The S-duality of the 4D super-Yang-Mills theory, proposed by Montonen and Olive in the late 70s, is one of the Holy Grails of modern Quantum Field Theory.  Mysteriously and surprisingly, the gauge group gets replaced under this duality by another group. The same dual group appeared a decade earlier in Mathematics in the work of Robert Langlands, in what has become known as the Langlands Program, unifying such fields as Number Theory, Harmonic Analysis and Geometry.

Recently, in a groundbreaking advance, Kapustin and Witten have linked the geometric Langlands correspondence and 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.

The goal of this Mini-Program is to study further the beautiful structures of these dualities from the point of view of both Physics and Mathematics. In particular, we will discuss recent results of Gaiotto and Witten on the S-duality of more general boundary conditions in the 4D super-Yang-Mills theories and their implications for the geometric Langlands correspondence. Closely related supersymmetric quantum field theories in four, three and two space-time dimensions will also be discussed.