Representation theory is a very broad subject. In a nutshell, it is a systematic study of how abstract groups (or algebras) can be represented by concrete linear transformations of a vector space. A guiding example is the symmetric group on four letters, which can be thought of as the rotational symmetries of a cube. Representation theory pervades diverse areas of mathematics, and even particle physics. In number theory the Langlands program posits a deep connection between representations of various Lie groups and representations of Galois groups, through the theory of L-functions.