Automorphic forms, Galois representations and the geometry of numbers
Automorphic forms are another class of analytic functions with deep connections to number theory. The best known examples are classical modular forms, whose connection with elliptic curves played a crucial role in the proof of Fermat's Last Theorem. This connection is but a special case in the Langlands Programme, a vast web of conjectures relating automorphic forms to arithmetic objects, such as algebraic varieties and representations of Galois groups. The group's research focuses on establishing and understanding this relationship in various contexts --- local and global, classical and p-adic. This bridge between analytic and arithmetic objects has far-reaching applications in number theory, strongly linked with the role of L-functions.