We start by recalling the essential geometry of wavelets, curvelets, ridgelets, Gabor, and how they all fit together in one picture. We will introduce a newcomer, 'wave atoms', which is a natural choice for the purpose of sparsely representing pseudo-differential and Fourier integral operators. We will then discuss the practical implementation of curvelets as numerically tight frames, as faithful to the continuous transform as possible, and in complexity 30 NlogN flops for both the direct and inverse transforms. The key step in the algorithm is the so-called 'wrapping' of selected Fourier samples taken from the same scale and angle. If there is time, we will discuss the implementation of wave atoms, and how it is possible to crack the infamous universal bound on time-frequency localization of wavelet packets.

(Joint work with Emmanuel Candes and Lexing Ying.)