In collaboration with G. Edwards we produce (local) Calabi-Yau metrics, in two complex dimensions, with cone singularities along intersecting complex lines, for cone angles that strictly violate the Troyanov condition. We identify the tangent cone at the origin as a product of two 2-cones. In the tangent cone limit, the line with the smallest cone angle remains apart while the other lines collide into a single cone factor.

We apply Menke’s JSJ decomposition for symplectic fillings to several families of contact 3-manifolds. Among other results, we complete the classification up to diffeomorphism of strong symplectic fillings of lens spaces. We reduce the classification of the strong symplectic fillings of large families of Seifert fibered spaces to that of lens spaces. We show that fillings of contact manifolds obtained by Legendrian surgery on certain Legendrian knots are the result of attaching a symplectic 2-handle to a filling of a lens space. This is joint work with Austin Christian.

In mathematics it is usually a routine way to decompose the mathematical object under consideration into some simpler ones which are easier to study. By analyzing how the pieces assemble the information is obtained about the original object. In homotopy theory, we usually need to decompose a topological space as the product or wedge of some smaller spaces. This method helps us to easily investigate the homotopy groups or homology groups of the original space via the ones of the smaller spaces. In this talk, I will simply introduce the method of homotopy decomposition and then give some our recent results on the homotopy decompositon of some looped Co-H spaces.

We recently improved our result on Kahler-Ricci type flow whose stationary points are cscK metrics, introduced by Yuan Yuan, Zhang YuGuang and the speaker.

In this talk, we will present some recent progress on the geometry of strictly nef vector bundles, with focus on the characterization of projective spaces.

In this talk, we will establish asymptotic behaviors at infinity of solutions of general fully nonlinear elliptic equations and then will use it to study some concrete equations, especially, to Monge-Ampere equations. The domain will be the whole spaces or the half spaces.

In this talk, I give an introduction on Sobolev and Moser-Trudinger type inequalities for the complex Monge-Ampere equation. In particular, I will present a PDE proof to these inequalities. These inequalities can be applied to a PDE approach to a priori estimates for solutions to the equation with the right-hand side in $L^p$ for any given $p>1$. Our proof uses various PDE techniques but not the pluri-potential theory.

In this talk, we will first recall some background and research history of Chern's conjecture, which asserts that a closed, minimally immersed hypersurface of the unit sphere Sn+1(1) with constant scalar curvature is isoparametric. Next, we introduce our progress in this conjecture. We proved that for a closed hypersurface Mn ⊂ Sn+1(1) with constant mean curvature and constant non-negative scalar curvature, if tr(Ak) are constants (k = 3,...,n−1) for shape operator A, then M is isoparametric, which generalizes the theorem of de Almeida and Brito in their 1990's paper in 《Duke Math. J. 》 for n = 3 to any dimension n, strongly supporting Chern’s conjecture