Mathematicians use groups to capture the idea of symmetry, and manifolds to describe the shapes of spaces—from the surface of the Earth to models of the universe. In this talk, I will show how drawing and visualizing groups can reveal surprising information about the shape of a space. Through pictures, examples, and geometric intuition, we will see how symmetry can constrain and sometimes completely determine the global structure of a manifold. The talk is designed to be accessible to students and non-experts.
