  A metric with positive sectional curvature on the Gromoll-Meyer exotic
7-sphere is constructed explicitly. The proof relies on a 2-parameter family of
left invariant metrics on Sp(2) and a one-parameter family of conformal
deformations via an isoparametric function F on it. One byproduct is a metric
with positive sectional curvature on a homotopy (but not diffeomorphic) $RP^7$.
