A macro-meso-microscopic view of disk packing
This event is part of the Condensed Matter Theory Seminar Series.
Everyone knows hexagonal close-packings are the most efficient way to pack unit disks in R^2. But what do we mean by the definite article "the" in the previous sentence? Are hexagonal close-packings the only optimal packings? If we measure optimality by density, the answer is, No. In fact, there are far too many density-optimal packings to classify in any meaningful way. This suggests that density is too coarse a notion to capture everything that we mean (or should mean!) by "efficient".
To study efficiency, we study deficiency, and seek ways to quantify defects in a regular packing. An obstacle here is that common kinds of defects inhabit disparate scales (e.g., point defects are infinitesimal compared to line defects, which in turn are infinitesimal compared to the bulk). This suggests we turn to extensions of the real numbers that include infinitesimal elements (or rather, as turns out to be more helpful, infinite elements).
We use a regularization trick to make sense of these ideas. This enables us to sharpen our notion of optimal packing so that the optimal disk-packings are provably the hexagonal close-packings and no others. Unfortunately, the notion of deficiency covered by this theorem is not the most natural one, nor is it the one most directly applicable to higher-dimensional problems. This is very much work-in-progress and I'm hoping others can help!