Carleson proved in 1966 that the Fourier series of any square integrable
function converges pointwise almost everywhere to the function, by establishing boundedness
of the maximally modulated Hilbert transform from L^2 into weak L^2. This
talk is about a generalization of his result, where the Hilbert transform
is replaced by a singular integral operator along a paraboloid. I will
review the history of extensions of Carleson's theorem, and then discuss
the two main ingredients needed to deduce our result: sparse bounds for
singular integrals along the paraboloid, and a square function argument
relying on the geometry of the paraboloid.