Non-commutative cluster algebras beyond surfaces

Since their introduction cluster algebras have been related to many aspects of mathematics, and notably Fock and Goncharov established a connection with higher Teichmüller theory. In trying to generalize their construction to the Theta-positive setting, one naturally encounters non-commutative cluster structures. I will give a quick overview of this, focused on the case of non-commutative surfaces as introduced by Berenstein and Retakh, before explaining ongoing work on Fock-Goncharov coordinates for representations of fundamental groups of bordered surfaces into Spin(p,q), and especially on understanding the underlying non-commutative cluster structure. This is joint work with Zack Greenberg, Dani Kaufman and Anna Wienhard.