Title: A cluster-theoretic perspective on Markov Numbers and Cohn Matrices
Abstract: Markov numbers, i.e. numbers which appear in solution triples to x^2 + y^2 + z^2 = 3xyz, first appeared in the context of Diophantine approximation. Cohn exhibited a connection between Markov numbers and the lengths of closed simple geodesics on the punctured torus. A byproduct is the family of Cohn matrices, which can be seen as a matrixization of Markov numbers. It is known that Markov numbers can be viewed as specializations of cluster variables in the cluster algebra from a once-punctured torus. We give a cluster-algebraic interpretation to Cohn matrices, using poset-theoretic formulas from Kantarcı Oguz-Yıldırım and Pilaud-Reading-Schroll. We also discuss variations of the Markov equation and cluster Cohn matrices, inspired by generalized cluster algebras in the sense of Chekhov-Shapiro. This is based on joint works with Yasuaki Gyoda and with Archan Sen.