Moduli spaces—informally, spaces that parametrize shapes—play a central role in geometry, with moduli spaces of hyperbolic surfaces providing a particularly rich example. The talk begins with puzzles and visualizations of related spaces, before lurching toward hyperbolic surfaces. Somewhat unexpectedly, these same surfaces also arise in the solution of certain graph drawing problems, revealing a bridge between surface geometry and visualization.
Parts of the talk are based on joint work with Alfredo Hubard and Arnaud de Mesmay. Although no prior background is assumed, feel free to prepare for the talk by playing Quadratis (http://quadratis.app) developed with together with Paul Turner, Mario Gutiérrez and Reyna Juárez.