We construct new families of Kähler-Ricci solitons on complex line bundles over ℂℙⁿ⁻¹, n ≥ 2. Among these are examples whose initial or final condition is equal to a metric cone ℂⁿ/ℤ_k. We exhibit a noncompact Ricci flow that shrinks smoothly and self-similarly for t < 0, becomes a cone at t = 0, and then expands smoothly and self-similarly for t > 0; this evolution is smooth in space-time except at a single point, at which there is a blowdown of a ℂℙⁿ⁻¹. We also construct certain shrinking solitons with orbifold point singularities.

This paper extends to dimension 4 the results in the article "Second order families of special Lagrangian 3-folds" by Robert Bryant. We consider the problem of classifying the special Lagrangian 4-folds in ℂ⁴ whose fundamental cubic at each point has a nontrivial stabilizer in SO(4). Points on special Lagrangian 4-folds where the SO(4)-stabilizer is nontrivial are the analogs of the umbilical points in the classical theory of surfaces. In proving existence for the families of special Lagrangian 4-folds, we used the method of exterior differential systems in Cartan-Kähler theory. This method is guaranteed to tell us whether there are any families of special Lagrangian submanifolds with a certain stabilizer type, but does not give us an explicit description of the submanifolds. To derive an explicit description, we looked at foliations by submanifolds and at other geometric particularities. In this manner, we settled many of the cases and described the families of special Lagrangian submanifolds in an explicit way.

We provide new examples of manifolds which admit a Riemannian metric with sectional curvature nonnegative, and strictly positive at one point. Our examples include the unit tangent bundles of ℂℙⁿ, ℍℙⁿ and \mathbb Oℙⁿ (the Cayley plane), and a family of lens space bundles over ℂℙⁿ.

We prove a remarkable formula for Hodge integrals conjectured by Mariño and Vafa, 2002, based on large N duality, using functorial virtual localization on certain moduli spaces of relative stable morphisms.