Caltech/UCLA Joint Analysis Seminar

Fourier restriction, Radon-like operators, and decoupling theory are three active areas of harmonic analysis which involve submanifolds of Euclidean space in a fundamental way. In each case, the mapping properties of the objects of study depend in a fundamental way on the "non-flatness" of the submanifold, but with the exception of certain extreme cases (primarily curves and hypersurfaces), it is not clear exactly how to quantify the geometry in an analytically meaningful way. In this talk, I will discuss a series of recent results which shed light on this situation using tools from an unusually broad range of mathematical sources.