In this talk I shall report a joint work with Xiao Wei on the graded decomposition numbers and inverse graded decomposition numbers for parabolic Verma modules in any integral blocks of a parabolic category of a semisimple Lie algebra. A sum formula for the radical filtrations of parabolic Verma modules is also obtained, which generalizes the classical Jantzen sum formula for the usual Verma modules.

We investigate the contracting curvature flow of closed, strictly convex axially symmetric hypersurfaces in R^n+1 and S^n+1 by sigma_k^\alpha, where \sigma_k is the k-th elementary symmetric function of the principal curvatures and \alpha\ge 1/k. We prove that for any n\geq3 and any fixed k with 1\leq k\leq n, there exists a constant c(n,k)>0 such that if \alpha lies in the interval [1/k,1/k+c(n,k)], then we have a nice curvature pinching estimate involving the ratio of the biggest principal curvature to the smallest principal curvature at every point of the flow hypersurface, and we prove that the properly rescaled hypersurfaces converge exponentially to the unit sphere. Our results provide an evidence for the general convergence result without initial curvature pinching conditions.

In this talk we will study the limit of a sequence of harmonic maps from surfaces with uniformly bounded energy. We define the generalized neck domain as the union of ghost bubbles and necks that are directly connected and give a uniform upper bound of the energy density on the generalized neck domain. As an application, we prove an inequality of the index and nullity of the sequence.

The i-quantum groups are certain quantum algebras arising from the study of quantum symmetric pairs. They are defined by generators and relations and are embedded as coideal subalgebras of the quantum linear groups. Moreover, the restriction of the famous quantum Schur-Weyl duality pair for quantum linear groups and Hecke algebras of type A gives a duality pair for an i-quantum group and the Hecke algebras of type B. This new duality was discovered by H. Bao and W. Wang. Moreover, Bao, Kujawa, Li and Wang developed a geometric setting of the duality, including some fundamental multiplication formulas and the modified i-quantum group. In this talk, I report a new realization for the i-quantum group $U^j(n)$ and it’s applications to representations of finite orthogonal groups. This is joint work with Yadi Wu.

研究生和大学生的差别是什么？研究生应具备怎样的素质？从大学生到研究生的过程就是从学习到研究、从集体到个体的过程。当一名研究生，要做好充分的准备，要有良好的心态。报告人根据自己三十多年来指导研究生的经验，结合个人成长经历，谈谈研究生阶段应该重视的方面，对研究生的成长提出建议。

We propose a flow to study the Chern-Yamabe problem and discuss the long time existence of the flow. In the balanced case we show that the Chern-Yamabe problem is the Euler-Lagrange equation of some functional. The monotonicity of the functional along the flow is derived. We also show that the functional is not bounded from below.

代数簇是代数几何的主要研究对象。双有理代数几何是代数几何的经典分支之一。光滑代数簇有很多好的性质，但有奇点的代数簇在双有理代数几何中自然出现。本次报告将介绍双有理代数几何关于奇点的基本问题，以及近期与Caucher Birkar在这方面的部分进展。

A formula of dimensions for the spaces of generalized theta functions on moduli spaces of parabolic bundles on a curve of genus g , the so called Verlinde formula, was predicted by Rational Conformal Field Theories. The proof of Verlinde formula by identifying the spaces of generalized theta functions with the spaces of conformal blocks from physics was given in last century mainly by Beauville and Faltings (so called infinite dimensional proof). Under various conditions, many mathematicians tried to give proofs of Verlinde formula without using of conformal blocks, which are called finite dimensional proofs by Beauville. In this talk, we give unconditionally a purely algebro-geometric proof of Verlinde formula.