The Markov Ordering of the Rationals

Explore the fascinating intersection of hyperbolic geometry and number theory in this 54-minute lecture by Jonah Gaster on the Markov ordering of rational numbers. Delve into the relationship between rational numbers and simple closed curves on a once-punctured torus with a complete hyperbolic metric. Discover how the "modular" torus, with its holonomy group conjugate to PSL(2,Z), reveals special arithmetic properties of curve lengths, connecting to Diophantine approximation. Learn about McShane's elegant proof of Aigner's conjectures regarding the partial ordering of rationals induced by hyperbolic length on the modular torus. Examine the characterization of monotonicity in this ordering along lines of varying slope in the (q,p)-plane, offering insights into the interplay between geometry and number theory.