In mathematics, vector bundles on algebraic curves may be studied as holomorphic vector bundles on compact Riemann surfaces, which is the classical approach, or as locally free sheaves on algebraic curves C in a more general, algebraic setting (which can for example admit singular points).
The result of Grothendieck (1957), that holomorphic vector bundles on the Riemann sphere are sums of line bundles, is now often called the Birkhoff–Grothendieck theorem, since it is implicit in much earlier work of Birkhoff (1909) on the Riemann–Hilbert problem.
Atiyah (1957) gave the classification of vector bundles on elliptic curves.
He was seeking a generalization of the Jacobian variety, by passing from holomorphic line bundles to higher rank.
This idea would prove fruitful, in terms of moduli spaces of vector bundles.