# Inverse Problems, Imaging and Nonlinear Algebra

Recognized by the 2017 Chemistry Nobel Prize, cryo-electron microscopy (cryo-EM) is an imaging technique to determine the 3-D shape of macromolecules from many noisy 2-D images. Mathematically, cryo-EM represents a particularly rich inverse problem, with unknown orientations, extreme noise, big data, and conformational heterogeneity.

This lecture presents mathematical developments arising from cryo-EM, notably involving nonlinear algebra. In particular, I will introduce a general framework for statistical estimation under compact group actions, which establishes a connection between information theory and group invariant theory. I will also discuss a novel tensor decomposition algorithm, combining ideas from Sylvester (19th century) and the power method, of broad applicability in data science. Specializing to cryo-EM, I will provide answers to domain questions that were raised 40 years ago, and present *ab initio* reconstruction results obtained using a numerical optimization-based solver. The approach is promising for other imaging modalities as well, though several mathematical and computational challenges remain.

*Joe Kileel is currently a Simons postdoc in the Program in Applied and Computational Mathematics, Princeton University working with Amit Singer. In 2017, he obtained his PhD in mathematics from UC Berkeley under the supervision of Bernd Sturmfels, where his thesis received the Bernard Friedman Memorial Prize for best in applied mathematics.*