Positively curved manifolds of fixed point cohomogeneity one are classified up to equivariant diffeomorphism.

We define a new spectrum for compact length spaces and Riemannian manifolds called the “covering spectrum” which roughly measures the size of the one dimensional holes in the space. More specifically, the covering spectrum is a set of real numbers δ>0 which identify the distinct δ covers of the space. We investigate the relationship between this covering spectrum, the length spectrum, the marked length spectrum and the Laplace spectrum. We analyze the behavior of the covering spectrum under Gromov–Hausdorff convergence and study its gap phenomenon.

On compact homogeneous spaces, we investigate the Hilbert action restricted to the space of homogeneous metrics of volume one. Based on a detailed understanding of the high energy levels of this action, we assign to a compact homogeneous space a simplicial complex, whose non-contractibility is a sufficient condition for the existence of homogeneous Einstein metrics.

We use the 2-loop term of the Kontsevich integral to show that there are (many) knots with trivial Alexander polynomial which do not have a Seifert surface whose genus equals the rank of the Seifert form. This is one of the first applications of the Kontsevich integral to intrinsically 3-dimensional questions in topology.

Our examples contradict a lemma of Mike Freedman, and we explain what went wrong in his argument and why the mistake is irrelevant for topological knot concordance.