a local smoothing estimate for the schrodinger equation we show that the schrodinger operator eit is bounded from to for all a 2n 1 2 1 q 2 q and q2 4 n 1 this is almost sharp with respect to the sobolev index we also show that the schrodinger maximal operator sup0 s0 if and only if it is bounded from to when s 2s0 a corollary is that sup0 3 4

langlands program trace formulas and their geometrization the langlands program relates galois representations and auto morphic representations of reductive algebraic groups the trace formula is a powerful tool in the study of this connection and the langlands functorial ity conjecture after giving an introduction to the langlands program and its geometric version which applies to curves over finite fields and over the complex field i give a survey of my recent joint work with robert langlands and ngo bao chau fln fn on a new approach to proving the functoriality conjecture using the trace formulas and on the geometrization of the trace formulas in particular i discuss the connection of the latter to the categorifi cation of the langlands correspondence

on a vanishing conjecture appearing in the geometric langlands correspondence 0 1 this paper should be regarded as a sequel to 7 there it was shown that the geometric langlands conjecture for gln follows from a certain vanishing conjecture the goal of the present paper is to prove this vanishing conjecture let x be a smooth projective curve over a ground field k let e be an m dimensional local system on x and let bunm be the moduli stack of rank m vector bundles on x the geometric langlands conjecture says that to e we can associate a perverse sheaf fe on bunm which is a hecke eigensheaf with respect to e the vanishing conjecture of 7 says that for all integers n m a certain functor ave depending on e and a parameter d z which maps the category d bunn to itself vanishes identically when d is large enough the fact that the vanishing conjecture implies the geometric langlands conjecture may be regarded as a geometric version of the converse theorem moreover as will be explained in the sequel the vanishing of the functor ave is analogous to the condition that the rankin selberg convolution of e viewed as an m dimensional galois representation and an automorphic form on gln with n m is well behaved both the geometric langlands conjecture and the vanishing conjecture can be formulated in any of the sheaf theoretic situations e g q adic sheaves when char k d modules when char k 0 and sheaves with coefficients in a finite field f again when char k when the ground field is the finite field fq and we are working with adic coefficients it was shown in 7 that the vanishing conjecture can be deduced from lafforgue s theorem that establishes the full langlands correspondence for global fields of positive characteristic cf 9

the study of a class of rational difference equations abstract in this paper we study the global behavior of the higher order nonlinear difference equation x n 1 x n k 1 x n 2 k 1 bx n k 1 cx n 2 k 1 n 0 1 where b c and are positive k 1 2 3 and the initial conditions x 2 k 1 x 1 x 0 are positive real numbers we investigate the character of semicycles periodicity global stability and convergence to 2 k cycle solutions of mentioned equation

global attractivity in the recursive sequence xn 1 α βxn γ xn 1 our aim in this paper is to investigate the global attractivity of the recursive sequencex n email protected bx email protected n 1 n 0 1 where a 0 b c 0 we show that one positive equilibrium point of the equation is a global attractor with a basin that depends on certain conditions posed on the coefficients

global attractivity in a second order nonlinear difference equation abstract consider the difference equation x n 1 x n x n 1 n 0 1 2 1 where the function satisfies the following conditions 0 0 and is nonincreasing in the equation 1 has a unique positive solution if denotes the unique positive solution of 1 then x x x x x 0 for x x then x is a global attractor of all positive solutions of eq 1

global attractivity in a rational recursive sequence our aim in this paper is to investigate the periodic character invariant intervals and the global attractivity of all positive solutions of the nonlinear difference equationx n email protected email protected email protected n 1 n 0 1 where a 0 b c 0 we show that the positive equilibrium of the equation is a global attractor with a basin that depends on certain conditions posed on the coefficients

the qualitative behavior of solutions of a nonlinear difference equation abstract this paper is concerned with the qualitative behavior of solutions to the difference equation x n 1 p qx n k 1 x n n 0 1 2 where the initial conditions x k x 1 x0 are non negative k 1 2 3 and the parameters p q are non negative we start by establishing the periodicity the character of semicycles the global stability and the boundedness of the above mentioned equation we also present solutions that have unbounded behavior it is worth to mention that this difference equation is a special case of an open problem was introduced by m r s kulenovic and g ladas dynamics of second order rational difference equations with open problems and conjectures chapman and hall crc boca raton 2002 several computational examples are given to support our theoretical discussions the presented numerical tests represent different types of qualitative behavior of solutions to our nonlinear difference equation

character sheaves v this paper is part of a series s 131 devoted to the study of a class g of irreducible perverse sheaves called character sheaves on a connected reductive algebraic group g the numbering of chapters sections and references will continue that of s 131 this paper is a step towards the classification of character sheaves on g one of the main results is the following one under certain assumptions there is a natural surjective map with finite fibers from g to the set of all pairs 9 c up to conjugacy by the weyl group where 9 is a tame local system on the maximal torus and c is a two sided cell in the stabilizer w of 9 in the weyl group the assumptions made on g are

a strong type 2 2 estimate for a maximal operator associated to the schrodinger equation soit t f x sup t 0 t t f x ou t t f 1 exp it 2 f 1 4 on montre que etant donne un intervalle fini i i t f 2 dx c i r f x 2 dx

a rational difference equation abstract we investigate the boundedness character the periodic nature and the global asymptotic stability of all positive solutions of the equation in the title with positive parameters and nonnegative initial conditions

global stability of a rational difference equation abstract in this paper we study the global stability of the difference equation x n 1 x n a a 0 x n a k x n k n 0 1 where the parameters a a i 0 for i 0 k x k x 1 0 and x 0 0 we prove that the unique positive equilibrium is globally asymptotically stable if and only if it is locally asymptotically also we provide sufficient condition for it to be globally asymptotically stable and our results solve the open problem proposed by kulenovic and ladas dynamics of second order rational difference equations with open problems and conjectures chapman hall crc boca raton 2002

dynamics of a higher order rational difference equation abstract our aim in this paper is to investigate the dynamics of difference equation y n 1 p qy n 1 y n k n 0 1 2 where the initial conditions y k y 1 y0 are non negative k 1 2 3 and the parameters p and q are non negative we study characteristics such as the global stability the boundedness of positive solutions and the character of semicycles of the above mentioned difference equation in particular our results solve the open problem introduced by kulenovic and ladas in their monograph m r s kulenovic g ladas dynamics of second order rational difference equations with open problems and conjectures chapman and hall crc boca raton 2002

scattering theory in the energy space for a class of non linear wave equations we study the asymptotic behaviour in time of the solutions and the theory of scattering in the energy space for the non linear wave equation square varphi f varphi 0 in n n 3 we prove the existence of the wave operators asymptotic completeness for small initial data and forn 4 asymptotic completeness for arbitrarily large data the assumptions onf cover the case wheref behaves slightly better than a single powerp 1 4 n 2 both near zero and at infinity see 1 5 1 6 and 1 8

a note on the periodic cycle of xn 2 1 xn 1 xn the behaviour of solutions of the difference equation x n 1 a n b nx nx n 1 n 0 1 are investigated where the sequences a n and b n are periodic

global asymptotical stability of a second order rational difference equation in this paper we investigate the boundedness invariant interval semicycle and global attractivity of all positive solutions of the equation x n email protected email protected n 1a bx n cx n 1 n 0 1 where the parameters a c a b email protected 0 and the initial conditions y 1 y 0 are nonnegative real numbers we show that if the equation has no prime period two solutions then the positive equilibrium of the equation is globally asymptotically stable our results solve partially the conjecture proposed by kulenovic and ladas in their monograph m r kulenovic g ladas dynamics of second order rational difference equations with open problems and conjectures chapman hall crc boca raton 2001

the dynamics of χn 1 α βχna bχn cxn 1 facts and conjectures abstract we investigate the global character of solutions of the equation in the title with non negative parameters and nonnegative initial conditions

smoothing property for schrodinger equations with potential superquadratic at infinity we prove a smoothing property for one dimensional time dependent schrodinger equations with potentials which satisfy at infinity k 2 as an application we show that the initial value problem for certain nonlinear schrodinger equations with such potentials is l2 well posed we also prove a sharp asymptotic estimate of the lp norm of the normalized eigenfunctions of h v for large energy

on the recursive sequence yn 1 p yn 1 qyn yn 1 abstract we investigate the periodic character and the global stability of solutions of the equation y n 1 p y n 1 qy n y n 1 with positive parameters and positive initial conditions

characterizations of character sheaves for complex reductive algebraic groups character sheaves are geometric objects associated to a reductive algebraic group they are closely related to irreducible characters of finite chevalley groups the existence of character sheaves was conjectured by g lusztig in the book characters of reductive groups over finite fields 1984 there he described what one should expect character sheaves to be for complex reductive groups a definition of character sheaves for an arbitrary characteristic quite different from this description was given by him later 1985 it is shown in this thesis in particular that the original 1984 lusztig s suggestion for describing character sheaves is equivalent to the later definition we use alternative characterizations of character sheaves for a connected complex reductive algebraic group g that are obtained in this thesis and also the one conjectured by g lusztig and g laumon that was obtained in 1988 by v ginsburg and independently by i mirkovic and k vilonen these characterizations involve microsupport and microlocalization of a sheaf a microsupport ss cal f of a complex cal f of sheaves on g is a closed conic subset of t sp g g times bf g it tells us about singularities of cal f a microlocalization mu sb m cal f of cal f along a submanifold m of g is a complex of sheaves on the conormal bundle t sbsp m g which allows us to analyze cal f on the infinitesimally small neighbourhood of m theorem let g be a complex connected reductive algebraic group cal f be a g equivariant for conjugation irreducible perverse sheaf on g let cal n be the nilpotent cone of the lie algebra g of g s may denote either g or the set of all semisimple elements of g the following are equivalent i cal f is a character sheaf ii ss cal f vert sb s subset s times cal n iii supp mu sb cal o cal f subset cal o times cal n where cal o subset s is a conjugacy class iv supp mu sb x cal f subset cal n x in s characterization ii when s g is the one given by ginsburg mirkovic and vilonen characterization iii with s being the set of semisimple elements was conjectured by j bernstein for arbitrary characteristic this kind of characterizations essentially simplifies the development and our understanding of the theory currently there is still no such description for positive characteristic an important intermediate result is the description of the microsupport of a doubc constructible sheaf cal f on a manifold x in terms of microlocalizations to the points of this manifold ss cal f overline bigcup limits sb x in x rm supp mu sb x cal f

global stability and asymptotics of some classes of rational difference equations in this note we prove that the positive solutions of some classes of rational difference equations are globally asymptotically stable using a berg s result we also find asymptotics of some solutions of these equations

dimension of divergence sets for the pointwise convergence of the schrodinger equation abstract in this note we consider the pointwise convergence along curves for the schrodinger equation and obtain estimates for the capacitary dimension of divergence sets which extend our previous result in 6

on the new solutions to the fuzzy difference equation xn 1 a bxn abstract in this paper using a generalization of division for fuzzy numbers we study the existence and global behavior of the fuzzy difference equation x n 1 a b x n n 0 1 where a and b are positive fuzzy numbers some examples are presented to illustrate the applicability of our results

dynamics of a rational difference equation using both theoretical and computational approaches this paper is concerned with the qualitative behavior of solutions to the difference equationx n 1 x n px n qx n k n 0 1 2 where the initial conditions x k x 1 x 0 are non negative email protected 1 2 3 and the parameters p q are non negative our concentration is on invariant intervals the character of semicycles the global stability and the boundedness of the above mentioned equation it is worth to mention that this difference equation was an open problem introduced by kulenovic and ladas in m r s kulenovic and g ladas dynamics of second order rational difference equations with open problems and conjectures chapman and hall crc boca raton 2002 our final comments are about informative examples

dynamics of a higher order rational difference equation abstract in this paper we will investigate a nonlinear rational difference equation of higher order our concentration is on invariant intervals periodic character the character of semicycles and global asymptotic stability of all positive solutions of x n 1 x n x n k bx n cx n k n 0 1 it is worth to mention that our results solve the open problem proposed by kulenvic and ladas in their monograph dynamics of second order rational difference equations with open problems and conjectures chapman hall crc boca raton 2002

global asymptotic stability of a second order rational difference equation abstract the main goal of this paper is to investigate the global asymptotic stability of the difference equation y n 1 p n y n p n y n 1 n 0 1 2 where p n if n is even if n is odd and 0 0 and the initial conditions y 1 y 0 0 we show that the unique equilibrium y 1 is globally asymptotically stable under the certain conditions

recent advances in the langlands program these are the notes for the lecture given by the author at the mathematical current events special session of the ams meeting in baltimore on january 17 2003 topics reviewed include the langlands correspondence for gl n in the function field case and its proof by v drinfeld and l lafforgue the geometric langlands correspondence for gl n and its proof by d gaitsgory k vilonen and the author and the work of a beilinson and v drinfeld on the quantization of the hitchin system and the langlands correspondence for an arbitrary semisimple algebraic group

smoothing properties and retarded estimates for some dispersive evolution equations smoothing properties in the form of space time integrability properties play an important role in the study of dispersive evolution equations a number of them follow from a combination of general arguments and specific estimates we present a general formulation which makes the separation between the two types of ingredients as clear as possible and we illustrate it with the examples of the schrodinger equation of the wave equation and of a class of 1 1 dimensional equations related to the benjamin ono equation of special interest for the cauchy problem are retarded estimates expressed in terms of those properties we derive a number of such estimates associated with the last example and we mention briefly an application of those estimates to the cauchy problem for the generalized benjamin ono equation

the schrodinger equation along curves and the quantum harmonic oscillator abstract we consider the schrodinger equation associated to the harmonic oscillator i t u h u where h x 2 with initial data in the sobolev space h s r d with d 2 and s 3 8 we prove almost everywhere convergence of the solution to its initial data as time tends to zero which improves a result of yajima 1990 30 to this end we consider almost everywhere convergence for the free schrodinger along curves as it turns out these problems are more or less equivalent to that of the free schodinger equation along vertical lines our results are obtained by showing the equivalence of related maximal estimates

intersection homology ii in 19 20 we introduced topological invariants ih x called intersection homology groups for the study of singular spaces x these groups depend on the choice of a perversity p a perversity is a function from 2 3 to the non negative integers such that both c and c 2 c are positive and increasing functions of c 2 1 the group ihr is defined for spaces x called pseudomanifolds a pseudomanifold of dimension n is a space that admits a stratification x x n x n z x n 3 x 1 x 0

properties of a certain lyness equation we investigate oscillation cycle length and extreme values for the difference equationxn 1 a k 1i 0 bixn i xn k whereaandbiare nonnegative numbers and a k 1i 0 bi 0 ifa 0 it is known from theorem 2 2 1 of v l kocic g ladas and i w rodrigues j math anal appl 173 1993 127 157 that every cycle has no more than 2k 1 terms ifa 0 we show this is not necessarily true a general statement concerning cycle length ifa 0 remains an open question

the asymptotic stability of xn 1 axn bxn k 0 we obtain necessary and sufficient conditions for the asymptotic stability of the difference equation xn 1 axn bxn k 0 n 0 1 2 where the coefficients a and b are real numbers and the delay k is a nonnegative integer

global attractivity in a nonlinear second order difference equation in this paper we obtain a global attractivity result for the positive equilibrium of a nonlinear second order difference equation of the form xn 1 f xn xn 1 n 0 1 the result applies to the difference equation xn 1 a bxn a n 1 n 0 1 where a b a 0 1996 john wiley sons inc

regularity of solutions to the schrodinger equation soit f dans l espace de schwartz s r n et soit s t f x u x t rn exp ix exp it 2 f d x r n t r f est la transformee de fourier de f on considere u solution de l equation de schrodinger u i u t on pose s f x sup 0 t 1 s t f x x r n on introduit des espaces de sobolev h s en posant h s f s f hd s s r ou f hd s rn 1 2 s f 2 d 1 2 on etudie pour quelles valeurs de s on a b s f x 2 dx 1 2 c b f hd s f s r n pour toutes les boucles b de r n

global asymptotic behavior of a rational difference equation abstract the main goal of this paper is to investigate the global asymptotic behavior of the difference equation y n 1 y n 1 p n y n n 0 1 2 where p n is a period two sequence p n if n is even if n is odd with 0 and the initial conditions y 1 y 0 0 we obtain some global attractivity results for the equilibrium of this system for different values of the parameters

characteristic cycles for the loop grassmannian and nilpotent orbits x which is a linear combination of closures of conormal bundles to submanifolds of x intuitively the microlocal multiplicities c me asurethesingularity of at in settings related to representation theory a group g acts on x is g equivariant and the microlocal multiplicities play a significant but only partially understood role in representation theory see ro sv abv and kasa e g in this paper we compute microlocal multiplicities for certain cases of interest in representation theory let g be a connected reductive group with loop group lg and let p bethesubgroup of lg consisting of loops with positive fourier coefficients then p orbits on theloop grassmannian lg p correspond to the irreducible representations l of thedual re ductivegroup g for dominant weights and of a torus of

on rational recursive sequences our goal in this chapter is to gain some understanding of the dynamics of rational recursive sequences of the form x n 1 frac a sum limits i 0 k a i x n i b sum limits i 0 k b i x n i n 0 1 3 0 1 where k is a positive integer and the coefficients a b a 0 a k b 0 b k are nonnegative real numbers

on the dimension of divergence sets of dispersive equations we refine results of carleson sjogren and sjolin regarding the pointwise convergence to the initial data of solutions to the schrodinger equation we bound the hausdorff dimension of the sets on which convergence fails for example with initial data in h 1 mathbb r 3 the sets of divergence have dimension at most one

dynamics of the difference equation xn 1 xn pxn kxn q we study the invariant interval the character of semicycles the global stability and the boundedness of the difference equation x n 1 x n px n kx n q n 0 where the initial conditions x k x 1 x 0 are nonnegative and p q 0

global stability of yn 1 p qyn ryn k1 yn in this paper we investigate the global stability of difference equation y n 1 p qy n ry n k1 y n n 0 1 2 where the initial conditions y k y 1 y 0 are non negative email protected 1 2 3 and the parameters p q and r are non negative we show that when rq 1 and k is odd then this equation possesses unbounded solutions

existence of solutions for schrodinger evolution equations we study the existence uniqueness and regularity of the solution of the initial value problem for the time dependent schrodinger equationi u t 1 2 u v t x u u 0 u0 we provide sufficient conditions onv t x such that the equation generates a unique unitary propagatoru t s and such thatu t s u0ec1 l2 c0 h2 n foru0eh2 n the conditions are general enough to accommodate moving singularities of type x 2 n 4 or x n 2 n 3

stability of the recursive sequence xn 1 α βxn γ xn 1 abstract in this paper we investigate the global asymptotic stability of the recursive sequence xn 1 xn xn 1 n 0 1 where 0 we show that the unique positive equilibrium point of the equation is a global attractor with a basin that depends on the conditions posed on the coefficients

on the difference equation xn p xn sqxn t xn s abstract we investigate the persistence and global attractivity of the difference equation x n p x n s qx n t x n s n 1 2 3 with positive parameters and positive initial conditions our results not only include some previously known results but apply to some difference equations that have not been investigated so far

dynamics of a rational difference equation in this paper we investigate the periodic character invariant intervals oscillation and global stability of all positive solutions of the equationx n 1 px n x n kq x n k n 0 1 where the parameters p and q and the initial conditions x k x 0 are nonnegative real numbers as might be expected the two cases p q give rise to different dynamic behaviors in particular our results solve the open problem proposed by kulenvic and ladas in their monograph dynamics of second order rational difference equations with open problems and conjectures chapman hall crc boca raton 2001

global asymptotic stability for a higher order nonlinear rational difference equations abstract in this paper we investigate the boundedness periodic character invariant intervals and the global asymptotic stability of all nonnegative solutions of the difference equation x n 1 ax n bx n k a bx n n 0 1 where a b a b are positive real numbers k 1 is a positive integer and the initial conditions x k x 1 x 0 are nonnegative real numbers it is shown that the zero equilibrium of this equation is globally asymptotically stable if b a a and the unique positive equilibrium is globally asymptotically stable if a a b a a the results obtained solve a open problem proposed by kulenovic and ladas m r s kulenovi g ladas dynamics of second order rational difference equations chapman hall crc boca raton fl 2002 p 129

coherence on fractals versus pointwise convergence for the schrodinger equation we consider carleson s problem regarding convergence for the schrodinger equation in dimensions d ge 2 we show that if the solution converges almost everywhere with respect to alpha hausdorff measure to its initial datum as time tends to zero for all data h s mathbb r d then s ge frac d 2 d 2 d 1 alpha this strengthens and generalises results of bourgain and dahlberg kenig

dynamics of a non linear difference equation abstract our aim in this paper is to investigate the dynamics of difference equation y n 1 y n py n k y n q n 0 1 2 where the initial conditions y k y 1 y0 are non negative k 1 2 3 and the parameters p and q are non negative we study characteristics such as periodicity invariant interval the character of semicycles the global stability and the boundedness of positive solutions of the above mentioned difference equation in particular our results solve the open problem introduced by kulenovic and ladas in their monograph dynamics of second order rational difference equations with open problems and conjectures chapman and hall crc boca raton 2002