In this talk we will show that the volume of compact Kahler manifold with positive Ricci curvature cannot be bigger than the volume of the complex projective space. The proof requires some construction from convex geometry that goes back to Okounkov. The entire argument is purely algebraic and is quite different from the analogous Bishop's sphere theorem in Riemannian geometry.

In this talk, we present the asymptotic behavior of the solutions for the out-flow problem to the one-dimensional compressible Navier-Stokes-Poisson equation. First, we construct the nonlinear wave, then we address the large-time behavior of the solution by energy method. This is a joint work with Prof. Peicheng Zhu.

Let M be an m dimensional closed Kaehler manifold. We will present certain eigenvalue and heat kernel estimates for the Hodge Laplacian acting on smooth sections of the canonical bundle of M, i.e., (m,0)-forms. The main results only rely on the bound of the Ricci curvature, and the volume and diameter of M, instead of the bound of the whole curvature tensor for general differential forms.

Recently, an Orlicz Aleksandrov problem has been posed and two existence results of solutions to this problem for symmetric measures have been established. In this talk, we will report the existence and uniqueness of solutions to the Orlicz Aleksandrov problem for non-symmetric measures.

I will talk about the rigidity of contact stationary Legendrian (CSL) submanifolds in the unit sphere based on the joint works with Prof. Luo Yong and Dr. Yin Jiabin.

In this talk, we introduce the evolution of space-like graphic hypersurfaces defined over a convex piece of hyperbolic plane〖 H〗^n (1), of center at origin and radius 1, in the (n+1)-dimensional Lorentz-Minkowski space R_1^(n+1) along the inverse mean curvature flow with the vanishing Neumann boundary condition, and show that this flow exists for all the time.

Let Ω be a bounded open domain on the Euclidean space R^n. In this talk, we would like to consider the eigenvalues of fractional Laplacian, and establish an inequality of eigenvalues with lower order under certain conditions. We remark that, our eigenvalue inequality is universal and generalizes the eigenvalue inequality for the poly-harmonic operators.

This is an introduction about the recent progress on some open problems and conjectures about the total squred mean curvature in a Cartan-Hadamard manifold. The integral of geodesic curvature of curves represents the beding energy of a spingy wire, the study of which was initiated at the birth of the calculus of variations by J. Bernoulli in 1690s, and was extensively studied by Euler in 1740s. The total squared mean curvature of surfaces, nowdays called the Willmore energy, naturally raised up in the study of vibrating properties of thin plates in the 1810s. We will talk about the relationship of this energy and the first eigenvalue of Laplacian of a submanifold in a negatively curved space..