Number Theory Seminar

General finiteness results for positive integral 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 for positive integral points in the form of a positive Siegel theorem for affine varieties of dimension at least 2. On the finiteness of positive integral points, I will highlight 7-dimensional and 8-dimensional affine varieties that arise in the proof of the Fontaine–Plamondon conjecture for Dynkin friezes. On the infinitude of positive integral points, I will highlight surfaces and threefolds that arise in the Mordell–Schinzel program.