In this talk, we will discuss recent progress on quasi-local mass in
general relativity focusing on the Wang-Yau quasi-local mass and
discuss how to define other quantities such as angular momentum based
on the ideas and techniques developed in the quasi-local mass. We
will also discuss properties and applications of these newly defined
quantities.

We prove the existence of weak solutions of complex m- Hessian equations on compact Hermitian manifolds for the nonnegative  right hand side belonging to $L^p, p>n/m$ ($n$ is the dimension of the manifold). For smooth, positive data the equation has been recently solved by Sz\'ekelyhidi and Zhang. We also give a stability result for such solutions.

We consider fully nonlinear elliptic equations on complex manifolds which depend on the gradient in some nontrivial ways. Some of these equations arise from interesting problems in complex geometry, such as a conjecture by Gauduchon which is a natural generalization of Calabi conjecture to the Hermitian setting, and finding balanced metrics on Hermitian manifolds. We shall discuss difficulties in solving such equations and present recent results in our attempt to overcome these difficulties.  Our goal is to establish some general existence results which we hope will find useful applications in complex geometry in the near future. We'll explain how our results provide a proof to the Gauduchon conjecture building on previous work of Tossati-Weinkove and others. The talk is based on joint work with Xiaolan Nie, Chunhui Qiu and Rirong Ruan. 

For a compact Riemannian manifold of dimension at least three, we know that positive Yamabe invariant implies the existence of a conformal metric with positive scalar curvature. As a higher order analogue, we seek for similar characterizations for the Paneitz operator and Q-curvature in higher dimensions. For a smooth compact Riemannian manifold of dimension at least six, we prove that the existence of a conformal metric with positive scalar and Q-curvature is equivalent to the positivity of both the Yamabe invariant and the Paneitz operator. In addition, we also study the relationship between different conformal invariants associated to the Q-curvature. This is joint work with Matt Gursky and Fengbo Hang.

I will talk about the construction of Kahler-Einstein metric under the assumption of negative holomorphic sectional curvature. This is based on the joint work with Yau.