetric the stability properties are discussed in detail for the three state model for asymmetric couplings limit cycle behaviour is shown to be possible

learning grey toned patterns in neural networks the problem of learning multi state patterns in neural networks is investigated an analysis of the space of couplings gardner approach yields the distribution of local fields the critical storage capacity alpha c and the minimum number of errors for an overloaded network for noisy local fields the classification error is minimized if the local fields of the patterns are allowed to lie in intervals of finite width a fast converging adaptive learning algorithm is presented which finds the coupling matrix of optimal stability

heegaard surfaces and measured laminations i the waldhausen conjecture we give a proof of the so called generalized waldhausen conjecture which says that an orientable irreducible atoroidal 3 manifold has only finitely many heegaard splittings in each genus up to isotopy jaco and rubinstein have announced a proof of this conjecture using different methods

euclidean decompositions of noncompact hyperbolic manifolds on introduit une methode pour diviser une variete hyperbolique non compacte de volume fini en morceaux euclidiens canoniques

embedded minimal surfaces exotic spheres and manifolds with positive ricci curvature let n be a three dimensional riemannian manifold let e be a closed embedded surface in n then it is a question of basic interest to see whether one can deform in its isotopy class to some canonical embedded surface from the point of view of geometry a natural canonical surface will be the extremal surface of some functional defined on the space of embedded surfaces the simplest functional is the area functional the extremal surface of the area functional is called the minimal surface such minimal surfaces were used extensively by meeks yau my21 in studying group actions on three dimensional manifolds in my2 the theory of minimal surfaces was used to simplify and strengthen the classical dehn s lemma loop theorem and the sphere theorem in the setting there one minimizes area among all immersed surfaces and proves that the extremal object is embedded in this paper we minimize area among all embedded surfaces isotopic to a fixed embedded surface in the category of these surfaces we prove a general existence theorem theorem 1 a particular consequence of this theorem is that for irreducible manifolds an embedded incompressible surface is isotopic to an embedded incompressible surface with minimal area we also prove that there exists an embedded sphere of least area enclosing a fake cell provided the complementary volume is not a standard ball and provided there exists no embedded one sided rp2 by making use of the last result and a cutting and pasting argument we are able to settle a well known problem in the theory of three dimensional manifolds we prove that the covering space of any irreducible orientable three dimensional manifold is irreducible it is possible to exploit our existence theorem to study

storing an extensive number of grey toned patterns in a neural network using multistate neurons grey toned patterns are pictures composed of pixels of several shades of grey the ability of neural networks using multistate neurons to store such patterns is systematically investigated if conventional generalizations of hopfield networks using analogue or soft neurons are considered it is impossible to stabilize these grey tones nevertheless it is shown that it can be done with networks that use neurons which have only a discrete set of possible activities this is demonstrated for the pseudo inverse rule for the synaptic couplings where only the stability of the patterns shrinks with increasing number q of grey tones one wants to store if the patterns are uncorrelated one can use the hebb rule and in this case the mean field theory is presented applying this rule the storage capacity decreases as q 2 with the number of grey tones

nilprogressions and groups with moderate growth abstract we show that doubling at some large scale in a cayley graph implies uniform doubling at all subsequent scales the proof is based on the structure theorem for approximate subgroups proved by green tao and the first author we also give a number of applications to the geometry and spectrum of finite cayley graphs for example we show that a finite group has moderate growth in the sense of diaconis and saloff coste if and only if its diameter is larger than a fixed power of the cardinality of the group we call such groups almost flat and show that they have a subgroup of bounded index admitting a cyclic quotient of comparable diameter we also give bounds on the cheeger constant first eigenvalue of the laplacian and mixing time this can be seen as a finite group version of gromov s theorem on groups with polynomial growth it also improves on a result of lackenby regarding property in towers of coverings another consequence is a universal upper bound on the diameter of all finite simple groups independent of the cfsg

on the parallel dynamics of the q state potts and q ising neural networks using a probabilistic approach the parallel dynamics of theq state potts andq ising neural networks are studied at zero and at nonzero temperatures evolution equations are derived for the first time step and arbitraryq these formulas constitute recursion relations for the exact parallel dynamics of the extremely diluted asymmetric versions of these networks an explicit analysis including dynamical capacity temperature diagrams and the temperature dependence of the overlap is carried out forq 3 both types of models are compared

stabilizing and destabilizing heegaard splittings of sufficiently complicated 3 manifolds let m1 and m2 be compact orientable 3 manifolds with incompressible boundary and m the manifold obtained by gluing with a homeomorphism phi partial m 1 to partial m 2 we analyze the relationship between the sets of low genus heegaard splittings of m1 m2 and m assuming the map phi is sufficiently complicated this analysis yields counter examples to the stabilization conjecture a resolution of the higher genus analogue of a conjecture of gordon and a result about the uniqueness of expressions of heegaard splittings as amalgamations

transference results on weighted lebesgue spaces we present some transference results for a convolution operator with kernel k which is bounded from lp0 w0 into lp1 w1 our results are a natural extension of theclassical results of coifman an

almost everywhere convergence of powers of some positive lp contractions abstract we extend the solution of burkholder s conjecture for products of conditional expectations obtained by delyon and delyon for l 2 and by cohen for l p 1 p to the context of badea and lyubich let t be a finite convex combination of operators t j which are products of finitely many conditional expectations then t n f converges a e for every f l p 1 p with sup n t n f l p the proof uses the work of le merdy and xu on positive l p contractions satisfying ritt s resolvent condition as another application of the work of le merdy and xu we extend a result of bellow jones and rosenblatt proving that if a probability a k k z has bounded angular ratio then for every positive invertible isometry s of an l p space 1 p the operator t k z a k s k is a positive l p contraction such that for every f l p t n f converges a e and sup n t n f l p if a k is supported on n the same result is true when s is only a positive contraction of l p similar results are obtained for averages of bounded continuous representations of a compact lca group by positive operators in one l p space 1 p for a positive contraction t on l p which satisfies ritt s condition and f i t l p 0 1 we prove that n t n f 0 a e and sup n n t n f l p

unstable minimal surfaces and heegaard splittings in recent years minimal surfaces have played an important role in the study of three dimensional manifolds least area surfaces which minimize area in their isotopy 15 homotopy 23 or homology 10 classes have been used to make progress towards the understanding of three manifolds which contain such surfaces such surfaces can be shown to exist and to have strong regularity properties under the assumption that the homotopy isotopy or homology class in which they minimize area is non trivial however these techniques tend not to be applicable to the study of surfaces which are not essential in one of the above classes or to the study of manifolds which contain only such nonessential surfaces recently developed techniques open the way for the use of a new class of surfaces in three dimensional manifolds namely the class of unstable embedded minimal surfaces pitts and rubinstein have established the existence of unstable minimal submanifolds which arise from a heegaard splitting in such a way that their genus is less than or equal to that of the heegaard splitting 19 in this paper we extend their result to obtain greater topological control of the resulting minimal surface and apply the resulting theorem to purely topological problems relating to heegaard splittings of a three manifold a compact manifold is a handlebody if it is homeomorphic to a regular neighborhood of a graph in r 3 a heegaard splitting of a three manifold m is a triple f h1 h2 where f is an embedded connected orientable surface in m h1 and h 2 are handlebodies embedded in m m h 1 w h 2 and h c h2 f o h i o h 2 f is called a heegaard surface we say that two heegaard splittings of m f h1 h2 and f h i h are equivalent if there is a homeomorphism h m m such that f f f f h o h i and f h2 h by considering neighborhoods of triangulations one can show that any closed orientable three manifold admits a heegaard splitting of particular interest are the splittings with heegaard surface of smallest possible genus the genus of such a splitting is called the heegaard genus of the three manifold

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

heegaard structure respects complicated jsj decompositions let m be a 3 manifold with torus boundary components t 1 and t 2 let phi t 1 rightarrow t 2 be a homeomorphism m phi the manifold obtained from m by gluing t 1 to t 2 via the map phi and t the image of t 1 in m phi we show that if phi is sufficiently complicated then any incompressible or strongly irreducible surface in m phi can be isotoped to be disjoint from t it follows that every heegaard splitting of a 3 manifold admitting a sufficiently complicated jsj decomposition is an amalgamation of heegaard splittings of the components of the jsj decomposition

weighted lorentz spaces and the hardy operator abstract we find a new expression for the norm of a function in the weighted loreritz space with respect to the distribution funetion and obtain as a simple consequence a generalization of the classical embeddings lp 1 lp lp and a new definition of the weak space p u w we also give some applications to the boundedness of the hardy operator s x0 from p0u0 w0 into p1u1 w1 with 0

the strong halfspace theorem for minimal surfaces d hoffman and w h meeks iii department of mathematics university of massachusetts amherst ma 01003 usa in 6 jorge and xavier constructed examples of complete nonplanar minimal surfaces contained between two parallel planes in 3 recently rosenberg and toubiana have found complete minimally immersed annuli that are proper in an open slab 1 12 on the basis of these and other results the question has arisen as to whether or not there could exist a properly immersed minimal surface in 3 that was contained between two parallel planes the answer is negative an immediate consequence of the following theorem

the classical theory of minimal surfaces we present here a survey of recent spectacular successes in classical minimal surface theory we highlight this article with the theorem that the plane the helicoid the catenoid and the one parameter family frtgt2 0 1 of riemann minimal examples are the only complete properly embedded minimal planar domains in r 3 the proof of this result depends primarily on work of colding

equivariant minimax and minimal surfaces in geometric three manifolds minimal surfaces in riemannian three dimensional manifolds have played a major role in recent studies of the geometry and topology of 3 manifolds in particular stable and least area minimal surfaces have been used extensively sy msy hs on the other hand explicit examples of unstable minimal surfaces have rarely been given for the 3 sphere s with the standard metric of constant sectional curvature there is the classical paper of lawson lh showing that closed orientable surfaces of every genus occur as unstable minimal surfaces embedded in 5 3 more recently karcher pinkall and sterling kps have constructed several new examples in s we have discovered new infinite families of minimal surfaces in s more generally we announce a number of new finite and infinite families of embedded minimal surfaces in geometric 3 manifolds using the minimax procedure described below geometric structures on 3 manifolds were introduced by thurston tw see also the excellent survey by scott sp1 there are eight geometries r s s x r nil h 2 x r s l r solv and h a geometric structure on a 3 manifold s is a representation of e as a quotient of one of the above eight spaces divided out by a covering transformation group acting isometrically equivalently e is locally isometric to one of these spaces with its natural homogeneous space structure the first six of these geometries give seifert fiber spaces and we are mainly interested in such examples the so 2 isometry actions associated with most seifert fiber spaces yield infinite classes of embedded minimal surfaces note that interesting examples can also be obtained in the other two geometries e g in hyperbolic geometry h 3 pr3 2 the minimax procedure has proved to be a versatile and powerful means of constructing unstable minimal surfaces in 3 manifolds for basic details of this technique see pj ss p r l and pr2 suppose that g is a finite group of isometries acting on a closed oriented riemannian 3 manifold e assume that a is a heegaard surface in e i e the closures of the components of e a are handlebodies k and k assume furthermore that a is g equivariant i e ga a for all g e g we consider one parameter smooth families at t 0 1 sweeping out e and having the following properties ao and ai are graphs at is isotopic to a for all 0 t 1 at is g equivariant for all t the handlebody kt is chosen so that the orientation on at is induced from that on

covering spaces of 3 orbifolds let o be a compact orientable 3 orbifold with non empty singular locus and a finite volume hyperbolic structure equivalently the interior of o is the quotient of hyperbolic 3 space by a lattice in psl 2 c with torsion then we prove that o has a tower of finite sheeted covers oi with linear growth of p homology for some prime p this means that the dimension of the first homology with mod p coefficients of the fundamental group of oi grows linearly in the covering degree the proof combines techniques from 3 manifold theory with group theoretic methods including the golodshafarevich inequality and results about p adic analytic pro p groups this has several consequences firstly the fundamental group of o has at least exponential subgroup growth secondly the covers oi have positive heegaard gradient thirdly we use it to show that a group theoretic conjecture of lubotzky and zelmanov would imply that o has large fundamental group this implication uses a new theorem of the author which will appear in a forthcoming paper these results all provide strong evidence for the conjecture that any closed orientable hyperbolic 3 orbifold with non empty singular locus has large fundamental group many of the above results apply also to 3 manifolds commensurable with an orientable finite volume hyperbolic 3 orbifold with non empty singular locus this includes all closed orientable hyperbolic 3 manifolds with rank two fundamental group and all arithmetic 3 manifolds

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

multiplicity of metastable retrieval phases in networks of multistate neurons it is demonstrated that networks of multistate neurons storing an ensemble of multistate patterns exhibit a multiplicity of metastable retrieval phases these phases are described by solutions of fixed point equations with characteristic retrieval errors they emerge if the gain of the neural input output relation is varied the number of these phases increases asq2 with the numberq of gray levels available to each neuron implications for the optimal gain function and basins of attraction are briefly discussed again networks endowed with pseudoinverse couplings are found to perform better than networks with hebbian couplings at moderate loading levels phases with retrieval errors are destabilized whereas the error free phase remains stable up to the theoretically possible maximum if the gain parameter is properly chosen

surface subgroups of kleinian groups with torsion we prove that every finitely generated kleinian group that contains a finite non cyclic subgroup either is finite or virtually free or contains a surface subgroup hence every arithmetic kleinian group contains a surface subgroup

0 efficient triangulations of 3 manifolds efficient triangulations of 3 manifolds are defined and studied it is shown that any triangulation of a closed orientable irreducible 3 manifold m can be modified to a 0 efficient triangulation orm can be shown to be one of the manifolds s 3 rp 3 or l 3 1 similarly any triangulation of a com pact orientable irreducible irreducible 3 manifold can be modified to a 0 efficient triangulation the notion of a 0 efficient ideal tr is de fined it is shown if m is a compact orientable irreducible irreducible 3 manifold having no essential annuli and distinct from the 3 cell then m admits an ideal triangulation furthermore it is shown that any ideal trian gulation of such a 3 manifold can be modified to a 0 efficient ideal triangula tion a 0 efficient triangulation of a closed manifold has only one vertex or the manifold is s 3 and the triangulation has precisely two vertices 0 efficient tri angulations of 3 manifolds with boundary and distinct from the 3 cell have all their vertices in the boundary and then just one vertex in each bound ary component as tools we introduce the concepts of barrier surface and shrinking as well as the notion of crushing a triangulation along a normal sur face a number of applications are given including an algorithm to construct an irreducible decomposition of a closed orientable 3 manifold an algorithm to construct a maximal collection of pairwise disjoint normal 2 spheres in a closed 3 manifold an alternate algorithm for the 3 sphere recognition prob lem results on edges of low valence in minimal triangulations of 3 manifolds and a construction of irreducible knots in closed 3 manifolds

heegaard splittings the virtually haken conjecture and property τ the behaviour of finite covers of 3 manifolds is a very important but poorly understood topic there are three increasingly strong conjectures in the field that have remained open for over twenty years the virtually haken conjecture the positive virtual b1 conjecture and the virtually fibred conjecture any of these would have profound ramifications for 3 manifold theory in this paper we explore the interaction of these conjectures with the following seemingly unrelated areas eigenvalues of the laplacian and heegaard splittings

the existence of embedded minimal surfaces and the problem of uniqueness it was in the early nineteen thirties that douglas and radd solved the plateau problem for a rectifiable jordan curve in euclidean space the minimal surfaces that they obtained are realized by conformal harmonic maps from the unit disk in 1948 morrey generalized the theorem of douglas and radd to minimal surfaces in a homogeneously regular riemannian manifold in both cases the solutions are defective as they may possess branch points it was not until 1968 that osserman 24 was able to prove that interior branch points do not exist on the douglas radd morrey solution osserman s theorem in this full generality was proved by alt 2 3 and gulliver 8 if the jordan curve is real analytic gulliver and lesley 9 showed that the solution surface is free of branch points even on its boundary hence at least in this case we know that the douglas radd morrey solution is an immersion and represents a classical surface however the douglas radd morrey solution may in general have selfintersections it was then a classical problem to find conditions on the jordan curve to guarantee that the solution is an embedded disk in 1932 radd was the first to produce a general condition of this type he proved that if the curve can be projected onto a convex curve in the plane then it bounds a unique embedded minimal disk which is a graph over this plane it was not until the seventies that there was further progress on the problem of embeddedness osserman had conjectured that if a jordan curve is extreme that is lies on the boundary of its convex hull then the douglas rad6 solution is embedded gulliver and spruck 10 proved this conjecture under the additional assumption that the absolute total curvature of the jordan curve is less than 4 z right after this work tomi and tromba 32 showed that an extreme curve must bound an embedded minimal disk which need not be stable in 1976 almgren and simon 1 established the existence for such a curve of an embedded stable minimal disk which need not be a douglas radd solution finally in the same year the authors 18 proved that any douglas rad6 solution must be embedded

embeddability in the 3 sphere is decidable we show that the following algorithmic problem is decidable given a 2 dimensional simplicial complex can it be embedded topologically or equivalently piecewise linearly in r3 by a known reduction it suffices to decide the embeddability of a given triangulated 3 manifold x into the 3 sphere s3 the main step which allows us to simplify x and recurse is in proving that if x can be embedded in s3 then there is also an embedding in which x has a short meridian that is an essential curve in the boundary of x bounding a disk in s3 b x with length bounded by a computable function of the number of tetrahedra of x

heegaard splittings of seifert fibered spaces with boundary we give the classification theorem for heegaard splittings of fiberwise orientable seifert fibered spaces with nonempty boundary a thin position argument yields a reducibility result which by induction shows that all heegaard splittings of such manifolds are vertical in the sense of lustig moriah algebraic arguments allow a classification of the vertical heegaard splittings

characterizing dehn surgeries on links via trisections we summarize and expand known connections between the study of dehn surgery on links and the study of trisections of closed smooth 4 manifolds in particular we propose a program in which trisections could be used to disprove the generalized property r conjecture including a process that converts the potential counterexamples of gompf scharlemann and thompson into genus four trisections of the standard 4 sphere that are unlikely to be standard we also give an analog of the casson gordon rectangle condition for trisections that obstructs reducibility of a given trisection

neural networks that use three state neurons the dynamics of a neural network which uses three state neurons 1 0 and 1 is solved exactly in the limit of non symmetric and highly dilute synapses recursion relations for the activity the fraction of non zero neurons and overlap of the network with a given pattern are derived which have three generic kinds of fixed points a retrieval fixed point a chaotic fixed point which corresponds to non zero activity but no overlap and a zero fixed point where all the neurons go to the 0 state as the non retrieval fixed points both have activities different from the retrieval fixed point one can easily tell whether a pattern has been recovered an analysis of which fixed points occur as a function of the thresholds and the storage ratio of the system yields remarkably rich phase diagrams optimising the threshold level can be very important especially when low activity patterns are stored a similar analysis can be applied to biased networks using two state 1 0 neurons finally one finds that mixture states which have an overlap with two patterns can be stabilised by a threshold in the networks using three state neurons this property allows larger higher activity memories to be naturally constructed out of smaller ones

on restriction of maximal multipliers in weighted settings we obtain restriction results of k de leeuw s type for maximal operators defined through fourier multipliers of either strong or weak type for weighted l p spaces with 1 p infinity applications to the case of hormander mihlin multipliers singular integral operators and bochner riesz sums are given

a finite size scaling investigation for q state hopfield models storage capacity and basins of attraction the storage capacity of a q state hopfield network is determined via finite size scaling for parallel dynamics and q or 8 the results are in good agreement with theoretical predictions by rieger the basins of attraction and other associative memory properties are discussed for q 4 6 a self controlling q state model with improved basins of attraction is proposed

on boundedness of discrete multilinear singular integral operators let m be a measurable locally bounded function defined in r2 let 1 p1 q1 p2 q2 such that pi 1 implies qi let also 0 p3 q3 and 1 p 1 p1 1 p2 1 p3 we prove the following transference result the operator cm f g x r rf g m e2 ix d d initially defined for integrable functions with compact fourier support extends to a bounded bilinear operator from lp1 q1 r lp2 q2 r into lp3 q3 r if and only if the family of operators dm t p a b n t1p 1212 1212p q m t t e2 in d d initially defined for finite sequences a ak1 k1 z b bk2 k2 z where p k1 zak1e 2 ik1 and q k2 zbk2e 2 ik2 extend to bounded bilinear operators from lp1 q1 z lp2 q2 z into lp3 q3 z with norm bounded by uniform constant for all t 0 we apply this result to prove boundedness of the discrete bilinear hilbert transforms and other related discrete multilinear singular integrals including the endpoints

effective finiteness of irreducible heegaard splittings of non haken 3 manifolds the main result is a short effective proof of tao li s theorem that a closed non haken hyperbolic 3 manifold n has at most finitely many irreducible heegaard splittings along the way we show that n has finitely many branched surfaces of pinched negative sectional curvature carrying all closed index 1 minimal surfaces this effective result together with the sequel with daniel ketover solves the classification problem for heegaard splittings of non haken hyperbolic 3 manifolds

optimal storage properties of neural network models the authors calculate the number p alpha n of random n bit patterns that an optimal neural network can store allowing a given fraction f of bit errors and with the condition that each right bit is stabilised by a local field at least equal to a parameter k for each value of alpha and k there is a minimum fraction fmin of wrong bits they find a critical line alpha c k with alpha c 0 2 the minimum fraction of wrong bits vanishes for alpha alpha c k the calculations are done using a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is locally stable in a finite region of the k alpha plane including the line alpha c k but there is a line above which the solution becomes unstable and replica symmetry must be broken

the space of embedded minimal surfaces of fixed genus in a 3 manifold iii planar domains this paper is the third in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed but arbitrary closed 3 manifold in cm3 cm5 we describe the case where the surfaces are topologically disks on any fixed small scale although the focus of this paper general planar domains is more in line with cm6 we will prove a result here namely corollary iii 3 5 below which is needed in cm5 even for the case of disks roughly speaking there are two main themes in this paper the first is that stability leads to improved curvature estimates this allows us to find large graphical regions these graphical regions lead to two possibilities

finite state neural networks a step toward the simulation of very large systems neural networks composed of neurons withq n states and synapses withq states are studied analytically and numerically analytically it is shown that these finite state networks are much more efficient at information storage than networks with continuous synapses in order to take the utmost advantage of networks with finite state elements a multineuron and multisynapse coding scheme is introduced which allows the simulation of networks having 1 0 109 couplings at a speed of 7 1 109 coupling evaluations per second on asingle processor of the cray ymp a local learning algorithm is also introduced which allows for the efficient training of large networks with finite state elements

eigenvalues of the laplacian the first betti number and the congruence subgroup problem theorem 1 1 let f be an arithmetic lattice in the real lie group so n 1 n 2 if n 7 then assume r does not come from the twisted forms 3 6d4 if n 3 and f comes from the units of a quaternion algebra over a number field l with a unique complex embedding then assume l has a subfield of index 2 then for some finite index congruence subgroup fo of r the commutator quotient is infinite equivalently 1 fo the first betti number of the manifold fo hn is nonzero when 1 so n 1 so n is the n dimensional hyperbolic space

on the complexity of the heegaard structure of hyperbolic 3 manifolds a well known result of waldhausen 22 states that all heegaard splittings of s 3 and of s 2 s 1 of a given genus g are homeomorphic and even isotopic recent work of various authors see 20 19 2 shows that there are more simple 3 manifolds where such a uniqueness holds however this is not true for most 3 manifolds see 12 15 pitts rubinstein 18 have announced a proof of the waldhausen conjecture 23 that for any 3 manifold m the maximal number hm g of non homeomorphic heegaard splittings is finite for any given genus g n this was shown by hass 10 for genus 2 manifolds and g 2 and by johannson 11 for haken manifolds it is expected that the number hm g grows for generic m i e not the above mentioned simple ones with the complexity of m in this note we consider heegaard splittings of minimal genus gmin m and show that for a large class of hyperbolic 3 manifolds hm gmin m grows exponentially theorem a let k s 3 be the montesinos knot m e 1 1 g g such that all i are mutually distinct odd primes and let g c d 1 g 22g 1 1 g let m be the closed 3 manifold obtained from k by m n surgery where m is even then for each m tupel e1 eg 1 1 g 1 1 there exists a minimal genus heegaard splitting y e1 eg and these are pairwise non homeomorphic

critical heegaard surfaces in this paper we introduce critical surfaces which are described via a 1 complex whose definition is reminiscent of the curve complex our main result is that if the minimal genus common stabilization of a pair of strongly irreducible heegaard splittings of a 3 manifold is not critical then the manifold contains an incompressible surface conversely we also show that if a non haken 3 manifold admits at most one heegaard splitting of each genus then it does not contain a critical heegaard surface in the final section we discuss how this work leads to a natural metric on the space of strongly irreducible heegaard splittings as well as many new and interesting open questions

new results on restriction of fourier multipliers we develop an extension of the transference methods introduced by r coifman and g weiss and apply it to study the problem of the restriction of fourier multipliers between rearrangement invariant spaces obtaining natural extensions of the classical de leeuw s result and its further extension to maximal fourier multipliers due to c kenig and p tomas

an unknotting result for complete minimal surfaces r3 in this note we prove theorem if s is a complete properly embedded minimal surface in r 3 with the euclidean metric and if a is a closed connected complementary domain of z then either a is metrically r 2 0 hi or up to diffeomorphism a is made from a countable or finite collection of building blocks by attaching a proper collection of closed 1 handles each building blocks is a closed 3 ball or an open annulus x 0 1 the first theorem of this kind was b lawson s i l unknottedness theorem for minimal surfaces in s 3 that result was obtained by considering the second variation of arc length because the flat euclidean metric of r 3 does not contribute a favorable curvature term as found in s 3 a dual approach employing stable minimal surfaces dual to lawson s spanning arcs is required the key technical result proved by an analysis of the second variation operator on surfaces is that geometric planes are the only stable minimal surfaces in r 3 c p and f c s from this starting point a more topological theory has been developed by frohman hoffman meeks yau and others our argument is based to a large degree on the methods used by c frohman and w meeks f m to classify 1 ended proper complete minimal imbeddings in r 3 we present four succinct topological assertions about any 3 manifold with boundary a s which is a connected closed complementary region of a complete proper minimal surface in r 3 these statements should be thought of as axioms for this emerging theory deduced from geometry they help to mark the boundary with topology the theorem follows quickly from the axioms axiom 1 occurs explicitly as corollary 3 2 i f m axiom 2 is an immediate consequence of the methods of f m and axiom 3 is proved by an extension of these methods axiom 4 is related to an idea on the ordering of ends m y due to w meeks and s t yau these are the axioms their verification actually only requires that the mean curvature vector of 2 be everywhere zero or pointing towards a

three dimensional manifolds kleinian groups and hyperbolic geometry 1 a conjectural picture of 3 manifolds a major thrust of mathematics in the late 19th century in which poincare had a large role was the uniformization theory for riemann surfaces that every conformai structure on a closed oriented surface is represented by a riemannian metric of constant curvature for the typical case of negative euler characteristic genus greater than 1 such a metric gives a hyperbolic structure any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions the exceptional cases the sphere and the torus have spherical and euclidean structures three manifolds are greatly more complicated than surfaces and i think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 or more except perhaps for the fact that so many 3 manifolds are beautiful the situation has changed so that i feel fairly confident in proposing the

least area incompressible surfaces in 3 manifolds let m be a riemannian manifold and let f be a closed surface a map f f m is called least area if the area of f is less than the area of any homotopic map from f to m note that least area maps are always minimal surfaces but that in general minimal surfaces are not least area as they represent only local stationary points for the area function the existence of least area immersions in a homotopy class of maps has been established when the homotopy class satisfies certain injectivity conditions on the fundamental group 18 17 in this paper we shall consider the possible singularities of such immersions our results show that the general philosophy is that least area surfaces intersect least meaning that the intersections and self intersections of least area immersions are as small as their homotopy classes allow when measured correctly one should note that evidence supporting this view had been found by meeks yau in their embedding theorems for minimal disks and 2 spheres 13 143 our main result asserts that if a least area immersion is homotopic to an embedding then it has no self intersections which clearly exemplifies the above philosophy the precise result is the following

averaging and finite size analysis for disorder the hopfield model abstract a finite size scaling study of the capacity problem for the hopfield model is presented questions of identifying the correct shape of the scaling function of corrections to finite size scaling and in particular the problem of properly dealing with disorder are carefully addressed at first order phase transitions like the one considered here relevant physical quantities typically scale exponentially with system size and it is argued that in disordered systems reliable information about the phase transition can therefore be obtained only by averaging their logarithm rather than by considering the logarithm of their average an issue reminiscent of the difference between quenched and annealed disorder but previously ignored in the problem at hand our data for the hopfield model yield c 0 141 0 0015 they are thus closer to the results of a recent one and two step replica symmetry breaking rsb analysis and disagree with that of an earlier one step rsb study with those of previous simulations and with that of a recent paper using an infinite step rsb scheme

word hyperbolic dehn surgery in the late 1970 s thurston dramatically changed the nature of 3 manifold theory with the introduction of his geometrisation conjecture and by proving it in the case of haken 3 manifolds 23 the conjecture for general closed orientable 3 manifolds remains perhaps the most important unsolved problem in the subject a weaker form of the conjecture 19 deals with the fundamental group of a closed orientable 3 manifold it proposes that either it contains z z as a subgroup or it is word hyperbolic in the sense of gromov 11 word hyperbolic groups are precisely those groups which satisfy a linear isoperimetric inequality they are of fundamental importance in geometric group theory and have very many useful properties

transference of strong type maximal inequalities by separation preserving representations let x lp ii where 1 p r is a uniformly bounded strongly continuous representation of g in x by separation preserving operators for instance by positivity pre serving operators we show that r transfers to x strong type maximal estimates for sequences of convolution operators defined on lp g if r also has a uniformly bounded version in l2 t then r will transfer to lp t strong type bounds for maximal operators defined by sequences of multipliers which are continuous on the dual group g these theo rems which cease to be valid if the separation preserving hypothesis is removed provide strong type maximal operator counterparts for the corresponding single operator theorems of coifman weiss transference theory one consequence of the second theorem described above is a maximal theorem counterpart of the homomorphism theorem for sin gle multiplier transforms

sharp lorentz space estimates for rough operators we demonstrate the h 1 l 1 2 or l p l p 2 mapping properties of several rough operators in all cases these estimates are sharp in the sense that the lorentz exponent 2 cannot be replaced by any lower number