If you give a mathematician a surface, their neighbor will give you another one. If their neighbor gives you another surface, you may start to wonder if they are the same. If you want to determine if two surfaces are the same, you will have algebraist, geometers, topologist and number theorists knocking at your door, asking them to show you how they think about surfaces. If you give me some time to talk, I’ll tell you about some fun ways to visualize and move between different representations of surfaces and the research inspired by these visualizations.

In this talk, I will describe some fun and intuitive ways to visualize Riemann surfaces and to move between their many representations. I will also discuss how these visual perspectives inspire new questions and directions in my research.