Abstract

In the first part of the talk I will discuss the 1958 algebro-geometric paper of Atiyah “On analytic surfaces with double points”, relating smoothings and resolutions of two-dimensional double point singularities. In the second part I will review the quaternion number system, differential-geometric hyperkahler structures on four-dimensional manifolds and the ALE spaces which connect with the first part. For the last part of the talk, I will introduce the octonion number system, the exceptional Lie group G{2} and the corresponding differential-geometric structures on seven-dimensional manifolds. I will discuss some parts of the 2001 paper of Atiyah and Witten “M-Theory dynamics on a manifold of G{2} holonomy”, and questions of current research interest concerning singularities of G_{2} structures.