Title: A positive dichotomy between Diophantine geometry and Dynkin friezes

Abstract: General finiteness results for positive integer solutions to Diophantine equations are rare. In this talk, I will describe how the theory of cluster algebras can be a remarkable source of finiteness & infinitude in the form of a Siegel theorem for positive integral points on affine varieties. On the finiteness of positive integral points, I will highlight 7-dimensional and 8-dimensional affine varieties that arise in the Fontaine–Plamondon conjecture on Dynkin friezes. On the infinitude of positive integral points, I will highlight surfaces and threefolds that arise in the Mordell–Schinzel program on Diophantine equations xyz = G(x, y).