Talk Abstracts

Many problems in Fourier analysis, like Fourier restriction theory and the regularity theory of Radon-like transforms, can be understood abstractly as the study of natural operators associated to submanifolds of Euclidean space. Often these operators are “bad” when the submanifolds lie in a proper affine subspace of the ambient space, but likewise it is often difficult to understand exactly which submanifolds are “well curved” in the ways that matter in each context. Through a series of examples where notions like curvature and transversality are important but poorly understood, we will see how framing the problems as reflecting and arising from affine symmetry illuminates new connections between Fourier analysis, differential geometry, and algebraic geometry (specifically, Geometric Invariant Theory).