Description:

Title: Geometric Optimization and Construction under Curvature Bounds

Abstract: My dissertation consists of two parts.

The first part is about optimal transport theory, which studies the optimization
problem of moving one measure to another with the minimal cost. My research work [Wan24] generalizes McCann’s theorem [McC01] about optimal transport on Riemannian manifolds to a submanifold setting, and obtains a corresponding nonlocal change of variable formula. As an application, this generalization is applied to prove the Michael-Simon inequality, an isoperimetric-type inequality for submanifolds, assuming the intermediate Ricci curvatures of the ambient manifold are bounded from below.

The second part is about Ricci limit spaces, which are Gromov-Hausdorff limits of a sequence of Riemannian manifolds with a uniform lower bound on the Ricci curvatures. In my joint work with E. Hupp and A. Naber [HNW25], we show that (volume) collapsing Ricci limit spaces can be topologically nontrivial everywhere, and thus admit no manifold structure anywhere. This should be compared to the regularity results by J. Cheeger and T. Colding in [CC97] and its sequels, where they showed that non-collapsing Ricci limit spaces admit a topological manifold structure on an open dense subset.

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