Title: Extensions in the Preprojective Algebras 

In 2001, Fomin and Zelevinsky introduced cluster algebras which can be used to describe many important varieties from Lie theory. Leclerc gives a cluster structure on coordinate rings of Richardson varieties using the representation theory of preprojective algebras. While his construction is very algebraic, we take a more combinatorial approach. We are aiming to find a combinatorial description for when modules in the preprojective algebra have extensions, which corresponds to describing when cluster variables are compatible. We will partially answer this question for certain types of modules and discuss future goals for further development.