  A 1930s conjecture of Hopf states that an even-dimensional compact Riemannian
manifold with positive sectional curvature has positive Euler characteristic.
We prove this conjecture under the additional assumption that the isometry
group has rank at least five. The fundamental new tool used to achieve this is
a reduction to, and structural results concerning, a representation theoretic
problem involving torus representations all of whose isotropy groups are
connected.
