sharp ill posedness and well posedness results for the kdv burgers equation the periodic case we prove that the kdv burgers is globally well posed in h 1 t with a solution map that is analytic from h 1 t to c 0 t h 1 t whereas it is ill posed in h s t as soon as s 0 small enough in view of the result of kappeler and topalov for kdv it thus appears that even if the dissipation part of the kdv burgers equation allows to lower the c infty critical index with respect to the kdv equation it does not permit to improve the c 0 critical index

on global existence and scattering for the wave maps equation we prove global existence and scattering for the wave maps equation in n 1 dimensions n 2 3 for initial data which is small in the scale invariant homogeneous besov space 2 1 n 2 x 2 1 n 2 1 this result was proved in an earlier paper by the author for n 4

on the well posedness ill posedness and norm inflation for a higher order water wave model on a periodic domain abstract in this work we are interested in the well posedness issues for the initial value problem associated with a higher order water wave model posed on a periodic domain t we derive some multilinear estimates and use them in the contraction mapping argument to prove the local well posedness for initial data in the periodic sobolev space h s t s 1 with some restriction on the parameters appeared in the model we use the conserved quantity to obtain the global well posedness for given data with sobolev regularity s 2 also we use splitting argument to improve the global well posedness result in h s t for 1 s 2 well posedness result obtained in this work is sharp in the sense that the flow map that takes initial data to the solution cannot be continuous for given data in h s t s 1 finally we prove a norm inflation result by showing that the solution corresponding to a smooth initial data may have arbitrarily large h s t norm with s 1 for arbitrarily short time

sharp ill posedness results for the kdv and mkdv equations on the torus abstract we establish a new a priori bound for l 2 bounded sequences of solutions to the mkdv equations on the torus this first enables us to construct weak solutions in l 2 for this equation and to check that the solutions constructed by kappeler and topalov in the defocusing case satisfy the equation in some weak sense in a second time we prove that the solution maps associated with the mkdv and the kdv equations are discontinuous for the h s t topology for respectively s 0 and s 1 these last results are sharp

rank 1 phenomena for mapping class groups we prove that every element of the mapping class group g has linear growth confirming a conjecture of n ivanov and that g is not boundedly generated we also provide restrictions on linear representations of g and its finite index subgroups

scattering for the quartic generalised korteweg de vries equation abstract we show that the quartic generalised kdv equation u t u x x x u 4 x 0 is globally well posed for data in the critical scale invariant space h x 1 6 r with small norm and locally well posed for large norm improving a result of grunrock a grunrock a bilinear airy estimate with application to gkdv 3 differential integral equations 18 12 2005 1333 1339 as an application we obtain scattering results in h x 1 r h x 1 6 r for the radiation component of a perturbed soliton for this equation improving the asymptotic stability results of martel and merle y martel f merle asymptotic stability of solitons for subcritical generalized kdv equations arch ration mech anal 157 3 2001 219 254

the automorphism group of a free group is not linear a linear group has ascending chain condition on centralizers malcev s p 511 finitely generated linear groups are residually finite malcev s p 453 a solvable linear group is nilpotent by abelian byfinite tits 5 pp 14551461 a linear group either contains a free group of rank two or is solvable by locally finite lubotzky 3 has characterized finitely generated linear groups over a field of characteristic zero by purely group theoretic conditions these conditions appear difficult to check although they have been used to show linearity for certain groups however we do not use his theorem the main result of this paper theorem 5 is that for n 2 3 the automorphism group of a free group of rank n is not a linear group the proof uses the representation theory of algebraic groups to show that a kind of diophantine equation between the irreducible representations of

global well posedness and inviscid limit for the korteweg de vries burgers equation abstract considering the cauchy problem for the korteweg de vries burgers equation u t u x x x x 2 u u 2 x 0 u 0 where 0 1 and u is a real valued function we show that it is globally well posed in h s s s and uniformly globally well posed in h s s 3 4 for all 0 1 moreover we prove that for any t 0 its solution converges in c 0 t h s to that of the kdv equation if tends to 0

sharp well posedness and ill posedness results for a quadratic non linear schrodinger equation abstract we establish that the quadratic non linear schrodinger equation iu t u xx u 2 where u r r c is locally well posed in h s r when s 1 and ill posed when s 1 previous work in c kenig g ponce l vega quadratic forms for the 1 d semilinear schrodinger equation trans amer math soc 346 1996 3323 3353 had established local well posedness for s 3 4 the local well posedness is achieved by an iteration using a modification of the standard x s b spaces the ill posedness uses an abstract and general argument relying on the high to low frequency cascade present in the non linearity and a computation of the first non linear iterate

unramified subextensions of ray class field towers abstract fix a prime l in this paper we explore various senses in which the ramification in an infinite l extension of number fields can be small in particular 1 we construct infinitely many pairs of primes p q distinct from l such that q admits an infinite l extension unramified outside p q and 2 we explore the possibility of finding infinite unramified subextensions inside infinite ramified l extensions of number fields

multilinear weighted convolution of l 2 functions and applications to nonlinear dispersive equations the x s b spaces as used by beals bourgain kenig ponce vega klainerman machedon and others are fundamental tools to study the low regularity behavior of nonlinear dispersive equations it is of particular interest to obtain bilinear or multilinear estimates involving these spaces by plancherel s theorem and duality these estimates reduce to estimating a weighted convolution integral in terms of the l 2 norms of the component functions in this paper we systematically study weighted convolution estimates on l 2 as a consequence we obtain sharp bilinear estimates for the kdv wave and schrodinger x s b spaces

polynomial identities in graded group rings restricted lie algebras and p adic analytic groups let g be any finitely generated group and let k be a field of characteristic p 0 it is shown that the graded group ring gr kg satisfies a nontrivial polynomial identity if and only if the pro p completion of g is p adic analytic i e can be given the structure of a lie group over the p adic field q p the proof applies theorems of lazard quillen and passman as well as results on engel lie algebras and on dimension subgroups in positive characteristic

the symmetric regularized long wave equation well posedness and nonlinear stability abstract the focus of the present work is the symmetric regularized long wave equation we prove that the initial value problem for this equation is locally and globally well posed in h per s h per s 1 and h s r h s 1 r if s 0 we also prove the existence and nonlinear stability of periodic travelling wave solutions of cnoidal type for the equation mentioned above

minimal two spheres and the topology of manifolds with positive curvature on totally isotropic two planes there has been much interest among differential geometers in finding relationships between curvature and topology of riemannian manifolds for the most part efforts have been directed towards understanding the implications of positive or negative sectional curvature ricci curvature or scalar curvature but there are other hypotheses on the curvature which also deserve investigation in this article we will consider the topological implications of a new curvature assumption positive curvature on totally isotropic two planes and we will prove via the sacks uhlenbeck theory of minimal two spheres that a compact simply connected riemannian manifold of dimension at least four with positive curvature on totally isotropic two planes is homeomorphic to a sphere as corollaries we will obtain a proof via minimal two spheres for riemannian manifolds of dimension at least four of the berger klingenberg rauch toponogov sphere theorem strengthened to pointwise quarter pinching as well as a proof that every simply connected compact riemannian manifold with positive curvature operators is homeomorphic to a sphere let m be an n dimensional riemannian manifold with tangent space tpm at the point p e m recall that the curvature operator at p is the self adjoint

on the cauchy problem for the zakharov system abstract we study the local cauchy problem in time for the zakharov system 1 1 and 1 2 governing langmuir turbulence with initial data u 0 n 0 t n 0 h k h lscr h l 1 in arbitrary space dimension we define a natural notion of criticality according to which the critical values of k l are 2 3 2 2 2 using a method recently developed by bourgain we prove that the zakharov system is locally well posed for a variety of values of k l the results cover the whole subcritical range for 4 for 3 they cover only part of it and the lowest admissible values are k l 1 2 0 for 2 3 and k l 0 1 2 for 1 as a by product of the one dimensional result we prove well posedness of the benney system 1 14 and 1 15 governing the interaction of short and long waves for the same values of k l

a group theoretic characterization of linear groups let r be a group when is f linear this is an old problem the first to study this question systematically was malcev in 1940 m who essentially reduced the problem to finitely generated groups note that a finitely generated group is linear over some field of characteristic zero if and only if it can be embedded in cl c for some n very little progress was made since that paper of malcev although as linear groups are a quite special type of group many necessary conditions were obtained e g r should be residually finite and even virtually residually p for almost all primes p f should be virtually torsion free and if not solvable by finite it has a free non abelian subgroup etc cf z of course none of these properties characterizes the finitely generated linear groups over in this paper we give such a characterization using the congruence structure of r first some definitions for a group h d h denotes the minimal number of generators for h

on groups of polynomial subgroup growth let be a finitely generated group andan the number of its subgroups of indexn we prove that assuming is residually nilpotent e g linear thenan grows polynomially if and only if is solvable of finite rank this answers a question of segal the proof uses a new characterization ofp adic analytic groups the theory of algebraic groups and the prime number theorem the method can be applied also to groups of polynomial word growth

asymptotics frequency modulation and low regularity ill posedness for canonical defocusing equations in a recent paper kenig ponce and vega study the low regularity behavior of the focusing nonlinear schrodinger nls focusing modified korteweg de vries mkdv and complex korteweg de vries kdv equations using soliton and breather solutions they demonstrate the lack of local well posedness for these equations below their respective endpoint regularities in this paper we study the defocusing analogues of these equations namely defocusing nls defocusing mkdv and real kdv all in one spatial dimension for which suitable soliton and breather solutions are unavailable we construct for each of these equations classes of modified scattering solutions which exist globally in time and are asymptotic to solutions of the corresponding linear equations up to explicit phase shifts these solutions are used to demonstrate lack of local well posedness in certain sobolev spaces in the sense that the dependence of solutions upon initial data fails to be uniformly continuous in particular we show that the mkdv flow is not uniformly continuous in the l 2 topology despite the existence of global weak solutions at this regularity finally we investigate the kdv equation at the endpoint regularity h and construct solutions for both the real and complex kdv equations the construction provides a nontrivial time interval t t and a locally lipschitz continuous map taking the initial data in h to a distributional solution u c 0 t t h which is uniquely defined for all smooth data the proof uses a generalized miura transform to transfer the existing endpoint regularity theory for mkdv to kdv

global wellposedness of kdv in h 1 t r by the inverse method we show that the korteweg de vries equation kdv tv x t x3v x t 6v x t xv x t x t t r is globally in time wellposed in the sobolev space of distributions h t r for any 1

stability of harmonic maps of kahler manifolds soit m n une application harmonique stable d une surface de riemann fermee dans un espace symetrique hermitien simple compact alors est holomorphe

a bilinear estimate with applications to the kdv equation u x 0 u0 x where u0 h r our principal aim here is to lower the best index s for which one has local well posedness in h r i e existence uniqueness persistence and continuous dependence on the data for a finite time interval whose size depends on u0 hs equation in 1 1 was derived by korteweg and de vries 21 as a model for long wave propagating in a channel a large amount of work has been devoted to the existence problem for the ivp 1 1 for instance see 9 10 the inverse scattering method applies to this problem and under appropriate decay assumptions on the data several existence results have been established see 5 6 14 28 33 another approach inherited from hyperbolic problems relies on the energy estimates and in particular shows that 1 1 is locally well posed in h r for s 3 2 see 2 3 12 29 30 31 using these results and conservation laws global in time well posedness in h r s 2 was established see 3 12 30 also global in time weak solutions in the energy space h r were constructed in 34 in 13 and 22 a local smoothing effect for solutions of 1 1 was discovered this combined with the conservation laws was used in 13 and 22 to construct global in time weak solutions with data in h r and even in l r in 16 we introduced oscillatory integral techniques to establish local well posedness of 1 1 in h r s 3 4 and hence global in time well posedness in h r s 1 in 16 we showed how to obtain the above mentioned result by picard iteration in an appropriate function space in 4 j bourgain introduced new function spaces adapted to the linear operator t 3 x for which there are good bilinear estimates for the nonlinear term x u 2 using these spaces bourgain was able to establish local well posedness of 1 1 in h r l r and hence by a conservation