My research is in the general area of geometry and topology, and focuses on understanding the geometric and topological structures of Riemannian manifolds with prescribed constraints on the Ricci curvature: manifolds with Ricci curvature bounded from below, Ricci flows and Ricci solitons.

Heuristically speaking, the Ricci curvature can be regarded as the Laplacian of the Riemannian metric. So we may think of Riemannian manifolds with non-negative Ricci curvature as the "sub-harmonic objects" in the category of all Riemannian manifolds, and Ricci-flat manifolds as the "harmonic objects". As the Ricci flow on a manifold is a smooth family of Riemannian metrics whose variation is governed by the Ricci curvature tensor, we can think of the Ricci flows as the "heat-diffusing objects" in the category of all Riemannian manifolds.

A general philosophy expects the (sub-) harmonic and the heat-diffusing objects to enjoy better regularity than the generic ones, and the extra regularity should provide more information on structure of the rough limits. This is indeed the case for the category of Riemannian manifolds. Gromov introduced the so-called Gromov-Hausdorff distance, which roughly describes how much the shape of one compact metric space differs from another, and defines a topology on the category of all compact metric spaces. Given a sequence of closed Riemannian m-manifolds with Ricci curvature uniformly bounded from below, the Gromov compactness theorem guarantees the existence of a metric space as the Gromov-Hausdorff limit (rough limit), and a more refined structural theory for such limit spaces has been successfully developed in the past three decades.

In the study of manifolds with prescribed Ricci curvature conditions, a crucial quantity is the volume of a geodesic ball at a fixed scale. If such quantities have a uniformly positive lower bound for a sequence of such manifolds, i.e. "volume non-collapsing", a relatively satisfactory structural theory of the Gromov-Hausdorff limits has been established. However, the volume non-collapsing assumption may fail in many natural settings, and the theme of my research is to understand the collapsing geometry of manifolds with prescribed Ricci curvature conditions. Here we say a sequence of m-dimensional Riemannian manifolds collapse to a metric space if it is the Gromov-Hausdorff limit of the sequence, and is of Hausdorff dimension strictly less than m. Intuitively, for those manifolds in the sequence that are sufficiently close to the Gromov-Hausdorff limit, they look, to the naked eyes, like the lower dimensional limiting metric space, whence the term "collapse".