Jean Bourgain

On lipschitz embedding of finite metric spaces in Hilbert space

It is shown that anyn point metric space is up to logn lipeomorphic to a subset of Hilbert space. We also exhibit an example of ann point metric space which cannot be embedded in Hilbert space with distortion less than (logn)/(log logn), showing that the positive result is essentially best possible. The methods used are of probabilistic nature. For instance, to construct our example, we make use of random graphs.

Global solutions of nonlinear Schrödinger equations

Introduction and summary An overview of results on the Cauchy problem for NLS Further comments 3D

Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case

H^1

Estimates for the Number of Sums and Products and for Exponential Sums in Fields of Prime Order

-critical defocusing NLS Global wellposedness below energy norm Nonlinear Schrodinger equation with periodic boundary conditions Growth of Sobolev norms in linear Schrodinger equations with smooth time dependent potential Zakharov systems References Index.

Explicit constructions of RIP matrices and related problems

is globally wellposed in time. More precisely, we obtain a unique solution u = uφ ∈ CH1([0,∞[) such that for all time, u(t) depends continuously on the data φ (in fact, the dependence is even real analytic here). Moreover, there is scattering for t→∞. The same statement holds for radial data φ ∈ H, s ≥ 1 and proves in particular global existence of classical solutions in the radially symmetric case. Also this issue was open. Thus this is the analogue for NLS of the result for the wave equation with quintic nonlinearity obtained by M. Struwe [Str] in the radial case (and by M. Grillakis [Gr], [S-S], in general). In the case of the wave equation, the proof is based on the following two different facts:

Invertibility of ‘large’ submatrices with applications to the geometry of Banach spaces and harmonic analysis

Our first result is a ‘sum-product’ theorem for subsets A of the finite field

On the maximal ergodic theorem for certain subsets of the integers

{{\mathbb F}_p}

On nonperturbative localization with quasi-periodic potential

, p prime, providing a lower bound on

Lifting in Sobolev spaces

\max (|A+A|, |A\cdot A|)

Continuity of the Lyapunov Exponent for Quasiperiodic Operators with Analytic Potential

. The second and main result provides new bounds on exponential sums \[\sum_{x_1,\dots,x_k\in A} \exp(2\pi ix_1\dotsc x_k\xi/p),\] where