Mark Gritter

This talk on Apollonian circle packings is great: https://youtu.be/3mU9d0cPgGM

The local-global conjecture said that other than the "obvious" congruence restrictions, every sufficiently large natural number appeared as the curvature in a circle packing. But this is false and the counterexample is not at all hard: they exhibit a circle packing that contains no squares, and a simple proof based on reciprocity.

The best part is that this started with a REU (an undergrad) doing some visualizations and finding an increasingly-clear pattern suggesting the conjecture couldn't be true.

Katherine E. Stange: The local-global conjecture for Apollonian circle packings is false (NTWS 235)

YouTube
Jan 29, 2025 at 8:08pmWeb
Show threadMark GritterJan 29, 2025

A couple captures from the stream (haven't read the paper yet!)

Left: the pattern found in the REU: fix a pair of curvatures (from admissible congruence classes) and plot a black dot if no packing has that pair. Local-global conjecture says this should "empty out", but doesn't seem to be happening!

Right: now that they know what to look for, another way of discovering a pattern: plot differences between missing curvatures and a striking structure emerges after the initial mess.

#MathVisualization

Show threadMark GritterJan 29, 2025
Paper is https://arxiv.org/abs/2307.02749
The local-global conjecture for Apollonian circle packings is false

In a primitive integral Apollonian circle packing, the curvatures that appear must fall into one of six or eight residue classes modulo 24. The local-global conjecture states that every sufficiently large integer in one of these residue classes will appear as a curvature in the packing. We prove that this conjecture is false for many packings, by proving that certain quadratic and quartic families are missed. The new obstructions are a property of the thin Apollonian group (and not its Zariski closure), and are a result of quadratic and quartic reciprocity, reminiscent of a Brauer-Manin obstruction. Based on computational evidence, we formulate a new conjecture.

arXiv.org