Behind many of the recent advances in number theory, like Fermat's last theorem or the Sato-Tate conjecture, are Galois representations and their associated L-functions. Galois representations are of great interest to number theorists since they encode the answers to many arithmetic questions.
In this talk, we will define the notion of a Galois representation, and discuss the relationship between various open problems related to their L-functions and questions arising from arithmetic.