We consider an ancient solution $g(•, t)$ of the Ricci flow on a compact surface that exists for $t\in (−\infty, T)$ and becomes spherical at time $t = T$. We prove that the metric $g(•, t)$ is either a family of contracting spheres, which is a type I ancient solution, or a King–Rosenau solution, which is a type II ancient solution.

We consider surfaces with parallel mean curvature vector (pmc surfaces) in complex space forms and introduce a holomorphic differential on such surfaces. When the complex dimension of the ambient space is equal to two we find a second holomorphic differential and then determine those pmc surfaces on which both differentials vanish. We also provide a reduction of codimension theorem and prove a non-existence result for pmc 2-spheres in complex space forms.

We extend Donaldson’s diagonalization theorem to intersection forms with certain local coefficients, under some constraints. This provides new examples of non-smoothable topological 4-manifolds.

In 1969, Vic Klee asked whether a convex body is uniquely determined (up to translation and reflection in the origin) by its inner section function, the function giving for each direction the maximal area of sections of the body by hyperplanes orthogonal to that direction. We answer this question in the negative by con- structing two infinitely smooth convex bodies of revolution about the $x_n$-axis in $\mathbb{R}^n, n\ge 3$, one origin symmetric and the other not centrally symmetric, with the same inner section function. Moreover, the pair of bodies can be arbitrarily close to the unit ball.

We use moduli spaces of instantons and Chern-Simons invariants of flat connections to prove that the Whitehead doubles of $(2, 2^n − 1)$ torus knots are independent in the smooth knot concordance group; that is, they freely generate a subgroup of infinite rank.

An expansion is developed for the Weil-Petersson Riemann curvature tensor in the thin region of the Teichmüller and moduli spaces. The tensor is evaluated on the gradients of geodesic lengths for disjoint geodesics. A precise lower bound for sectional curvature in terms of the surface systole is presented. The curvature tensor expansion is applied to establish continuity properties at the frontier strata of the augmented Teichmüller space. The curvature tensor has the asymptotic product structure already observed for the metric and covariant derivative. The product structure is combined with the earlier negative sectional curvature results to establish a classification of asymptotic flats. Furthermore, tangent subspaces of more than half the dimension of Teichmüller space contain sections with a definite amount of negative curvature. Proofs combine estimates for uniformization group exponential-distance sums and potential theory bounds.