Mathematics Colloquium - Markov numbers and integer geometry

Speaker: Matty Van-Son (The Open University)

Abstract: We discuss the history of Markov numbers, which are solutions to the equation x^2+y^2+z^2=3xyz. These solutions can be arranged to form a tree, and we show that similar trees of SL(2,Z) matrices, quadratic forms, and sequences of positive integers relate very closely to Markov numbers. We use the tree structure of sequences, along with a geometric property of the minimal value of forms at integer points, to propose an extension to Markov numbers. We then study this extension.

This is a joint work with Oleg Karpenkov (University of Liverpool).