Discrete restriction, exponential sum and their application in nonlinear dispersive equations

Bourgain introduced some methods in number theory to study periodic Schordinger equations and built the bridge between partial differential equations and number theory. In the framework on Stein-Thomas, the Strichartz estimates for Schordinger equations were established with the aid of Weyl summation and Hardy-Littlewood’s circle methods, etc. However, the estimates are proved for only partial integrable indices in the higher dimensional case. It was surprising that Bourgain and Demeter developed the decoupling inequality, which not only helped proved the Strichartz estimates nonlinear Schordinger equations, but also became a new method to study some famous conjectures in harmonic analysis, analytic number theory, and geometric measure theory, etc. This talk will present some methods in number theory, exponential sum based on decoupling method, and their application in nonlinear periodic dispersive equations.