My research interest is in geometry, particularly,interaction between
topology and differential geometry. For instance, I am studying the
nonexistence problems of Einstein metrics and Ricci flow solutions on
4-manifolds by using the Seiberg-Witten equations. I am also interested
in the geomery of the Yamabe invariant. The computation of the Yamabe
invariant for a given manifold is a difficult problem in general. By
using the Seiberg-Witten equations, I determined the exact value of the
Yamabe invariant for a large class of 4-manifolds which includes complex
surfaces as special cases. Furthermore, I am also interested in both the
Ricci flow in higher dimension and some generalized versions of the
Ricci flow like the Ricci Yang-Mills flow. The Ricci flow was first
introduced by R. Hamilton in 1981 and used as the main tool in G.
Perelman's solution of the Poincare conjecture in 2002. Perelman
introduced many new and remarkable ideas to prove the conjecture. The
theory developed by Hamilton and Perelman is now called the
Hamilton-Perelman theory. One of my recent interests is to investigate
geometric analytical properties of the generalized versions of the Ricci
flow from the Hamilton-Perelman theoretical point of view.