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).