Describing geometry and symmetry of cryo-EM datasets using algebra

Cryo-electron microscopy (cryo-EM) is a microscopy technique used to discover the 3D structure of molecules from very noisy 2D images. This thesis presents two projects that use algebra to describe cryo-EM datasets. In the first part, we obtain an algebraic description of common lines datasets. Common lines are lines of intersection between cryo-EM images in 3D. They are an important ingredient in some 2D to 3D reconstruction algorithms, and they can be characterized by polynomial equalities and inequalities. In the second part, we show how 3D symmetries of a molecule can be detected from only 2D cryo-EM images, without performing full 3D reconstruction.