Ezgi Kantarcı Oğuz (Galatasaray University) – A poset model for $q$-deformed Markov Numbers

The positive integers that come up in the solutions of the Diophantine equation $x^2+y^2+z^2=3xyz$ are called Markov numbers. They play an important role in the theory of rational expansions and come with a 100+ year old open problem called the Uniqueness conjecture. Recently discovered connections to cluster algebras have revitalized the theory, leading to generalizations and deformations motivated by the cluster model. In this talk, we will use oriented posets to construct a combinatorial model for $q$-deformed Markov numbers. We will also discuss future directions and further avenues of research.