Carleson's operator is a useful device in the study of pointwise convergence of Fourier series of a general $L^2$ function. Stein and Wainger studied a variant of Carleson's operator, where a linear phase function is replaced by a polynomial phase. In this talk, we study a variant of the result of Stein and Wainger, where the maximal operator is replaced by its variational norm counterpart. Connections will be made to square function estimates and local smoothing estimates of the linear Schrodinger equation. This is joint work with Shaoming Guo and Joris Roos.