A quiver is another name for a directed graph, usually with a finite number of vertices and arrows. A representation of a quiver consists of a vector space at each vertex and a linear map for each arrow.

The fundamental problem is to classify such representations. This is an impossible problem in general. Nevertheless one may in simple cases classify the representations: for example, one vertex and one arrow, a loop from that vertex to itself, leads to the Jordan Normal Form classification of matrices. If one takes a Dynkin diagram of type A, D, or E , and puts an arrow on the edge one gets a quiver and these are exactly the quivers with the following property: there is only a finite number of representations that can not be written as direct sums of smaller representations (an indecomposable representation) and every representation is a direct sum of copies of these indecomposable ones. Moreover, in that case, the indecomposables are in bijection with the positive roots of the root system corresponding to the Dynkin diagram.

For more complicated quivers there are infinitely many indecomposable representations and these usually come in families that are parameterized by interesting geometric objects. We will give some examples.

Representation theory of quivers interacts with a wide range of other branches of mathematics (even string theory). There are a number of reasons for this but one fundamental reason is that in many areas of mathematics one is interested in collections of objects and maps between them: one then gets a quiver by assigning a vertex to each object and an arrow to each map.

This talk will be a gentle introduction to the subject.