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X-ray crystallography is a widely used technology for reconstructing the lattice structure of crystallized protein molecules from the diffraction patterns produced when the crystal is bombarded with X-rays. This technology plays a critical role in drug discovery and medicine. At the heart of this method lies the so-called "phase problem": if we treat the protein's electron density as the signal to be reconstructed, the diffraction pattern fails to captures any phase information of this signal. The crystallographic phase retrieval problem therefore seeks to recover an unknown signal from its crystallographic phase-less measurements. In this talk, we will introduce the crystallographic phase retrieval problem, discuss its well-posed-ness, and outline a few results that determine if and when recovery is possible. To establish these recovery results, we will outline how the crystallographic phase retrieval problem is a special instance of a problem seeking to invert the second moment measurements of a more general framework known as the Multi-Reference Alignment (MRA) model.

This talk is based on joint works with Dan Edidin, Nadav Dym and Tamir Bendory.