an assignment problem at high temperature physicists have studied the stochastic assignment problem using ideas from statistical mechanics for a version of this problem we give at high enough temperature a complete proof of the existence of the structure they predict

modular elliptic curves and fermat s last theorem when andrew john wiles was 10 years old he read eric temple bell s the last problem and was so impressed by it that he decided that he would be the first person to prove fermat s last theorem this theorem states that there are no nonzero integers a b c n with n 2 such that an bn cn the object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular fermat s last theorem follows as a corollary by virtue of previous work by frey serre and ribet

on the rate of convergence in wasserstein distance of the empirical measure let mu n be the empirical measure associated to a n sample of a given probability distribution mu on mathbb r d we are interested in the rate of convergence of mu n to mu when measured in the wasserstein distance of order p 0 we provide some satisfying non asymptotic l p bounds and concentration inequalities for any values of p 0 and d ge 1 we extend also the non asymptotic l p bounds to stationary rho mixing sequences markov chains and to some interacting particle systems

extra heads and invariant allocations let n be an ergodic simple point process on e d and let n be its palm version thorisson ann probab 24 1996 2057 2064 proved that there exists a shift coupling of and n that is one can select a random point y of n such that translating n by y yields a configuration whose law is that of we construct shift couplings in which y and are functions of and prove that there is no shift coupling in which n is a function of the key ingredient is a deterministic translation invariant rule to allocate sets of equal volume forming a partition of r d to the points of n the construction is based on the gale shapley stable marriage algorithm amer math monthly 69 1962 9 15 next let be an ergodic random element of 0 1 zd and let be r conditioned on r 0 1 a shift coupling x of r and is called an extra head scheme we show that there exists an extra head scheme which is a function of r if and only if the marginal e r 0 is the reciprocal of an integer when the law of is product measure and d 3 we prove that there exists an extra head scheme x satisfying e exp c x d this answers a question of holroyd and liggett ann probab 29 2001 1405 1425

on arithmetic fundamental lemmas we present a relative trace formula approach to the gross zagier formula and its generalization to higher dimensional unitary shimura varieties as a crucial ingredient we formulate a conjectural arithmetic fundamental lemma for unitary rapoport zink spaces we prove the conjecture when the rapoport zink space is associated to a unitary group in two or three variables

an algorithm for the n lyapunov exponents of an n dimensional unknown dynamical system abstract an algorithm for estimating lyapunov exponents of an unknown dynamical system is designed the algorithm estimates not only the largest but all lyapunov exponents of the unknown system the estimation is carried out by a multivariate feedforward network estimation technique we focus our attention on deterministic as well as noisy system estimation the performance of the algorithm is very satisfactory in the presence of noise as well as with limited number of observations

on the modularity of elliptic curves over q wild 3 adic exercises in this paper building on work of wiles wi and of wiles and one of us r t tw we will prove the following two theorems see 2 2 theorem a if e q is an elliptic curve then e is modular theorem b if gal q q gl2 f5 is an irreducible continuous representation with cyclotomic determinant then is modular we will first remind the reader of the content of these results and then briefly outline the method of proof if n is a positive integer then we let 1 n denote the subgroup of sl2 z consisting of matrices that modulo n are of the form 1 0 1

clustered motion in symplectic coupled map systems clustered motion of particles is found in hamiltonian dynamics of symplectic coupled map systems particles assemble and move with strong correlation the motion is chaotic but is distinguishable from random chaotic motion lyapunov analysis distinguishes global instability from local fluctuations clustered motions have finite lifetime they have fractal geometric structure in the phase space as the orbits are trapped to ruins of kam tori and islands

continuum limit of total variation on point clouds we consider point clouds obtained as random samples of a measure on a euclidean domain a graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points they connect our goal is to develop mathematical tools needed to study the consistency as the number of available data points increases of graph based machine learning algorithms for tasks such as clustering in particular we study when the cut capacity and more generally total variation on these graphs is a good approximation of the perimeter total variation in the continuum setting we address this question in the setting of convergence we obtain almost optimal conditions on the scaling as the number of points increases of the size of the neighborhood over which the points are connected by an edge for the convergence to hold taking of the limit is enabled by a transportation based metric which allows us to suitably compare functionals defined on different point clouds

diagonal cycles and euler systems ii the birch and swinnerton dyer conjecture for hasse weil artin l functions this article establishes new cases of the birch and swinnerton dyer conjecture in analytic rank 0 for elliptic curves over mathbb q viewed over the fields cut out by certain self dual artin representations of dimension at most 4 when the associated l function vanishes to even order ge 2 at its central point two canonical classes in the corresponding selmer group are constructed and shown to be linearly independent assuming the non vanishing of a garrett hida p adic l function at a point lying outside its range of classical interpolation the key tool for both results is the study of certain p adic families of global galois cohomology classes arising from gross kudla schoen diagonal cycles in a tower of triple products of modular curves

a pde approach to a 2 dimensional matching problem we prove asymptotic results for 2 dimensional random matching problems in particular we obtain the leading term in the asymptotic expansion of the expected quadratic transportation cost for empirical measures of two samples of independent uniform random variables in the square our technique is based on a rigorous formulation of the challenging pde ansatz by caracciolo et al phys rev e 90 012118 2014 that linearizes the monge ampere equation

a test of non identifying restrictions and confidence regions for partially identified parameters abstract we propose an easily implementable test of the validity of a set of theoretical restrictions on the relationship between economic variables which do not necessarily identify the data generating process the restrictions can be derived from any model of interactions allowing censoring and multiple equilibria when the restrictions are parameterized the test can be inverted to yield confidence regions for partially identified parameters thereby complementing other proposals primarily chernozhukov et al chernozhukov v hong h tamer e 2007 estimation and confidence regions for parameter sets in econometric models econometrica 75 1243 1285

rates of convergence for partial mass problems we consider a class of partial mass problems in which a fraction of the mass of a probability measure is allowed to be changed trimmed to maximize fit to a given pattern this includes the problem of optimal partial transportation of mass where a part of the mass need not be transported and also trimming procedures which are often used in statistical data analysis to discard outliers in a sample the data with lowest agreement to a certain pattern this results in a modified trimmed version of the original probability which is closer to the pattern we focus on the case of the empirical measure and analyze to what extent its optimally trimmed version is closer to the true random generator in terms of rates of convergence we deal with probabilities on mathbb r k and measure agreement through probability metrics our choices include transportation cost metrics associated to optimal partial transportation and the kolmogorov distance we show that partial transportation as opposed to classical complete transportation results in a sharp decrease of costs only in low dimension in contrast for the kolmogorov metric this decrease is seen in any dimension

hida families and rational points on elliptic curves let e be an elliptic curve over q of conductor n mp having a prime p n of multiplicative reduction because e is modular it corresponds to a normalised weight two eigenform on 0 n whose q expansion is denoted f n anq letx hom z p z p z p 1 z zp which contains z as a dense subset by associating to k z the character x xk 2 denote by a u the ring of cp valued p adic analytic functions on a compact open subset u of x hida s theory associates to f a neighborhood u of 2 x which can be assumed for simplicity to be contained in the residue class of 2 modulo p 1 and a formal q expansion

persistent chaos in high dimensions an extensive statistical survey of universal approximators shows that as the dimension of a typical dissipative dynamical system is increased the number of positive lyapunov exponents increases monotonically and the number of parameter windows with periodic behavior decreases a subset of parameter space remains where noncatastrophic topological change induced by a small parameter variation becomes inevitable a geometric mechanism depending on dimension and an associated conjecture depict why topological change is expected but not catastrophic thus providing an explanation of how and why deterministic chaos persists in high dimensions

inference for empirical wasserstein distances on finite spaces summary the wasserstein distance is an attractive tool for data analysis but statistical inference is hindered by the lack of distributional limits to overcome this obstacle for probability measures supported on finitely many points we derive the asymptotic distribution of empirical wasserstein distances as the optimal value of a linear programme with random objective function this facilitates statistical inference e g confidence intervals for sample based wasserstein distances in large generality our proof is based on directional hadamard differentiability failure of the classical bootstrap and alternatives are discussed the utility of the distributional results is illustrated on two data sets

quasi genericity of bifurcations to high dimensional invariant tori for maps we consider a family of maps in a banach spacee near the situation when the derivative at the fixed point has two pairs of complex eigenvalues lying on the unit circle the other part of the spectrum being strictly inside the unit disc we focus our attention on the region of the parameter space where the truncated normal form of the maps shows a bifurcation of a family of invariantt 1 circles into a family of invariantt 2 tori we show that this problem needs a 3 dimensional parameter unfolding and that for the complete maps bifurcation occurs at points where is the rotation number on the non normally hyperbolict 1 circle ande 2i are the eigenvalues of the constant matrix conjugated to the non contracting part of the linearization on the normal fiber bundle overt 1 making some non resonance and diophantine assumptions on leading to a positive measure cantor set int 2 we show that in paraboloidal regions of the 3 dim parameter space we have clean bifurcations as for the truncated normal form the complement of these regions forms a set of bubbles such as the ones obtained by chenciner in chen for a codimension 2 problem for maps in 2 the main tool here is a generalization for a matrix function ont 1 close to a constant of the quasi conjugacy to a constant modulo a minimum of additional parameters moved quasi conjugacy for the infinite dimensional case we use ac decoupling result on the angular dependent linear parts into a contraction still angular dependent and another part quasi conjugated to a constant matrix this type of analysis applies for a wide range of problems where truncated normal forms of the maps give bifurcations fromt n tot n 1 tori and this needs a n 1 dimensional parameter unfolding

an efficient qr based method for the computation of lyapunov exponents abstract an efficient and numerically stable method to determine all the lyapunov characteristic exponents of a dynamical system is developed numerical experiments are presented highlighting some aspects of convergence accuracy and efficiency in the computation of the lyapunov characteristic exponents

on the non ordinary locus in hilbert blumenthal surfaces let l q d d a square free integer be a totally real quadratic field and letmdl n be the moduli space in characteristic p 3 parameterizing principally polarized abelian surfaces a in characteristic p together with a symplectic level n 3 structure and an embedding of rings ol end a hereend a are the endomorphisms fixed by the rosati involution associated to we refer the reader to dp ra and vg for further details let v vdl n be the complement of the ordinary locus inm and let s sdl n be the supersingular locus this paper is about v the analysis and the results are divided according to the decomposition of p in l we establish the following results 1 if p is inert or ramified in l then v s every component of s is a smooth rational curve and the number of components is

tate cycles on some quaternionic shimura varieties mod p let f be a totally real field in which a prime number p 2 is inert we continue the study of the generalized goren oort strata on quaternionic shimura varieties over finite extensions of fp we prove that when the dimension of the quaternionic shimura variety is even the tate conjecture for the special fiber of the quaternionic shimura variety holds for the cuspidal isotypical component as long as the two unramified satake parameters at p are not differed by a root of unity

approximate bayesian computation with the wasserstein distance a growing number of generative statistical models do not permit the numerical evaluation of their likelihood functions approximate bayesian computation has become a popular approach to overcome this issue in which one simulates synthetic data sets given parameters and compares summaries of these data sets with the corresponding observed values we propose to avoid the use of summaries and the ensuing loss of information by instead using the wasserstein distance between the empirical distributions of the observed and synthetic data this generalizes the well known approach of using order statistics within approximate bayesian computation to arbitrary dimensions we describe how recently developed approximations of the wasserstein distance allow the method to scale to realistic data sizes and we propose a new distance based on the hilbert space filling curve we provide a theoretical study of the method proposed describing consistency as the threshold goes to 0 while the observations are kept fixed and concentration properties as the number of observations grows various extensions to time series data are discussed the approach is illustrated on various examples including univariate and multivariate g and k distributions a toggle switch model from systems biology a queuing model and a levy driven stochastic volatility model

uber die lokale zetafunktion von shimuravarietaten monodromiefiltration und verschwindende zyklen in ungleicher charakteristik

hirzebruch zagier cycles and twisted triple product selmer groups let e be an elliptic curve over mathbb q and a another elliptic curve over a real quadratic number field we construct a mathbb q motive of rank 8 together with a distinguished class in the associated bloch kato selmer group using hirzebruch zagier cycles that is graphs of hirzebruch zagier morphisms we show that under certain assumptions on e and a the non vanishing of the central critical value of the twisted triple product l function attached to e a implies that the dimension of the associated bloch kato selmer group of the motive is 0 and the non vanishing of the distinguished class implies that the dimension of the associated bloch kato selmer group of the motive is 1 this can be viewed as the triple product version of kolyvagin s work on bounding selmer groups of a single elliptic curve using heegner points

a finiteness theorem for zero cycles over p adic fields let r be a henselian discrete valuation ring let x be a regular projective flat scheme over spec r with generalized semistable reduction we prove a bijectivity theorem for etale cycle class maps of the chow group of 1 cycles on x as an application we prove a finiteness theorem for the chow group of 0 cycles on a projective smooth variety over a p adic field

rates of convergence for the empirical quantization error for n k n and r 0 let e n r p k r inf 1 k k i 1 x i f x i r where the infimum is taken over all measurable maps f r d r d with f r d n and x 1 x k are i i d r d valued random variables we analyse the asymptotic a s behaviour of the nth empirical quantization error e n r p k

characteristic exponents and invariant manifolds in hilbert space the multiplicative ergodic theorem and the construction almost everywhere of stable and unstable manifolds pesin theory are extended to differentiable dynamical systems on hilbert manifolds under some compactness assumptions the results apply to partial differential equations of evolution and also to non invertible maps of compact manifolds

partial hyperbolicity lyapunov exponents and stable ergodicity we present some results and open problems about stable ergodicity of partially hyperbolic diffeomorphisms with non zero lyapunov exponents the main tool is local ergodicity theory for non uniformly hyperbolic systems

probability of local bifurcation type from a fixed point a random matrix perspective results regarding probable bifurcations from fixed points are presented in the context of general dynamical systems real random matrices time delay dynamical systems companion matrices and a set of mappings known for their properties as universal approximators neural networks the eigenvalue spectrum is considered both numerically and analytically using previous work of edelman et al based upon the numerical evidence various conjectures are presented the conclusion is that in many circumstances most bifurcations from fixed points of large dynamical systems will be due to complex eigenvalues nevertheless surprising situations are presented for which the aforementioned conclusion does not hold e g real random matrices with gaussian elements with a large positive mean and finite variance

majorizing measures the generic chaining majorizing measures provide bounds for the supremum of stochastic processes they represent the most general possible form of the chaining argument going back to kolmogorov majorizing measures arose from the theory of gaussian processes but they now have applications far beyond this setting the fundamental question is the construction of these measures this paper focuses on the tools that have been developed for this purpose and in particular the use of geometric ideas applications are given to several natural problems where entropy methods are powerless

minimax grid matching and empirical measures in this article we solve the minimax grid matching problem in dimensions greater than two as a by product we settle a long open problem involving the glivenko cantelli convergence of empirical measures

trilinear forms and the central values of triple product l functions we give an explicit formula for certain global trilinear forms that appear in jacquet s conjecture in terms of local trilinear forms and the central values of triple product l functions

optimal transport from lebesgue to poisson this paper is devoted to the study of couplings of the lebesgue measure and the poisson point process we prove existence and uniqueness of an optimal coupling whenever the asymptotic mean transportation cost is finite moreover we give precise conditions for the latter which demonstrate a sharp threshold at d 2 the cost will be defined in terms of an arbitrary increasing function of the distance the coupling will be realized by means of a transport map allocation map which assigns to each poisson point a set cell of lebesgue measure 1 in the case of quadratic costs all these cells will be convex polytopes

large memory capacity in chaotic artificial neural networks a view of the anti integrable limit in the literature it was reported that the chaotic artificial neural network model with sinusoidal activation functions possesses a large memory capacity as well as a remarkable ability of retrieving the stored patterns better than the conventional chaotic model with only monotonic activation functions such as sigmoidal functions this paper from the viewpoint of the anti integrable limit elucidates the mechanism inducing the superiority of the model with periodic activation functions that includes sinusoidal functions particularly by virtue of the anti integrable limit technique this paper shows that any finite dimensional neural network model with periodic activation functions and properly selected parameters has much more abundant chaotic dynamics that truly determine the model s memory capacity and pattern retrieval ability to some extent this paper mathematically and numerically demonstrates that an appropriate choice of the activation functions and control scheme can lead to a large memory capacity and better pattern retrieval ability of the artificial neural network models

on the compuation of lyapunov exponents for continuous dynamical systems in this paper we consider discrete and continuous qr algorithms for computing all of the lyapunov exponents of a regular dynamical system we begin by reviewing theoretical results for regular systems and present general perturbation results for lyapunov exponents we then present the algorithms give an error analysis of them and describe their implementation finally we give several numerical examples and some conclusions

routes to chaos in high dimensional dynamical systems a qualitative numerical study this paper examines the most probable route to chaos a high dimensional dynamical systems function space time delay neural networks endowed with a probability measure in a computational setting the most probable route to chaos relative to the measure we impose on the function space as the dimension is increased is observed to be a sequence of neimark sacker bifurcations into chaos the analysis is composed of the study of an example dynamical system followed by a probabilistic study of the ensemble of dynamical systems from which the example was drawn a scenario depicting the decoupling of the stable manifolds of the torus leading up to the onset of chaos in high dimensional dissipative dynamical systems is also presented c 2006 elsevier b v all rights reserved

how many zeros of a random polynomial are real we provide an elementary geometric derivation of the kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients we show that the expected number of real zeros is simply the length of the moment curve 1 t tn projected onto the surface of the unit sphere divided by the probability density of the real zeros is proportional to how fast this curve is traced out we then relax kac s assumptions by considering a variety of random sums series and distributions and we also illustrate such ideas as integral geometry and the fubini study metric

on modular representations of bar q q arising from modular forms where g is the galois group gai i q and f is a finite field of characteristic i 3 suppose that p is modular of level n i e that it arises from a weight 2 newform of level dividing n and trivial nebentypus character then p is an odd representation the matrix p c where c is a complex conjugation in g has eigenvalues 1 1 since 1 and 1 are distinct in f p is absolutely irreducible and has a model over every subfield of f containing the set trace p we assume that f has been chosen so that it is generated by this set assume that p is a prime which exactly divides n we write p i n and restrict p to a decomposition group of g for the prime p view the restriction as a representation p p of gal t p qp we say that p is finite at p 34 or 35 w if there is a finite flat f vector space scheme h over zp for which the action of gal t p qp on the f vector space h 1 p gives pp i f p 9 is finite at p if and only if pp is unramified it is clear that p is finite at p whenever p is already modular of level n p in w of 34 cf 35 serre conjectured the converse i e that if p is finite then p is modular of level n p soon thereafter mazur 1 20 proved this conjecture in the case

ring theoretic properties of certain hecke algebras the purpose of this article is to provide a key ingredient of w2 by establishing that certain minimal hecke algebras considered there are complete intersections as is recorded in w2 a method going back to mazur m allows one to show that these algebras are gorenstein but for the complete intersection property a new approach is required the methods of this paper are related to those of chapter 3 of w2 the methods of section 3 of this paper are based on a previous approach of one of us a w we would like to thank henri darmon fred diamond and gerd faltings for carefully reading the first version of this article gerd faltings has also suggested a simplification of our argument as well as of the argument of chapter 3 of w2 and we would like to thank him for allowing us to reproduce these in the appendix to this paper r t would like to thank a w for his invitation to collaborate and for sharing his many insights into the questions considered r t would also like to thank princeton university universite de paris 7 and harvard university for their hospitality during this collaboration a w was supported by an nsf grant

mean field equations bifurcation map and route to chaos in discrete time neural networks abstract we investigate the dynamical behaviour of neural networks with asymmetric synaptic weights in the presence of random thresholds we inspect low gain dynamics before using mean field equations to study the bifurcations of the fixed points and the change of regime that occurs when varying control parameters we infer different areas with various regimes summarized by a bifurcation map in the parameter space we numerically show the occurence of chaos that arises generically by a quasi periodicity route we then discuss some features of our system in relation with biological observations such as low firing rates and refractory periods

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow