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..

In 1989, Kusner proposed the generalized Willmore conjecture which states that the Lawson minimal surfaces $\xi_{g,1}$ minimizes uniquely the Willmore energy for all immersions in the 3-sphere with genus g>0. We show that it holds under some symmetric assumption. That is, the conjecture holds if $f:M\rightarrow S^3$ is of genus $g>1$ and is symmetric under the symmetric group $G_{g,1}$ action. Here $G_{g,1}$ denote the symmetric group of $\xi_{g,1}$ generated by reflections of circles of $S^3$, used in Lawson's original construction of $\xi_{g,1}$. This is based on joint works with Prof. Kusner.

In these talks, I will briefly survey some development of results on hyperbolic Dehn fillings. I will discuss works of I.Agol and M.Lackenby related to bounds on exceptional Dehn fillings of cusped hyperbolic 3-manifolds.

Cone spherical metrics are constant curvature +1 conformal metrics with finitely many cone singularities on compact Riemann surfaces. Their existence has been an open problem since 1980s. The speaker will talk about the recent progresses on this problem joint with Qing Chen, Yu Feng, Bo Li, Lingguang Li, Yiqian Shi, Jijian Song and Yingyi Wu.

A modular tensor category is pointed if every simple object is a simple current. We show that any pointed modular tensor category is equivalent to the module category of a lattice vertex operator algebra. Moreover, if the pointed modular tensor category C is the module category of a twisted Drinfeld double associated to a finite abelian group G and a 3-cocycle with coefficients in U(1), then there exists a selfdual positive definite even lattice L such that G can be realized an automorphism group of lattice vertex operator algebra $V_L,$ $V_L^G$ is also a lattice vertex operator algebra and C is equivalent to the module category of $V_L^G.$ This is a joint work with S. Ng and L. Ren.