Timothy Candy

Harmonic Analysis and Dispersive PDE

A central object of study in harmonic analysis is the Fourier transform. The Fourier transform provides a powerful tool for studying waves and oscillations, and so it is unsurprising that much of the progress in dispersive partial differential equations (PDE) over the past decades has relied crucially on developments in harmonic analysis. In these two lectures we explore the deep interplay between harmonic analysis and dispersive PDE. We begin with the Fourier restriction problem and the role of curvature in oscillatory integrals. We then turn to nonlinear applications, highlighting Strichartz estimates, bilinear restriction estimates, and some of the ideas used in the large data theory of nonlinear dispersive PDE. In particular, we attempt to illustrate how tools from harmonic analysis have played a key role in understanding the behaviour solutions to nonlinear dispersive PDE.