the optimal trilinear restriction estimate for a class of hypersurfaces with curvature abstract in 4 nearly optimal l 1 trilinear restriction estimates in r n 1 are established under transversality assumptions only in this paper we show that the curvature improves the range of exponents by establishing l p estimates for any p 2 n 4 3 n 2 in the case of double conic surfaces the exponent 2 n 4 3 n 2 is shown to be the universal threshold for the trilinear estimate

on the essential spectrum of the transfer operator for expanding markov maps the essential spectrum of the transfer operator for expanding markov maps of the interval is studied in detail to this end we construct explicityly an infinite set of eigenfunctions which allows us to prove that the essential spectrum inc k is a disk whose radius is related to the free energy of the liapunov exponent

consistency conditions for brane tilings abstract given a brane tiling on a torus we provide a new way to prove and generalise the recent results of szendr i mozgovoy and reineke regarding the donaldson thomas theory of the moduli space of framed cyclic representations of the associated algebra using only a natural cancellation type consistency condition we show that the algebras are 3 calabi yau and calculate donaldson thomas type invariants of the moduli spaces two new ingredients to our proofs are a grading of the algebra by the path category of the associated quiver modulo relations and a way of assigning winding numbers to pairs of paths in the lift of the brane tiling to the universal cover these ideas allow us to generalise the above results to all consistent brane tilings on k 1 surfaces we also prove a converse no consistent brane tiling on a sphere gives rise to a 3 calabi yau algebra

sasaki einstein manifolds and volume minimisation we study a variational problem whose critical point determines the reeb vector field for a sasaki einstein manifold this extends our previous work on sasakian geometry by lifting the condition that the manifolds are toric we show that the einstein hilbert action restricted to a space of sasakian metrics on a link l in a calabi yau cone x is the volume functional which in fact is a function on the space of reeb vector fields we relate this function both to the duistermaat heckman formula and also to a limit of a certain equivariant index on x that counts holomorphic functions both formulae may be evaluated by localisation this leads to a general formula for the volume function in terms of topological fixed point data as a result we prove that the volume of a sasaki einstein manifold relative to that of the round sphere is always an algebraic number in complex dimension n 3 these results provide via ads cft the geometric counterpart of a maximisation in four dimensional superconformal field theories we also show that our variational problem dynamically sets to zero the futaki invariant of the transverse space the latter being an obstruction to the existence of a kahler einstein metric

on the erdős distinct distances problem in the plane in this paper we prove that a set of n points in r 2 has at least c n log n distinct distances thus obtaining the sharp exponent in a problem of erd os we follow the setup of elekes and sharir which in the spirit of the erlangen program allows us to study the problem in the group of rigid motions of the plane this converts the problem to one of point line incidences in space we introduce two new ideas in our proof in order to control points where many lines are incident we create a cell decomposition using the polynomial ham sandwich theorem this creates a dichotomy either most of the points are in the interiors of the cells in which case we immediately get sharp results or alternatively the points lie on the walls of the cells in which case they are in the zero set of a polynomial of suprisingly low degree and we may apply the algebraic method in order to control points incident to only two lines we use the ecnode polynomial of the rev george salmon to conclude that most of the lines lie on a ruled surface then we use the geometry of ruled surfaces to complete the proof

on the multilinear restriction and kakeya conjectures we prove d linear analogues of the classical restriction and kakeya conjectures in rd our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians closely related to heat flow we conclude by giving some applications to the corresponding variable coefficient problems and the so called joints problem as well as presenting some n linear analogues for n d

period doubling bifurcations for families of maps on ℝ n infinite sequences of period doubling bifurcations in one parameter families 1 pf of maps enjoy very strong universality properties this is known numerically in a multitude of cases and has been shown rigorously for certain 1 pf of maps on the interval these bifurcations occur in 1 pf of analytic maps at values of the parameter tending to a limit with the asymptotically geometric ratio 1 4 6692 in this paper we indicate the main steps of a proof that the same is true for 1 pf of analytic maps from n to n whose restriction to n is real

consistency and derangements in brane tilings brane tilings describe lagrangians vector multiplets chiral multiplets and the superpotential of four dimensional supersymmetric gauge theories these theories written in terms of a bipartite graph on a torus correspond to worldvolume theories on n d3 branes probing a toric calabi yau threefold singularity a pair of permutations compactly encapsulates the data necessary to specify a brane tiling we show that geometric consistency for brane tilings which ensures that the corresponding quantum field theories are well behaved imposes constraints on the pair of permutations restricting certain products constructed from the pair to have no one cycles permutations without one cycles are known as derangements we illustrate this formulation of consistency with known brane tilings counting formulas for consistent brane tilings with an arbitrary number of chiral bifundamental fields are written down in terms of delta functions over symmetric groups

renormalization group analysis of some dynamical systems with noise we formulate a renormalization group analysis for the study of the accumulation of period doubling in the presence of noise the main tool is a renormalization of the time evolution of the noise the critical indices depend on the nature of the noise but are given by thermodynamic quantities describing the large deviations of the lyapunov exponent of the linearized random renormalization

produits aleatoires d operateurs matrices de transfert nous etudions le comportement asymptotique de produits aleatoires d operateurs de ruelle perron frobenius nous etendons le travail de ruelle obtenu dans le cas homogene au cas aleatoire

a bilinear approach to the restriction and kakeya conjectures the purpose of this paper is to investigate bilinear variants of the restriction and kakeya conjectures to relate them to the standard formulations of these conjectures and to give applications of this bilinear approach to existing conjectures the methods used are based on several observations and results of bourgain see 2 6 together with some refinements by moyua vargas and vega 17 18 this paper is organized as follows in the first section we discuss bilinear restriction estimates and show how one can pass back and forth between these estimates and the standard restriction estimates we also generalize the 12 7 bilinear restriction estimate of 18 to higher dimensions in the second section we give analogues of the above results for the kakeya operator in particular we give a bilinear improvement to wolff s kakeya theorem in arbitrary dimension in the third section we give applications of these bilinear estimates in three dimensions for example we are able to improve the 42 11 exponent in wolff s restriction theorem to 34 9 we are also able to prove a sharp lp lq restriction theorem which improves on the classical l2 l4 tomas stein theorem and also give some concrete progress on a bilinear restriction conjecture of klainerman and machedon we also give a non bilinear approach to these estimates which gives weaker results but is more direct and probably has a wider range of application

scaling for period doubling sequences with correlated noise the scaling behaviour of the period doubling sequences perturbed by correlated noises is investigated using a renormalisation group the renormalisation transformation is achieved by a path integral formalism which allows a necessary generalisation of the random perturbation notion for a stationary and weakly correlated in a sense specified in this paper gaussian random perturbation it is proved that the scaling behaviour depends only on the same universal constant k 6 619 as for uncorrelated perturbations

slow sweep through a period doubling cascade delayed bifurcations and renormalisation abstract we investigate analytically the effect on a period doubling cascade of slowly sweeping the bifurcation parameter by means of asymptotic calculations first we analyse the behaviour of the orbits when sweeping through one period doubling bifurcation there is still an effective bifurcation called a dynamic bifurcation but which is delayed we compute the delay sweeping in forward and backward directions yields qualitatively different results the presence of noise in the system even of small intensity turns out to play a key role we analyse the various behaviours as a function of the relative sizes of the sweep rate and the noise intensity in order to tackle the sweep through a whole cascade we extend the renormalisation theory for period doubling of one dimensional maps to non autonomous maps the particular we show that a slow linear of the parameter results in a new unstable eigenvalue which provides a scaling law for the sweep rate taking also into account the effect of noise we then derive an asymptotic scaling for the successive period doubling bifurcations as a result of the sweep and the noise only a finite number of doublings can be observed we estimate this maximum number we also show how to estimate the location of the real bifurcation of the real points from results of experiments with non zero sweep velocity when the functions involved to be analytic we show that the delay of a period doubling bifurcation due to sweep depends crucially on their degree of smoothness in both the analytic and non analytic cases we give insight into the geometry of some invariant curves reminiscent of an accordion our analysis explains numerical and experimental results on the problem

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

remark on exit times from cones in mathbb r n of brownian motion our purpose is to show how the asymptotics in corollary 1 3 of 2 can be obtained under much weaker hypotheses it turns out the problem essentially reduces to showing that if r s is a bessel process u 0 and 0 then p left mathop f limits 0 1 r s 2 ds leqq u right o t alpha as t we provide a simple proof of this fact

restriction estimates using polynomial partitioning ii we improve the estimates in the restriction problem in dimension n geqslant 4 to do so we establish a weak version of a k linear restriction estimate for any k the exponents in this weak k linear estimate are sharp for all k and n

global complexity and essential simplicity a conjectural picture of the boundary of chaos for smooth endomorphisms of the interval abstract mixing a symbolic approach to the dynamics of the period doubling operators recent results by sullivan on the renormalization for real analytic maps and some confidence a global picture emerges for the structure of the boundary of positive topological entropy in spaces of smooth endomorphisms of the interval