Compact manifolds with positive sectional curvature have not been classified. So far one knows of only two general obstruction
theorems which describe topological properties of such a manifold. However, on the other hand, there are only very few known
examples. One tool to attack the classification is the use of isometric group actions. This is the background for the following
two talks. In the first talk we will cover some preliminaries and interesting examples; in the second talk we will describe a
beautiful result, namely Hsiang-Kleiner’s theorem (1989) which classifies compact, simply connected, positively curved 4-manifolds
that admit isometric circle actions.