Appears in collection : The Pentagram Map and Ramifications

The classical cross-ratio "crosses" both classical and modern mathematics, linking geometry with combinatorics, dynamics, and mathematical physics. It provides natural coordinates on many moduli spaces. In this talk, I will introduce a new q-deformed, or ``quantum'' cross-ratio: a PSL(2,Z)-invariant of ordered quadruples of distinct points on the rational projective line. The construction is based on the theory of q-deformed rational numbers developed jointly with Sophie Morier-Genoud. I will discuss several properties of this new invariant, with its close connection to q-deformed Coxeter frieze patterns taking centre stage.