Abstract: In this talk, I will discuss a couple general mathematical frameworks that have arisen in the study of the cryo-electron microscopy problem. I will start by recalling that cryo-EM is about determining the 3D shape of a protein from many noisy tomographic images of randomly oriented particles. Motivated by this application, I will mostly focus on the use of non-Euclidean norms in manifold learning. Synthetic results show, that when applying diffusion maps to volumetric data, it can be advantageous to base affinities on the Earth Mover’s Distance rather the Euclidean norm. I will characterize the limiting differential operator that we obtain when we use non-Euclidean norms, and contrast this to the classical Laplace-Beltrami operator. Time permitting, I may also mention the problem of parameter estimation in statistics in the presence of group actions, and some unexpected connections to invariant theory. Based on joint works with several coauthors who will be named precisely during the talk.