published on May 19, 2026
Sonification in number theory: listening to the Riemann zeta function
By Jonathan Love
Connectivity of Markoff and Nielsen graphs
Appears in collection : 2026 - T1 - WS3 - Integrating Research and Illustration in Number Theory
The generalized Markoff equation gives rise to a dynamical system via the Markoff group action on its solution set. Over finite fields, the action produces graphs that are conjectured by Bourgain, Gamburd, and Sarnak to form an expander family. This conjecture has implications in both number theory (strengthening the affine linear sieve for Markoff numbers) and computational group theory (bounding runtime of the Product Replacement Algorithm for $SL_2(F_p)$). In this talk, we discuss recent progress toward proving connectivity of Markoff graphs and related results on Nielsen graphs of matrix pairs from $SL_2(F_p)$.
