Geometric Langlands correspondence

The Langlands program, is a web of connections between number theory and geometry. 

The geometric Langlands program is a reformulation of the Langlands correspondence. 

The geometric Langlands correspondence relates algebraic geometry to representation theory. 

More about it:

The correspondence is between:

• Algebraic geometry: A D-module is roughly a geometric object that encodes a system of differential equations. Sheafs are a way to systematically keep track of data on a space. Sheaves are like local weather maps, which contain data about a fixed region, however can be combined with other maps (other sheaves) to produce a kind of global map. D-modules are sheaves. D-modules satisfy a special property, the Hecke eigensheaf condition. It's a key property that picks out special sheaves in the Geometric Langlands correspondence. This geometry of sheaves lives on the moduli stack of principle G-bundles, lives over a smooth algebraic curve X. This global geometry is described by this side of the correspondence. 

• Number theory: This side is about representations of this geometry. This is more linear theoretic data. That very curve that I spoke of on the geometrical side of the analysis has a corresponding Langlands dual group. Number theory describes the local systems or flat connections for these dual Langlands groups. These local systems are vector bundles with flat connections. Evariste Galois introduced the idea encoding arithmetic problems using symmetry groups. Langlands is going to elevate this idea bringing together number theory and representation theory with algebraic geometry. Langland's insight was that certain Galois representations correspond with autmomorphic forms (the D-modules