Title: The symplectic packing numbers
Abstract: The k-th "symplectic packing number" of a finite volume symplectic manifold is the proportion of the volume that can be filled by k disjoint symplectically embedded balls. In the first part of my talk, I will explain what we know about these numbers, and give some examples of what we'd like to know. In particular, I will explain the following remarkable stability property, which sharply contrasts the Riemannian case: for many symplectic manifolds, the packing number is 1 for sufficiently large k. This "packing stability" phenomenon was discovered by Biran in the 90s. In the second part, I will explain some recent work aimed at understanding how characteristic packing stability actually is. In fact, it has been conjectured that it holds for all symplectic manifolds, but we will show that it does not. A symplectic "fractal Weyl law" plays a major role in our proof.