Atiyah’s work on holomorphic vector bundles and gauge theories

The first part of the article surveys Atiyah’s work in algebraic geometry during the 1950s, mainly on holomorphic vector bundles over curves. In the second part we discuss his work from the late 1970s on mathematical aspects of gauge theories, involving differential geometry, algebraic geometry, and topology.

Michael Atiyah’s work in algebraic topology

In 1960 algebraic topology was at the centre of the mathematical stage, but Michael Atiyah burst into the field and changed its focus and its language. I describe his work of the following decade and its influence, keeping to the themes of K K -theory and generalized cohomology to minimise the overlap with Dan Freed’s account of Atiyah’s work on index theory, which also appears in this issue.