Colloquium: Dr. Jonah Gaster

Many of the 21st century spectacular advances in geometry and topology involve the study of simple closed geodesics on hyperbolic surfaces (Thurston, Mirzakhani, etc). I will tell you about one difficult little aspect of this story that I’ve been thinking about recently: How many simple closed geodesics on a hyperbolic surface may have precisely the same length? I will sketch some of the background ideas that indicate why this question is deep, and related to other things that sound like they have nothing to do with hyperbolic geometry (like modular curves, Diophantine approximation, and solutions to polynomial equations).