Geometry of planar surfaces and exceptional fillings

The Tammes problem considers the densest disk packing on the surface of a 2-sphere embedded in $\mathbb{R}^3$.  There is a natural analog of this problem in hyperbolic geometry considering the densest equal area cusp packing on n-punctured spheres (with $n\geq 3$). After providing the relevant background, we will discuss progress on this question. As part of this talk, we will also describe via an embedding argument that reduces a problem in 3-manifold topology to this 2-dimensional question about packing punctured spheres. This is joint work with Jessica Purcell.