An overview of the Geometric Langlands Program

Robert Langlands in the 1960's proposed a program, which is a
culmination of the sequence of theorems in number theory beginning
with the quadratic reciprocity law, proved by Gauss around 1800.
Alexander Beilinson and Vladimir Drinfeld suggested a geometric
version of the Langlands program bringing the ideas of Langlands
together with tools and ideas from other areas of mathematics and
physics: algebraic geometry, sheaf theory, moduli spaces, integrable
systems and many others.

The Categorical Langlands Duality is a conjectural equivalence of two
categories associated to a reductive group and a smooth projective
curve; it is the strongest form of the Geometric Langlands
Correspondence. I will formulate this equivalence and explain what is
currently known. Then I will discuss a certain semi-classical limit of
the conjecture, which is a duality for Hitchin systems. I will survey
known results. Time permits, I'll discuss the Quantum Langlands
Correspondence.