Curvature, Combinatorics, and the Fourier Transform, Volume 48, Number 6

I n their 1978 paper “Problems in harmonic analysis related to curvature” [SteinWainger78], E. M. Stein and S. Wainger studied a variety of operators defined over hypersurfaces and other lower-dimensional submanifolds of Rd . The recurring theme is that the behavior of these operators is governed, to various degrees, by the Gaussian curvature of the underlying manifold— the determinant of the differential of the Gauss map taking a point on the manifold to the unit normal at that point. In recent years these ideas have undergone further development, not only in the context of harmonic analysis, but also in related problems in combinatorics and analytic number theory. This article is not intended to be an exhaustive survey of the most recent advances in these areas. Rather, the purpose of this article is to describe some of these connections by way of examples accessible to a wide range of mathematicians.