delay independence of mutual information rate of two symbolic sequences introduced by shannon as a rate of actual transmission mutual information rate mir is an extension of mutual information to a pair of dynamical processes we show a delay independence theorem according to which mir is not sensitive to a time shift between the two processes numerical studies of several benchmark situations confirm that this theoretical asymptotic property remains valid for realistic finite sequences estimations based on block entropies and a causal state machine algorithm perform better than an estimation based on a lempel ziv compression algorithm provided that block length and maximum history length respectively can be chosen larger than the delay mir is thus a relevant index for measuring nonlinear correlations between two experimental or simulated sequences when the transmission delay in input output devices or dephasing in coupled systems is variable or unknown

cut set bound is loose for gaussian relay networks the cut set bound developed by cover and el gamal in 1979 has since remained the best known upper bound on the capacity of the gaussian relay channel we develop a new upper bound on the capacity of the gaussian primitive relay channel which is tighter than the cut set bound our proof uses gaussian measure concentration to establish geometric relations satisfied with high probability between the n letter random variables associated with a reliable code for communicating over this channel we then translate these geometric relations into new information inequalities that cannot be obtained with classical methods combined with a tensorization argument proposed by courtade and ozgur in 2015 our result also implies that the current capacity approximations for gaussian relay networks which have linear gap to the cut set bound in the number of nodes are order optimal and lead to a lower bound on the pre constant

maximal parabolic regularity for divergence operators including mixed boundary conditions abstract we show that elliptic second order operators a of divergence type fulfill maximal parabolic regularity on distribution spaces even if the underlying domain is highly non smooth the coefficients of a are discontinuous and a is complemented with mixed boundary conditions applications to quasilinear parabolic equations with non smooth data are presented

concentration of measure inequalities for markov chains and phi mixing processes we prove concentration inequalities for some classes of markov chains and f mixing processes with constants independent of the size of the sample that extend the inequalities for product measures of talagrand the method is based on information inequalities put forward by marton in case of contracting markov chains using a simple duality argument on entropy our results also include the family of logarithmic sobolev inequalities for convex functions applications to bounds on supremum of dependent empirical processes complete this work

equations d evolution lineaires associees a des semi groupes de contractions dans les espaces lp on demontre l existence d une solution continue dans les espaces l p sous certaines hypotheses

parabolic control problems in measure spaces with sparse solutions optimal control problems in measure spaces lead to controls that have small support which is desirable e g in the context of optimal actuator placement for problems governed by parabolic partial differential equations well posedness is guaranteed in the space of square integrable measure valued functions which leads to controls with a spatial sparsity structure a conforming approximation framework allows one to derive numerically accessible optimality conditions as well as convergence rates in particular although the state is discretized the control problem can still be formulated and solved in the measure space numerical examples illustrate the structural features of the optimal controls

second order sufficient optimality conditions for parabolic optimal control problems with pointwise state constraints in this paper we study optimal control problems governed by semilinear parabolic equations where the spatial dimension is two or three moreover we consider pointwise constraints on the control and on the state we formulate first order necessary and second order sufficient optimality conditions we make use of recent results regarding elliptic regularity and apply the concept of maximal parabolic regularity to the occurring partial differential equations

measure concentration for euclidean distance in the case of dependent random variables let qn be a continuous density function in n dimensional euclidean space we think of qn as the density function of some random sequence xn with values in bbb r n for i 1 n let xi denote the collection of coordinates xi i i and let widebar x i denote the collection of coordinates xi i i we denote by q i x i bar x i the joint conditional density function of xi given widebar x i we prove measure concentration for qn in the case when for an appropriate class of sets i i the conditional densities q i x i bar x i as functions of xi uniformly satisfy a logarithmic sobolev inequality and ii these conditional densities also satisfy a contractivity condition related to dobrushin and shlosman s strong mixing condition

the second homology group of the mapping class group of an orientable surface in i 7 mumford shows that the picard group p ic is isomorphic to h2 f 2 and conjectures the latter is rank one g 3 we prove this below for g 5 another interpretation of this theorem may be obtained by identifying h2 f as bordism classes of fiber bundles f w4 t where t is a closed oriented surface sect 0 when f is closed every such bundle is bordant to f w t a bundle admitting a section s t w the theorem then says that

some inequalities for information divergence and related measures of discrimination inequalities which connect information divergence with other measures of discrimination or distance between probability distributions are used in information theory and its applications to mathematical statistics ergodic theory and other scientific fields we suggest new inequalities of this type often based on underlying identities as a consequence we obtain certain improvements of the well known pinsker inequality our study depends on two measures of discrimination called capacitory discrimination and triangular discrimination the discussion contains references to related research and comparison with other measures of discrimination e g ali silvey csiszar 1996 1966 divergences and in particular the hellinger distance

semiconjugacies between kleinian group actions on the riemann sphere we consider the problem of characterizing topologically the action of a kleinian group on the riemann sphere we prove that certain geometrically infinite kleinian group actions on can be obtained from geometrically finite ones by a semiconjugacy that is determined by the end invariants of the geometrically infinite group this turns out to be related to the problem of continuous extensions of maps of hyperbolic 3 space h 3 to maps of its boundary at infinity along the way we consider the general problem of extending maps to the boundary at infinity in gromov hyperbolic metric spaces we give criteria for extending to the boundary a map h x y between hyperbolic spaces if it extends to the boundary of certain subsets of x

exponential inequalities for dynamical measures of expanding maps of the interval we prove an exponential inequality for the absolutely continuous invariant measure of a piecewise expanding map of the interval as an immediate corollary we obtain a concentration inequality we apply these results to the estimation of the rate of convergence of the empirical measure in various metrics and also to the efficiency of the shadowing by sets of positive measure

concentration inequalities for functions of gibbs fields with application to diffraction and random gibbs measures we derive useful general concentration inequalities for functions of gibbs fields in the uniqueness regime we also consider expectations of random gibbs measures that depend on an additional disorder field and prove concentration w r t the disorder field both fields are assumed to be in the uniqueness regime allowing in particular for non independent disorder fields the modification of the bounds compared to the case of an independent field can be expressed in terms of constants that resemble the dobrushin contraction coefficient and are explicitly computable on the basis of these inequalities we obtain bounds on the deviation of a diffraction pattern created by random scatterers located on a general discrete point set in euclidean space restricted to a finite volume here we also allow for thermal dislocations of the scatterers around their equilibrium positions extending recent results for independent scatterers we give a universal upper bound on the probability of a deviation of the random scattering measures applied to an observable from its mean the bound is exponential in the number of scatterers with a rate that involves only the minimal distance between points in the point set

extending sobolev functions with partially vanishing traces from locally e δ domains and applications to mixed boundary problems we prove that given any k n for each open set rn and any closed subset d of such that is locally an e domain near d there exists a linear and bounded extension operator ek d mapping for each p 1 the space wdk p into wdk p rn here with o denoting either or rn the space wdk p o is defined as the completion in the classical sobolev space wk p o of restrictions to o of functions from cc rn whose supports are disjoint from d in turn this result is used to develop a functional analytic theory for the class wdk p including intrinsic characterizations boundary traces and extensions results interpolation theorems among other things which is then employed in the treatment of mixed boundary value problems formulated in locally e domains finally we also prove extension results on the scales of besov and bessel potential spaces on e domains with partially vanishing traces on ahlfors regular sets and explore some of the implications of such extension results

the capacity of the relay channel solution to cover s problem in the gaussian case consider a memoryless relay channel where the relay is connected to the destination with an isolated bit pipe of capacity c 0 let c c 0 denote the capacity of this channel as a function of c 0 what is the critical value of c 0 such that c c 0 first equals c infty this is a long standing open problem posed by cover and named the capacity of the relay channel in open problems in communication and computation springer verlag 1987 in this paper we answer this question in the gaussian case and show that c c 0 cannot equal to c infty unless c 0 infty regardless of the snr of the gaussian channels this result follows as a corollary to a new upper bound we develop on the capacity of this channel instead of single letterizing expressions involving information measures in a high dimensional space as is typically done in converse results in information theory our proof directly quantifies the tension between the pertinent n letter forms this is done by translating the information tension problem to a problem in high dimensional geometry as an intermediate result we develop an extension of the classical isoperimetric inequality on a high dimensional sphere which can be of interest in its own right

bounding bar d distance by informational divergence a method to prove measure concentration there is a simple inequality by pinsker between variational distance and informational divergence of probability measures defined on arbitrary probability spaces we shall consider probability measures on sequences taken from countable alphabets and derive from pinsker s inequality bounds on the bar d distance by informational divergence such bounds can be used to prove the concentration of measure phenomenon for some nonproduct distributions

bounded geometry for kleinian groups we show that a kleinian surface group or hyperbolic 3 manifold with a cusp preserving homotopy equivalence to a surface has bounded geometry if and only if there is an upper bound on an associated collection of coefficients that depend only on its end invariants bounded geometry is a positive lower bound on the lengths of closed geodesics when the surface is a once punctured torus the coefficients coincide with the continued fraction coefficients associated to the ending laminations

a unified framework for parabolic equations with mixed boundary conditions and diffusion on interfaces abstract in this paper we consider scalar parabolic equations in a general non smooth setting emphasizing interface conditions and mixed boundary conditions in particular we study dynamics and diffusion on a lipschitz interface and on the boundary where the diffusion coefficients are only assumed to be bounded measurable and positive semidefinite in the bulk we consider diffusion coefficients which may degenerate towards a lipschitz surface for this problem class we introduce a unified functional analytic framework based on sesquilinear forms and show maximal l p regularity and bounded h calculus for the corresponding operator providing well posedness for a large class of initial conditions and external forces

kantorovich duality for general transport costs and applications we introduce a general notion of transport cost that encompasses many costs used in the literature including the classical one and weak transport costs introduced by talagrand and marton in the 90 s and prove a kantorovich type duality theorem as a by product we obtain various applications in different directions we give a short proof of a result by strassen on the existence of a martingale with given marginals we characterize the associated transport entropy inequalities together with the log sobolev inequality restricted to convex concave functions some explicit examples of discrete measures satisfying weak transport entropy inequalities are also given

on maximal parabolic regularity for non autonomous parabolic operators abstract we consider linear inhomogeneous non autonomous parabolic problems associated to sesquilinear forms with discontinuous dependence of time we show that for these problems the property of maximal parabolic regularity can be extrapolated to time integrability exponents r 2 this allows us to prove maximal parabolic l r regularity for discontinuous non autonomous second order divergence form operators in very general geometric settings and to prove existence results for related quasilinear equations

the immersed interface multigrid methods for interface problems new multigrid methods are developed for the maximum principle preserving immersed interface method applied to second order linear elliptic and parabolic pdes that involve interfaces and discontinuities for elliptic interface problems the multigrid solver developed in this paper works while some other multigrid solvers do not for linear parabolic equations we have developed the second order maximum principle preserving finite difference scheme in this paper we use the crank nicolson scheme to deal with the diffusion part and an explicit scheme for the first order derivatives numerical examples are also presented

on rate of convergence of statistical estimation of stationary ergodic processes stationary ergodic processes with finite alphabets are approximated by finite memory processes based on an n length realization of the process under the assumptions of summable continuity rate and non nullness a rate of convergence in d distance is obtained with explicit constants asymptotically as n the result is near the optimum

concentration for multidimensional diffusions and their boundary local times we prove that probability laws of certain multidimensional semimartingales which includes time inhomogenous diffusions under suitable assumptions satisfy quadratic transportation cost inequality under the uniform metric from this we derive concentration properties of lipschitz functions of process paths that depend on the entire history in particular we estimate concentration of boundary local time of reflected brownian motions on a polyhedral domain we work out explicit applications of consequences of measure concentration for the case of brownian motion with rank based drifts

strong data processing inequalities and phi sobolev inequalities for discrete channels the noisiness of a channel can be measured by comparing suitable functionals of the input and output distributions for instance the worst case ratio of output relative entropy to input relative entropy for all possible pairs of input distributions is bounded from above by unity by the data processing theorem however for a fixed reference input distribution this quantity may be strictly smaller than one giving the so called strong data processing inequalities sdpis the same considerations apply to an arbitrary phi divergence this paper presents a systematic study of optimal constants in the sdpis for discrete channels including their variational characterizations upper and lower bounds structural results for channels on product probability spaces and the relationship between the sdpis and the so called phi sobolev inequalities another class of inequalities that can be used to quantify the noisiness of a channel by controlling entropy like functionals of the input distribution by suitable measures of input output correlation several applications to information theory discrete probability and statistical physics are discussed

transportation cost information inequalities and applications to random dynamical systems and diffusions we first give a characterization of the l1 transportation cost information inequality on a metric space and next find some appropriate sufficient condition to transportation cost information inequalities for dependent sequences applications to random dynamical systems and diffusions are studied

iteration of mapping classes and limits of hyperbolic 3 manifolds let mod s be an element of the mapping class group of a surface s we classify algebraic and geometric limits of sequences q i x y i 1 of quasi fuchsian hyperbolic 3 manifolds ranging in a bers slice when has infinite order with finite order restrictions there is an essential subsurface d s so that the geometric limits have homeomorphism type s d 0 typically has pseudo anosov restrictions and d has components with negative euler characteristic these components correspond to new asymptotically periodic simply degenerate ends of the geometric limit we show there is an s 1 depending on and bounded in terms of s so that q si x y i 1 converges algebraically and geometrically and we give explicit quasi isometric models for the limits

on continuous extensions of grafting maps the definition of the grafting operation for quasifuchsian groups is extended by bromberg preprint to all b groups in this paper we show that the extended grafting maps behave as continuous maps for every sequence which converges standardly to a boundary group although the maps are not continuous in general as a consequence of this result we extend goldman s grafting theorem for quasifuchsian groups to all boundary b groups

cusps are dense we show cusps are dense in bers boundary for teichmuller space the proof rests on an estimate for the algebraic effect of a unit quasiconformal deformation supported in the thin part of a hyperbolic riemann surface

projective structures with degenerate holonomy and the bers density conjecture we prove the bers density conjecture for singly degenerate kleinian surface groups without parabolics

ranks of mapping tori via the curve complex is generated by nielsen transformations this isequivalent to requiring that our two generating sets for g are related by asequence of the following moves 1 if a6 bare generators replace awith ab 2 if a6 bare generators switch their places in the ordering 3 if ais a generator replace it with a

spectral theory and complex interpolation abstract some relations between the spectral theory for linear operators and the complex theory of interpolation for families of banach spaces are studied an invertibility theorem is proven and size estimates for the spectra of operators on the interpolation spaces are given

solvability of parabolic equations in divergence form with partially bmo coefficients abstract we prove the h p 1 solvability of second order parabolic equations in divergence form with leading coefficients a i j measurable in t x 1 and having small bmo bounded mean oscillation semi norms in the other variables additionally we assume a 11 is measurable in x 1 and has small bmo semi norms in the other variables the corresponding results for the cauchy problem are also established parabolic equations in sobolev spaces h q p 1 with mixed norms are also considered under the same conditions of the coefficients

non autonomous maximal regularity for forms given by elliptic operators of bounded variation abstract we show maximal l p regularity for non autonomous cauchy problems provided the trace spaces are stable in some parameterized sense and the time dependence is of bounded variation in particular on l 2 for lipschitz domains and under mixed boundary conditions we obtain maximal l p regularity for all p 1 2 for elliptic operators with coefficients a i j c satisfying a i j x bv uniformly in x

stability bounds for stationary φ mixing and β mixing processes most generalization bounds in learning theory are based on some measure of the complexity of the hypothesis class used independently of any algorithm in contrast the notion of algorithmic stability can be used to derive tight generalization bounds that are tailored to specific learning algorithms by exploiting their particular properties however as in much of learning theory existing stability analyses and bounds apply only in the scenario where the samples are independently and identically distributed in many machine learning applications however this assumption does not hold the observations received by the learning algorithm often have some inherent temporal dependence this paper studies the scenario where the observations are drawn from a stationary mixing or mixing sequence a widely adopted assumption in the study of non i i d processes that implies a dependence between observations weakening over time we prove novel and distinct stability based generalization bounds for stationary mixing and mixing sequences these bounds strictly generalize the bounds given in the i i d case and apply to all stable learning algorithms thereby extending the use of stability bounds to non i i d scenarios we also illustrate the application of our mixing generalization bounds to general classes of learning algorithms including support vector regression kernel ridge regression and support vector machines and many other kernel regularization based and relative entropy based regularization algorithms these novel bounds can thus be viewed as the first theoretical basis for the use of these algorithms in non i i d scenarios

measure concentration for a class of random processes let x x i i be a stationary random process with a countable alphabet and distribution q let q x k 0 denote the conditional distribution of x x 1 x 2 x n given the k length past write d 1 x 1 0 if 1 x 1 and d 1 x 1 1 otherwise we say that the process x admits a joining with finite distance u if for any two past sequences k 0 k 1 0 and x k 0 x k 1 x 0 there is a joining of q k 0 and q x k 0 say dist 0 x 0 k 0 x k 0 such that the main result of this paper is the following inequality for processes that admit a joining with finite distance

concentration inequalities for dependent random variables via the martingale method the martingale method is used to establish concentration inequalities for a class of dependent random sequences on a countable state space with the constants in the inequalities expressed in terms of certain mixing coefficients along the way bounds are obtained on martingale differences associated with the random sequences which may be of independent interest as applications of the main result concentration inequalities are also derived for inhomogeneous markov chains and hidden markov chains and an extremal property associated with their martingale difference bounds is established this work complements and generalizes certain concentration inequalities obtained by marton and samson while also providing different proofs of some known results

harmonic maps into hyperbolic 3 manifolds high energy degeneration of harmonic maps of riemann surfaces into a hyperbolic 3 manifold target is studied generalizing results of m 1 in which the target is two dimensional the hopf foliation of a high energy map is mapped to an approximation of its geodesic representative in the target and the ratio of the squared length of that representative to the extremal length of the foliation in the domain gives an estimate for the energy the images of harmonic maps obtained when the domain degenerates along a teichmuller ray are shown to converge generically to pleated surfaces in the geometric topology or to leave every compact set of the target when the limiting foliation is unrealizable

operator valued fourier multiplier theorems and maximal l_p regularity we prove a mihlin type multiplier theorem for operator valued multiplier functions on umd spaces the essential assumption is r boundedness of the multiplier function as an application we give a characterization of maximal l p regularity for the generator of an analytic semigroup t t in terms of the r boundedness of the resolvent of a or the semigroup t t

ricci curvature of markov chains on metric spaces abstract we define the coarse ricci curvature of metric spaces in terms of how much small balls are closer in wasserstein transportation distance than their centers are this definition naturally extends to any markov chain on a metric space for a riemannian manifold this gives back after scaling the value of ricci curvature of a tangent vector examples of positively curved spaces for this definition include the discrete cube and discrete versions of the ornstein uhlenbeck process moreover this generalization is consistent with the bakry emery ricci curvature for brownian motion with a drift on a riemannian manifold positive ricci curvature is shown to imply a spectral gap a levy gromov like gaussian concentration theorem and a kind of modified logarithmic sobolev inequality the bounds obtained are sharp in a variety of examples

concentration inequalities for random fields via coupling we present a new and simple approach to concentration inequalities in the context of dependent random processes and random fields our method is based on coupling and does not use information inequalities in case one has a uniform control on the coupling one obtains exponential concentration inequalities if such a uniform control is no more possible then one obtains polynomial or stretched exponential concentration inequalities our abstract results apply to gibbs random fields both at high and low temperatures and in particular to the low temperature ising model which is a concrete example of non uniformity of the coupling

the virtual cohomological dimension of the mapping class group of an orientable surface let f f r be the mapping class group of a surface f of genus g with s punctures and r boundary components the purpose of this paper is to establish cohomology properties of f parallel to those of the arithmetic groups if g is a linear algebraic group defined over q x is the symmetric space associated to g and a is an arithmetic subgroup of g then a is virtually torsion free and acts properly discontinuously on x the rational homology of a is the same as that of x a furthermore there is a bordification of x bs to a manifold with corners 3 and an extension of the action of a to a properly discontinuous action on jf so that the quotient x a is compact the boundary of jf is homotopy equivalent to a wedge of spheres say of dimension d and the virtual cohomological dimension of a is n d 1 where n is the dimension of x in the case of the mapping class group there is no analog for g ivanov unpublished has proven that f is not arithmetic nevertheless f acts properly discontinuously on teichmiiller space r which is homeomorphic to euclidean space of dimension 6 g 6 2 s 3 will play the role of the symmetric space the quotient of 9by f is the moduli space of curves whose rational homology is then identified with that of e harvey har has constructed a borel serre bordification j of 3 by analytic methods in the case where f has punctures we will build g by a different combinatorial method in addition we will explicitly describe inside y a cell complex y of dimension 4 g 4 s onto which j may be f equivariantly retracted thus establishing an analog of the constructions for sl n by serre soul6 sol and ash a this complex will be of the lowest possible dimension because we will use j to prove our main result

a simple proof of the blowing up lemma corresp the blowing up lemma says that if the probability with respect to a product measure of a set a subseteq cal x n cal x finite n large is not exponentially small then its l n neighborhood has probability almost one for some l n o n here an information theoretic proof of the blowing up lemma generalizing it to continuous alphabets is given

on the density of geometrically finite kleinian groups the density conjecture of bers sullivan and thurston predicts that each complete hyperbolic 3 manifold m with finitely generated fundamental group is an algebraic limit of geometrically finite hyperbolic 3 manifolds we prove that the conjecture obtains for each complete hyperbolic 3manifold with no cusps and incompressible ends

the classification of kleinian surface groups i models and bounds we give the first part of a proof of thurston s ending lamination conjecture in this part we show how to construct from the end invariants of a kleinian surface group a lipschitz model for the thick part of the corresponding hyperbolic manifold this enables us to describe the topological structure of the thick part and to give a priori geometric bounds

a new look at independence the concentration of measure phenomenon in product spaces is a farreaching abstract generalization of the classical exponential inequalities for sums of independent random variables we attempt to explain in the simplest possible terms the basic concepts underlying this phenomenon the basic method to prove concentration inequalities and the meaning of several of the most useful inequalities

measure valued directional sparsity for parabolic optimal control problems a directional sparsity framework allowing for measure valued controls in the spatial direction is proposed for parabolic optimal control problems it allows for controls which are localized in space where the spatial support is independent of time well posedness of the optimal control problems is established and the optimality system is derived it is used to establish structural properties of the minimizer an a priori error analysis for finite element discretization is obtained and numerical results illustrate the effects of the proposed cost functional and the convergence results

hitting times functional inequalities lyapunov conditions and uniform ergodicity the use of lyapunov conditions for proving functional inequalities was initiated in 5 it was shown in 4 30 that there is an equivalence between a poincare inequality the existence of some lyapunov function and the exponential integrability of hitting times in the present paper we close the scheme of the interplay between lyapunov conditions and functional inequalities by showing that strong functional inequalities are equivalent to lyapunov type conditions showing that these lyapunov conditions are characterized by the finiteness of generalized exponential moments of hitting times we also give some complement concerning the link between lyapunov conditions and in tegrability property of the invariant probability measure and as such transportation inequalities and we show that some unbounded lyapunov conditions can lead to uniform ergodicity and coming down from infinity property

the classification of punctured torus groups thurston s ending lamination conjecture proposes that a flnitely generated kleinian group is uniquely determined up to isometry by the topology of its quotient and a list of invariants that describe the asymptotic geometry of its ends we present a proof of this conjecture for punctured torus groups these are free two generator kleinian groups with parabolic commutator which should be thought of as representations of the fundamental group of a punctured torus as a consequence we verify the conjectural topological description of the deformation space of punctured torus groups including bers conjecture that the quasi fuchsian groups are dense in this space and prove a rigidity theorem two punctured torus groups are quasi conformally conjugate if and only if they are topologically conjugate

monge kantorovitch measure transportation and monge ampere equation on wiener space let w h be an abstract wiener space assume two i i 1 2 probabilities on w w we give some conditions for the wasserstein distance between 1 and 2 with respect to the cameron martin space to be finite where the infimum is taken on the set of probability measures on w w whose first and second marginals are 1 and 2 in this case we prove the existence of a unique cyclically monotone map t iw with w h such that t maps 1 to 2 moreover if 2 then t is stochastically invertible i e there exists s w w such that s t iw 1 a s and t s iw 2 a s if in addition 1 then there exists a 1 convex function in the gaussian sobolev space such that these results imply that the quasi invariant transformations of the wiener space with finite wasserstein distance from can be written as the composition of a transport map t and a rotation i e a measure preserving map we give also 1 convex sub solutions and ito type solutions of the monge ampere equation on w

optimal control for the thermistor problem this paper is concerned with the state constrained optimal control of the two dimensional thermistor problem a quasi linear coupled system of a parabolic and elliptic pde with mixed boundary conditions this system models the heating of a conducting material by means of direct current existence uniqueness and continuity for the state system are derived by employing maximal elliptic and parabolic regularity by similar arguments the linearized state system is discussed while the adjoint system involving measures is investigated using a duality argument these results allow us to derive first order necessary conditions for the optimal control problem

the classification of kleinian surface groups ii the ending lamination conjecture thurston s ending lamination conjecture states that a hyperbolic 3manifold n with nitely generated fundamental group is uniquely determined by its topological type and its end invariants in this paper we prove this conjecture for kleinian surface groups the general case when n has incompressible ends relative to its cusps follows readily the main ingredient is a uniformly bilipschitz model for the quotient of h 3 by a kleinian surface group

on the closedness of the sum of two closed operators o t t 1 where e g a is the infinitesimal generator of an analytic semigroup in a complex banach space x we find that in general it is false that for every fe lp 0 t x problem 1 has a solution uewi p o t x c lp o t a usually it is necessary to impose some further regularity conditions on f consider for instance the way in which problem 1 is studied in the paper 8 and compare th 3 7 and 3 14 which are applied in w 4 of the same paper to solve problem 1 we find that in general we have only strong solutions while in the hilbert space setting we have strict solutions this corresponds to the fact that in general the operator a b where b u u is only closable while in a hilbert space it is closed this remark suggests that the better result may depend on the nicer geometric properties of a hilbert space with respect to a general banach space and it is natural to ask which conditions should be imposed on the banach space to force the sum of two suitable closed opera tors to be closed the above considerations motivate the manifold aim of the present paper a to find conditions on a banach space x and on closed linear operators a and b acting in x which ensure that a b is closed b to apply the result of a to problem 1 it seems that the convenient property for the banach space x is convexity or otherwise stated the umd condition for information about these equiva lent conditions we refer to 4 in which it is also proved that if x is convex and l p oe then the hilbert transform is continuous from lp r x into lp r x on the other hand it is also known see 2 that if there exists pe 1 oe such that the hilbert transform is continuous from lp r x into itself

norms on the cohomology of hyperbolic 3 manifolds we study the relationship between two norms on the first cohomology of a hyperbolic 3 manifold the purely topological thurston norm and the more geometric harmonic norm refining recent results of bergeron engun and venkatesh as well as older work of kronheimer and mrowka we show that these norms are roughly proportional with explicit constants depending only on the volume and injectivity radius of the hyperbolic 3 manifold itself moreover we give families of examples showing that some but not all qualitative aspects of our estimates are sharp finally we exhibit closed hyperbolic 3 manifolds where the thurston norm grows exponentially in terms of the volume and yet there is a uniform lower bound on the injectivity radius

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

l p maximal regularity for non autonomous evolution equations let a 0 l d x be strongly measurable and bounded where d x are banach spaces such that d x we assume that the operator a t has maximal regularity for all t 0 then we show under some additional hypothesis viz relative continuity that the non autonomous problem

a large deviation approach to some transportation cost inequalities new transportation cost inequalities are derived by means of elementary large deviation reasonings their dual characterization is proved this provides an extension of a well known result of s bobkov and f gotze their tensorization properties are investigated sufficient conditions and necessary conditions too for these inequalities are stated in terms of the integrability of the reference measure applying these results leads to new deviation results concentration of measure and deviations of empirical processes

on the growth of l 2 invariants for sequences of lattices in lie groups we study the asymptotic behaviour of betti numbers twisted torsion and other spectral invariants of sequences of locally symmetric spaces our main results are uniform versions of the degeorge wallach theorem of a theorem of delorme and various other limit multiplicity theorems a basic idea is to adapt the notion of benjamini schramm convergence bs convergence originally introduced for sequences of finite graphs of bounded degree to sequences of riemannian manifolds and analyze the possible limits we show that bs convergence of locally symmetric spaces g k implies convergence in an appropriate sense of the normalized relative plancherel measures associated to l2 g this then yields convergence of normalized multiplicities of unitary representations betti numbers and other spectral invariants on the other hand when the corresponding lie group g is simple and of real rank at least two we prove that there is only one possible bs limit i e when the volume tends to infinity locally symmetric spaces always bs converge to their universal cover g k this leads to various general uniform results when restricting to arbitrary sequences of congruence covers of a fixed arithmetic manifold we prove a strong quantitative version of bs convergence which in turn implies upper estimates on the rate of convergence of normalized betti numbers in the spirit of sarnak xue an important role in our approach is played by the notion of invariant random subgroups for higher rank simple lie groups g we exploit rigidity theory and in particular the nevo stuck zimmer theorem and kazhdan s property t to obtain a complete understanding of the space of irs s of g 2017 department of mathematics princeton university

solvability and maximal regularity of parabolic evolution equations with coefficients continuous in time abstract we establish maximal regularity of type lp for a parabolic evolution equation u t a t u t f t with a c 0 t l d a 0 x and construct the corresponding evolution family on the underlying banach space x our proofs are based on the operator sum method and the use of evolution semigroups the results are applied to parabolic partial differential equations with continuous coefficients

the kato square root problem for mixed boundary conditions we consider the negative laplacian subject to mixed boundary conditions on a bounded domain we prove under very general geometric assumptions that slightly above the critical exponent 12 its fractional power domains still coincide with suitable sobolev spaces of optimal regularity in combination with a reduction theorem recently obtained by the authors this solves the kato square root problem for elliptic second order operators and systems in divergence form under the same geometric assumptions thereby we answer a question posed by j l lions in 1962 30