Enumerating frieze patterns via Diophantine geometry

*Note: unusual location

Frieze patterns are infinite arrays of integers introduced by Coxeter as combinatorial objects with deep geometric and algebraic roots. Classical work of Conway–Coxeter and later developments from cluster algebra theory have produced many enumeration results over the positive integers and finite fields. I will explain how the combinatorial enumeration problem for friezes can be translated into solving families of Diophantine equations, and how tools from arithmetic geometry then allow a complete classification of positive integral friezes and enumerations over more general integral domains. This settles a conjecture of Fontaine–Plamondon, extends previous enumerations of friezes over finite fields beyond the classical cases, and gives partial (positive) answers to Diophantine questions raised by Mordell, Schinzel, and Kollár–Li. No background in arithmetic geometry will be assumed.

For information about the Richard P. Stanley Seminar in Combinatorics, visit… https://math.mit.edu/combin/