There is a trend, for some people, to study representations of quivers. The setting of the problem is undoubtedly natural, but representations of quivers are present in the literature for already >40 years.


Are there any connections of this trend with other Maths? For, it seems like it is a self-contained topic and basically I wonder why people study quivers so much -- in a sense everything becomes clear after the initial results of Gabriel and the old result of Yuri Drozd about wild/tame dichotomy, and these things ought to become boring.


BTW, about the dichotomy theorem -- is it really necessary to study so hard whether a given problem is of tame or wild (representation) type? In particular, why some people try to lift tame/wild things to curves and surfaces -- would that really yield something interesting in geometry?


(I am currently attending lectures about these things and unfortunately we were not told a single word about motivation, and when I tried to learn from the lecturer if this is really "top" Maths as he claims, he basically replied "this is important because I am doing this")



Are quivers useful outside of Representation Theory?
In addition to being a nice example for abelian, $A\_{\infty}$ and Calabi–Yau categories, and being a prototypical example for [Generalized Donaldson–Thomas Invariants and the Wall Crossing Phenomenon](https://arxiv.org/abs/0811.2435), quivers have a lot of applications in various different fields. Since the question is asking for applications in addition to representation theory, I'm listing a few cases.


Most prominent is Algebraic Geometry, particularly Moduli problems and GIT (read motivations in [Reineke's article](https://arxiv.org/abs/0802.2147)) and Video lectures on quivers by Reineke at Newton Institute, Cambridge.


Recently, a correspondence has been proposed between Gromov–Witten invariants and Quivers. ([Pandharipande–Gross](https://arxiv.org/abs/0909.5153)).


Also in physics applications in [String Theory](https://arxiv.org/abs/hep-th/9603167), [Supersymmetry](https://arxiv.org/abs/hep-th/0006189), [Black Holes](https://arxiv.org/abs/hep-th/0206072) and [Particle physics](https://arxiv.org/abs/hep-th/0007173).


Relation with quantum dilogarithm, number theory, and cluster algebras, read e.g. [this](https://arxiv.org/abs/1102.4148) review by Keller.


Also through the work of Nakajima, there is a relation to Instantons of Yang–Mills theory, Hitchin Moduli spaces and the theory of Hyperkähler manifolds.


