Analysis, Spectra and Number Theory, a Conference in Honor of Peter Sarnak’s 61st Birthday

A well-known theorem called the Sylvester-Gallai theorem states that, given n points in R^2, either they all lie on a line or else there is an ordinary line - a line through precisely two of the points. I will discuss joint work with Tao establishing a sharp lower bound on how many ordinary lines there must be, at least for n sufficiently large. Our argument uncovers algebraic structure in point configurations with few ordinary lines (or equivalently many collinearities). I will speculate on other situations where combinatorial information about collinearity might imply algebraic structure.