Curves on algebraic surfaces and hyperbolic codes

I will talk about joint work with Ted Chinburg where we turn a problem about finiteness results for algebraic curves of negative self-intersection on an algebraic surface into a packing problem in hyperbolic space. Specifically, we develop the notion of a "hyperbolic code", which mirrors the theory of spherical codes on Euclidean spheres. After briefly describing the motivation from algebraic surfaces, I will spend the rest of the talk explaining the geometry of hyperbolic codes, the hyperbolic kissing number, and what bounds we can prove for hyperbolic kissing numbers using a mix of hyperbolic and spherical geometry.