Langland's Duality for Hitchin Systems

The talk will be devoted to one of the manifestations of the geometric Langlands duality. Consider a complex smooth projective curve X (a.k.a. a compact Riemann surface). I will discuss rank n vector bundles on X as well as their moduli space Bun_n(X). I will explain that the cotangent bundle T*Bun_n(X) possesses a very rich structure (known as the Hitchin system). The Langlands duality for Hitchin systems is a conjectural auto-equivalence of T*Bun_n(X). Roughly speaking, this means that the points of T*Bun_n(X) parameterize certain objects on T*Bun_n(X) (precisely, the generalized line bundles
on the Hitchin fibers). In the simplest case, this reduces to the statement that the line bundles on an elliptic curve are parameterized by the elliptic curve itself.

I will only assume a basic knowledge of complex or algebraic geometry.