Dear All!
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")
Best Answer
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, 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) 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).
Also in physics applications in String Theory, Supersymmetry, Black Holes and Particle physics.
Relation with quantum dilogarithm, number theory, and cluster algebras, read e.g. this 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.