Introduction and summary Ⅰ.An overview of results on the Cauchy problem for NLS 1.Equations 2.Wellposedness of the Cauchy problem 3.Scattering results 4.Estimates on the linear group 5.Solving the Cauchy problem 6.Derivative nonlinear Schr?dinger equations Ⅱ.Further comments 1.Construction of blowup solutions for conformal NLS from the groundstate 2.Behaviour of higher Sobolev norms 3.Fourier restriction theory beyond L2 4.L2-concentration phenomenon 5.The Schr?dinger maximal function 6.Derivative nonlinear Schr?dinger equation Ⅲ.3D H-critical defocusing NLS 1.Consider 3D NLS 2.Fix a time interval I = (0,T] 3.Sketch of the argument 4.A concentration property 5.A version of Morawetz inequality 6.Construction of an appropriate time interval 7.Details on the perturbative analysis 8.A variant of the method Ⅳ.Global wellposedness below energy norm 1.Description of the method 2.The example of the NLW 3.The case of the nonlinear Schr?dinger equation 4.Symplectic capacities and symplectic Hilbert spaces 5.Global wellposedness of the NLW (4.41) Ⅴ.Nonlinear Schrodinger equation with periodic boundary conditions 1.Introduction 2.Results on the Cauchy problem 3.Periodic Strichartz inequalities 4.Sketch of proof of Theorems 2.1 and 2.7 5.Invariant Gibbs measures (1D) 6.Invarinat Gibbs measures (D＞1) 7.Invariant Gibbs measures (unbounded domains) 8.Quasi-periodic solutions Appendix 1.Growth of Sobolev norms in linear Schrodinger equations with smooth time dependent potential Appendix 2.Zakharov systems References Index
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