on the spectral gap for finitely generated subgroups of su 2 we prove the spectral gap property for free subgroups of su 2 generated by elements satisfying a noncommutative diophantine property in particular for free subgroups generated by elements with algebraic entries

the cyclotomic trace and the k theoretic analogue of novikov s conjecture abstract a trace construction the cyclotomic trace is given it associates to algebraic k theory of a group ring or better to waldhausen s a theory equivariant stable homotopy classes of the free loop space of its classifying space the cyclotomic trace detects the borel classes in algebraic k theory of the integers it is used to prove for a wide class of groups that the k theory assembly map is rationally injective this is the k theoretic analogue of novikov s conjecture

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

the coarse baum connes conjecture for spaces which admit a uniform embedding into hilbert space corollary 1 2 let be a finitely generated group if as a metric space with a word length metric admits a uniform embedding into hilbert space and its classifying space b has the homotopy type of a finite cw complex then the strong novikov conjecture holds for i e the index map from k b to k c r is injective corollary 1 2 follows from theorem 1 1 and the descent principle 23 by index theory the strong novikov conjecture implies the novikov conjecture on the homotopy invariance of higher signatures cf 8 for an excellent

on cyclic fixed points of spectra for a finite p group g and a bounded below g spectrum x of finite type mod p the g equivariant segal conjecture for x asserts that the canonical map x g rightarrow x hg from g fixed points to g homotopy fixed points is a p adic equivalence let c p n be the cyclic group of order p n we show that if the c p equivariant segal conjecture holds for a c p n spectrum x as well as for each of its geometric fixed point spectra varphi c p e x for 0 e n then the c p n equivariant segal conjecture holds for x similar results also hold for weaker forms of the segal conjecture asking only that the canonical map induces an equivalence in sufficiently high degrees on homotopy groups with suitable finite coefficients

nonpositive curvature is not coarsely universal we prove that not every metric space embeds coarsely into an alexandrov space of nonpositive curvature this answers a question of gromov geometric group theory cambridge university press cambridge 1993 and is in contrast to the fact that any metric space embeds coarsely into an alexandrov space of nonnegative curvature as shown by andoni et al ann sci ec norm super 4 51 3 657 700 2018 we establish this statement by proving that a metric space which is q barycentric for some q in 1 infty has metric cotype q with sharp scaling parameter our proof utilizes nonlinear metric space valued martingale inequalities and yields sharp bounds even for some classical banach spaces this allows us to evaluate the bi lipschitz distortion of the ell infty grid m infty n 1 ldots m n vert cdot vert infty into ell q for all q in 2 infty from which we deduce the following discrete converse to the fact that ell infty n embeds with distortion o 1 into ell q for q o log n a rigidity theorem of ribe ark mat 14 2 237 244 1976 implies that for every n in mathbb n there exists m in mathbb n such that if m infty n embeds into ell q with distortion o 1 then q is necessarily at least a universal constant multiple of log n ribe s theorem does not give an explicit upper bound on this m but by the work of bourgain geometrical aspects of functional analysis 1985 86 springer berlin 1987 it suffices to take m n and this was the previously best known estimate for m we show that the above discretization statement actually holds when m is a universal constant

property t and rigidity for actions on banach spaces we study property t and the fixed point property for actions on lp and other banach spaces we show that property t holds when l2 is replaced by lp and even a subspace quotient of lp and that in fact it is independent of 1 p we show that the fixed point property for lp follows from property t when 1 p 2 e for simple lie groups and their lattices we prove that the fixed point property for lp holds for any 1 p if and only if the rank is at least two finally we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive banach spaces

class ii neurons display a higher degree of stochastic synchronization than class i neurons we describe the relationship between the shape of the phase resetting curve prc and the degree of stochastic synchronization observed between a pair of uncoupled general oscillators receiving partially correlated poisson inputs in addition to inputs from independent sources we use perturbation methods to derive an expression relating the shape of the prc to the probability density function pdf of the phase difference between the oscillators we compute various measures of the degree of synchrony and cross correlation from the pdf s and use the same to compare and contrast differently shaped prcs with respect to their ability to undergo stochastic synchronization since the shape of the prc depends on underlying dynamical details of the oscillator system we utilize the results obtained from the analysis of general oscillator systems to study specific models of neuronal oscillators it is shown that the degree of stochastic synchronization is controlled both by the firing rate of the neuron and the membership of the prc type i or type ii it is also shown that the circular variance for the integrate and fire neuron and the generalized order parameter for a hippocampal interneuron model have a nonlinear relationship to the input correlation

synchronization of uncoupled oscillators by common gamma impulses from phase locking to noise induced synchronization nonlinear oscillators can mutually synchronize when they are driven by common external impulses two important scenarios are i synchronization resulting from phase locking of each oscillator to regular periodic impulses and ii noise induced synchronization caused by the poisson random impulses but their difference has not been fully quantified here we analyze a pair of uncoupled oscillators subject to common random impulses with gamma distributed intervals which can be smoothly interpolated between the regular periodic and the random poisson impulses their dynamics are characterized by phase distributions frequency detuning lyapunov exponents and information theoretic measures which clearly reveal the differences between the two synchronization scenarios

on the phase reduction and response dynamics of neural oscillator populations weundertake a probabilistic analysis of the response of repetitively firing neural populations to simple pulselike stimuli recalling and extending results from the literature we compute phase response curves prcs valid near bifurcations to periodic firing for hindmarsh rose hodgkin huxley fitzhugh nagumo and morris lecar models encompassing the four generic codimension one bifurcations phase density equations are then used to analyze the role of the bifurcation and the resulting prc in responses to stimuli in particular we explore the interplay among stimulus duration baseline firing frequency and population level response patterns we interpret the results in terms of the signal processing measure of gain and discuss further applications and experimentally testable predictions

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

uniqueness of invariant means for measure preserving transformations for some compact abelian groups x e g t n 2 and hii 1 z2 the group g of topological automorphisms of x has the haar integral as the unique g invariant mean on l x ax this gives a new characterization of lebesgue measure on the bounded lebesgue measurable subsets 8 of rn n 3 it is the unique normalized positive finitely additive measure on 8 which is invariant under isometries and the transformation of rn xi xn xi x2 x2 i x other examples of as well as necessary and sufficient conditions for the uniqueness of a mean on l q x 3 p which is invariant by some group of measure preserving transformations of the probability space x 3 p are described 0 introduction let 8 be the ring of bounded lebesgue measurable sets in rn or in sn the n dimensional unit sphere in rn and let xn be the lebesgue measure on 8 normalized by xn j 1 where j 0 1 or by xn sn 1 respectively the classical characterization by lebesgue of is that it is the unique positive real valued functionf on 8 which satisfies these three conditions a f jn 1 orf sn 1 respectively b f is invariant under isometries c f is countably additive in 1923 banach 1 studied the question of ruziewicz as to whetherf is still unique when c is replaced by c0 f is finitely additive banach gave a negative answer to this question for r 1 r2 and s1 but for r n 3 or sn n 2 this question is still unanswered in this paper we will prove a theorem which comes close to a solution of this problem for rn n 3 theorem 3 10 if f satisfies a b and c0 and if f is invariant under the transformation of r n n 3 given by xi x2 x f xi x2 x2 9 xn then f xn our method of proof will give several other theorems similar to this one in theorem 3 10 we do not need the full strength of b we need only know that f is absolutely continuous with respect to x i e for n e 3 f n 0 whenever n 0 the fact that for rn n 3 or sn n 2 a b and c0 imply this absolute continuity of f is an observation of tarski which is proved like this received by the editors march 3 1980 presented to the society january 10 1981 ams mos subject classifications 1970 primary 28a70 43a07 43a40 1981 american mathematical society 0002 9947 81 0000 0268 04 50

synchronization and desynchronization of self sustained oscillators by common noise we consider the effect of external noise on the dynamics of limit cycle oscillators the lyapunov exponent becomes negative under influence of small white noise what means synchronization of two or more identical systems subject to common noise we analytically study the effect of small nonidentities in the oscillators and in the noise and derive statistical characteristics of deviations from the perfect synchrony large white noise can lead to desynchronization of oscillators provided they are nonisochronous this is demonstrated for the van der pol duffing system

type ii phase resetting curve is optimal for stochastic synchrony the phase resetting curve prc describes the response of a neural oscillator to small perturbations in membrane potential its usefulness for predicting the dynamics of weakly coupled deterministic networks has been well characterized however the inputs to real neurons may often be more accurately described as barrages of synaptic noise effective connectivity between cells may thus arise in the form of correlations between the noisy input streams we use constrained optimization and perturbation methods to prove that the prc shape determines susceptibility to synchrony among otherwise uncoupled noise driven neural oscillators prcs can be placed into two general categories type i prcs are non negative while type ii prcs have a large negative region here we show that oscillators with type ii prcs receiving common noisy input synchronize more readily than those with type i prcs

a spectral gap theorem in simple lie groups we establish the spectral gap property for dense subgroups generated by algebraic elements in any compact simple lie group generalizing earlier results of bourgain and gamburd for unitary groups

correlation transfer in stochastically driven neural oscillators over long and short time scales in the absence of synaptic coupling two or more neural oscillators may become synchronized by virtue of the statistical correlations in their noisy input streams recent work has shown that the degree of correlation transfer from input currents to output spikes depends not only on intrinsic oscillator dynamics but also on the length of the observation window over which the correlation is calculated in this paper we use stochastic phase reduction and regular perturbations to derive the correlation of the total phase elapsed over long time scales a quantity that provides a convenient proxy for the spike count correlation over short time scales we derive the spike count correlation directly using straightforward probabilistic reasoning applied to the density of the phase difference our approximations show that output correlation scales with the autocorrelation of the phase resetting curve over long time scales we also find a concise expression for the influence of the shape of the phase resetting curve on the initial slope of the output correlation over short time scales these analytic results together with numerical simulations provide new intuitions for the recent counterintuitive finding that type i oscillators transfer correlations more faithfully than do type ii over long time scales while the reverse holds true for the better understood case of short time scales

an etale approach to the novikov conjecture we show that the rational novikov conjecture for a group of finite homological type follows from the mod 2 acyclicity of the higson compactification of an e we then show that for groups of finite asymptotic dimension the higson compactification is mod p acyclic for all p and deduce the integral novikov conjecture for these groups 2007 wiley periodicals inc

isomorphism conjectures in algebraic k theory duff in this paper we are concerned with the four functors y y3 if and which map from the category of topological spaces x to the category diff of q spectra the functor 3 or if maps the space x to the qspectrum of stable topological or smooth pseudoisotopies of x the spaces diff dif 3i x id9 x i 0 are hatcher s deloopings of 970 x yd x cf 29 the functor j maps a path connected space x to the algebraic ktheoretic s2 spectrum for the integral group ring z7r1x the spaces x x i 0 are the gersten wagoner deloopings of z o x cf 27 52 5 x is also defined on a nonpath connected space x to be x1 where xi i e i are the path components of x and ii xi indicates the direct limit of all finite products of the i xi i e i if x is path connected then the functor maps x to the l surgery classifying spaces for oriented surgery problems with fundamental group r i1x identified by their fourfold periodicity k 90 x k 4j x cf 35 39 44 47 in addition if x is not path connected then we set y77 x tiei5 fj xi where xi i e i are the path components of x results obtained by the authors over the past five years cf 15 25 and in particular 19 16 a 1 a 18 together with many earlier results and conjectures which are reviewed in 1 6 1 1 6 6 suggest that for any connected cwcomplex x and for 5 denoting any of the above functors the q spectrum x should be computable in a simple way from the values taken by 9 on covering spaces of x which have very simple fundamental groups in more detail we let 9 x denote the collection of all subgroups h c ir ix which are either finite or virtually infinite cyclic a group h is virtually infinite cyclic if there is a short exact sequence of groups 1 z h f 1 with f equal to a finite group and z equal to the infinite cyclic group for each such group we let xh x denote the covering space projection corresponding to h then the q spectrum 5 x should be computable in a simple way from the q spectra 5 xh h e 9 x the purpose of this paper is to formulate precisely a conjecture along these lines and to verify the conjecture for any x

discrete fundamental groups of warped cones and expanders in this paper we compute the discrete fundamental groups of warped cones as an immediate consequence this allows us to show that there exist coarsely simply connected expanders and superexpanders this also provides a strong coarse invariant of warped cones and implies that many warped cones cannot be coarsely equivalent to any box space

rigidity of warped cones and coarse geometry of expanders abstract we study the geometry of warped cones over free minimal isometric group actions and related constructions of expander graphs we prove a rigidity theorem for the coarse geometry of such warped cones namely if a group has no abelian factors then two such warped cones are quasi isometric if and only if the actions are finite covers of conjugate actions as a consequence we produce continuous families of non quasi isometric expanders and superexpanders the proof relies on the use of coarse topology for warped cones such as a computation of their coarse fundamental groups

optimal entrainment with smooth pulse and square signals in weakly forced nonlinear oscillators abstract a physical limit of entrainability of nonlinear oscillators is considered for an external weak signal forcing this limit of entrainability is characterized by the optimization problem maximizing the width of the arnold tongue the frequency locking range versus forcing magnitude under certain practical constraints here we show a solution to this optimization problem thanks to a direct link to holder s inequality this solution defines an ideal forcing realizing the entrainment limit and as the result a fundamental limit of entrainment is clarified as follows for 1 1 entrainment we obtain i a construction of the global optimal forcing and a condition for its uniqueness in l p space with p 1 and ii a construction of the global optimal pulse like forcings in l 1 space and for m n entrainment m n some informations about the non existence of the ideal forcing iii in addition we establish definite algorithms for obtaining the global optimal forcings for 1 p and these pulse like forcings for p 1 these theoretical findings are verified by systematic extensive numerical calculations and simulations

synchrony of limit cycle oscillators induced by random external impulses the mechanism of phase synchronization between uncoupled limit cycle oscillators induced by common random impulsive forcing is analyzed by reducing the dynamics of the oscillator to a random phase map it is shown that phase synchronization generally occurs when the oscillator is driven by weak random impulsive forcing in the limit of large interimpulse intervals the case where the interimpulse intervals are finite is also analyzed perturbatively for small impulse intensity for weak poisson impulses it is shown that the phase synchronization persists up to the first order approximation

on the topological hochschild homology of bu i the purpose of this paper and its sequel is to determine the homotopy groups of the spectrum thh here p is an odd prime is the adams summand of p local connective k theory see for example 25 and thh is the topological hochschild homology construction introduced by bokstedt in 3 in the present paper we will determine the mod p homotopy groups of thh and also the integral homotopy groups of thh l where l denotes the periodic adams summand in the sequel we will investigate the integral homotopy groups of thh using our present results as a starting point the thh construction appears to be of basic importance in algebraic k theory because it combines two useful properties it can be used to construct good approximations to the algebraic k theory functor and it is very accessible to calculation we shall review what is known about the first property in a moment the second property was demonstrated by bokstedt s calculation in his paper 4 of the homotopy groups of thh hz p and thh hz here hz p and hz denote the evident eilenberg mac lane spectra it is natural to ask about thh r for other popular ring spectra r and our work is a first step in this direction we pay special attention to the connective case because this is the case which is likely to be relevant in applications see subsection 1 4 below the calculation which we present in this paper is a homotopy theoretic one which uses the adams spectral sequence hereafter abbreviated ass this calculation has several interesting features in particular it is a pleasing example of an ass calculation in which although there are infinitely many differentials it is still possible to get the complete answer here is a summary of the contents of the paper in section 2 we review the facts we need to know about ordinary hochschild homology in section 3 we do the same

invariant measures for algebraic actions zariski dense subgroups and kazhdan s property t let k be any locally compact non discrete field we show that finite invariant measures for k algebraic actions are obtained only via actions of compact groups this extends both borel s density and fixed point theorems over local fields for semisimple solvable groups resp we then prove that for k algebraic actions finitely additive finite invariant measures are obtained only via actions of amenable groups this gives a new criterion for zariski density of subgroups and is shown to have representation theoretic applications the main one is to kazhdan s property t for algebraic groups which we investigate and strengthen

superrigidity of lattices in solvable lie groups let be a closed cocompact subgroup of a simply connected solvable lie groupg such that adg has the same zariski closure as adg if gln is any finite dimensional representation of we show that virtually extends to a representation ofg by combining this with work of margulis on lattices in semisimple groups we obtain a similar result for lattices in many groups that are neither solvable nor semisimple furthermore we show that if is isomorphic to a closed cocompact subgroup of another simply connected solvable lie groupg then any isomorphism from to extends to a crossed isomorphism fromg tog in the same vein we prove a more concrete form of mostow s theorem that compact solvmanifolds with isomorphic fundamental groups are diffeomorphic