Peter Newstead and I started working on vector bundles at about the same time. We have shared many ideas for over four decades although we never actually collaborated. I am happy to contribute this summary of some of these areas of common interest to this Festschrift on the occasion of his turning sixty-five.
The theory of vector bundles has many ramifications. One can study it from number theoretic, algebraic geometric and differential geometric points of view. It has also proved useful to mathematical physicists interested in Conformal Field theory, String theory, etc. In this account, I will mainly deal with the geometric aspects, both algebraic and differential, and will confine myself to just a few remarks on the number theoretic point of view.
The classical theory of abelian class fields seeks to understand Galois extensions of a number field in terms of the number theoretic behaviour of the corresponding integral extensions of the ring of integers in the number field. This has a geometric analogy. Consider any compact Riemann surface. Any finite covering of the surface gives a (finite) extension of the field of meromorphic functions on it. The attempt to try and understand abelian extensions of this field in terms of geometric data on the Riemann surface leads to the theory of Jacobians of Riemann surfaces.