The theory of holomorphic vector bundles on a compact Riemann surface is a vast "non-abelian" generalisation of the classical theory of the Jacobian variety. The Jacobian, a complex torus, arose in the work of Abel, Jacobi and Riemann on abelian integrals on compact Riemann surfaces. The non-abelian generalisation is achieved by replacing the homology group by the fundamental group of the surface and considering unitary representations of the fundamental group.
The theory is related to Algebraic Geometry, Differential Geometry, Number Theory and Theoretical Physics. The Speaker will give an overview of some aspects of the subject.
Some of the topics to be dealt with are :
- Algebro-geometric notions of stability of vector bundles and moduli.
- Relation to Yang-Mills and conformal field theories.
- Geometric Hecke correspondence, which plays a significant role in Geometric Langlands Theory.