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.

Let V be a vertex operator superalgebra and G a finite automorphism group of V containing the canonical automorphism such that V^G is regular.We classify the irreducible V^G -modules appearing in twisted V -modules and prove that these are all the irreducible V^G -modules. Moreover, the quantum dimensions of irreducible V^G -modules are determined, a global dimension formula for V in terms of twisted modules is obtained and a super quantum Galois theory is established. In addition, the S-matrix of V^G is computed.

I will discuss the result on the non-uniqueness of solutions to the dual Minkowski problem. In particular we show that for the problem with constant right hand side, when $q>2n$ the solution is non-unique.