The goal of this project is to study the Hitchin--Kobayashi correspondence which relates solutions of certain differential equations to certain geometric objects with algebraic structure. The differential equations of interest are related to those studied in particle physics. By viewing them from an algebraic perspective, we can often find properties of the space of solutions which would otherwise have been difficult to see. This correspondence has been proved for the compact and finite volume cases, but while some specific cases outside these have been worked out, the correspondence in the infinite volume case is still relatively not well-understood, and this is an active area of research.