By definition, a quiver is a directed graph.  

In this talk, we will recall the notion of a quiver representation.

The category of quiver representations can be highly non-trivial, e.g. the category of modules over any finitely-dimensional algebra over a closed field is equivalent to the category of quiver representations for some quiver.  

However, for some quivers, their representations have only finitely many indecomposables (meaning they are much easier to study). Famously, due to Gabriel, those are precisely the simply-laced Dynkin quivers: A_n, D_n, E_6, E_7, E_8.

I will explain how the corresponding combinatorics of root systems arise from representation theory via Bernstein—Gelfand—Ponomarëv reflection functors and prove Gabriel's theorem.