Cryo-electron microscopy is a promising imaging technique in structural biology, yielding a large number of very noisy images of a macromolecule in different, unknown rotations. The computational task of reconciling these images into a 3D model of the molecule has proven both mathematically rich and challenging, leading to a mathematical formulation of "synchronization" problems: the learning task of aligning rotated objects based on noisy measurements of their pairwise relative rotations. We present an algorithm following the framework of approximate message passing, which statistical physics suggests may yield the optimal efficient reconstruction. Our approach leverages the representation theory of compact groups to give a unified, general theory for problems with various conceptual 'rotations' or 'alignments'. (Joint work with Amelia Perry, Afonso Bandeira, and Ankur Moitra.)