  The Langlands Program was launched in the late 60s with the goal of relating
Galois representations and automorphic forms. In recent years a geometric
version has been developed which leads to a mysterious duality between certain
categories of sheaves on moduli spaces of (flat) bundles on algebraic curves.
Three years ago, in a groundbreaking advance, Kapustin and Witten have linked
the geometric Langlands correspondence to the S-duality of 4D supersymmetric
gauge theories. This and subsequent works have already led to striking new
insights into the geometric Langlands Program, which in particular involve the
Homological Mirror Symmetry of the Hitchin moduli spaces of Higgs bundles on
algebraic curves associated to two Langlands dual Lie groups.
