Mordell-Schinzel surfaces and cluster algebras

Abstract: The set of positive integer points of the celebrated Markov surface admits the structure of a 3-regular tree.

My objective in this talk is to unveil a similar phenomenon for Mordell-Schinzel surfaces; namely that the set of positive integer points of each such surface admits the structure of a 2-regular graph. The vertices of each graph naturally correspond to clusters in a suitable (generalised) cluster algebra.

I will then explain how the structure theory of cluster algebras translates into a resolution of the positive Mordell-Schinzel problem.

This is partly based on ongoing joint work with Robin Zhang (MIT).

algebraic geometrynumber theory

Audience: researchers in the topic

Boston University Number Theory Seminar