From sphere packings to extremal problems on hyperbolic surfaces

Dr. Maxine Fortier Bourque
Professor of Mathematics
Université de Montréal

What proportion of space can be occupied by balls of radius 1 that do not overlap? How many such balls can be tangent to a central ball? We know the answer to the two questions in dimensions 1, 2, 3, 8, and 24, and to the second question in dimension 4 as well. In most cases, the best known upper bounds are obtained through analysis rather than geometry, via a method called “linear programming” developed by Delsarte. In this talk, I will discuss joint work with Bram Petri in which we adapt these linear programming methods to obtain inequalities on several invariants associated with closed hyperbolic surfaces. The resulting upper bounds are the best known to date except in a handful of genera. The same methods allowed us to discover a new optimizer in genus 3 and two counterexamples to a 37-year-old conjecture of Colin de Verdière.