I will discuss the Kervaire sphere, which in many dimensions is a famous example of an exotic sphere. It carries a large group action on it (cohomogeneity one) and the problem is to decide whether it has a metric of nonnegative sectional curvature invariant under the group action. We will show that this is not the case, which was a big surprise for me ( I wanted to prove existence). This is joint work with Grove,Verdiani,and Wilking.