The circle has a one-dimensional hole. The (2-dimensional) sphere has a 2-dimensional hole. A 3-dimensional sphere -- you can't visualize these -- has a 3-dimensional hole. No surprise yet. But it turns out the 2-dimensional sphere also has a 3-dimensional hole. We'll start by understanding what all these statements mean. We'll then see how to visualize the (surprising) 3-dimensional hole in the 2-sphere, and sketch a proof that it's really there. Knots and complex numbers will make guest appearances.
Knowing the definition of a group and having a feeling for what continuity means will help. Although the talk is about topology, no prior knowledge of topology will be assumed.