Published: Aug. 21, 2023
Apollonian circle packing coloured mod 3

This summer, as part of the Department's Summer Research Experience for Undegraduates and First-Year Graduate Students, graduate student Summer Haag, undergraduate Clyde Kertzer, postdoc James Rickards and professor Katherine E. Stange had a surprise discovery;  the twenty-year old Local-Global Conjecture for Apollonian circle packings is false! Integer Apollonian circle packings, like the one shown at right, have circles whose curvatures are integers.  The image at right colours the circles according to the curvature modulo 3, showing that only integers congruent to 0 or 2 modulo 3 can appear.  The conjecture held that, aside from finitely many exceptions and some congruence conditions like the mod 3 restriction in the picture, every sufficiently large integer should appear as a curvature in a primitive integral Apollonian circle packing.  The team proved the existence of some new obstructions:  for example, some packings cannot admit square curvatures.  The proof depends on quadratic and quartic reciprocity.

The result was covered in Quanta Magazine.