Let $M_1$ and $M_2$ be two compact connected orientable 3-manifolds, $F_i/subset /partial M_i$ a compact connected surface, $i=1,2$, and $h:F_1/rightarrow F_2$ a homeomorphism. We call the 3-manifold $M=M_1/cup_h M_2$, obtained by gluing $M_1$ and $M_2$ together via $h$, a {/em surface sum} of $M_1$ and $M_2$. In the talk, I will introduce a recent result which gives out a sufficient and necessary condition for a surface sum of two handlebodies to be a handlebody. This is a joint work with He Liu, Fengling Li and Andrei Vesnin.

本报告将介绍Schoen-Marques-Neves 猜测及其进展情况，并讨论与该猜测相关的问题。

最近十几年来，非凯勒复几何是一个非常活跃的研究领域。这个报告将主要回顾我们曾研究过的复几何某些方面的进展.

In this work, we developed FEM to solve space fractional PDEs on irregular domains with unstructured mesh. The analytical calculation formula of fractional derivatives of finite element basis functions is given and a path searching method is developed to find the integra-tion paths corresponding to the Gaussian points. Moreover, a template matrix is introduced to speed up the procedures. As an application of the algorithm, we solved the time and space fractional diffusion equations. The stability and convergence of the fully discrete scheme are also analyzed. In addition, some remarks of the implementation will be given.

We will introduce our recent works on two transformations of complex structures: deformation and blow-up. We prove that the p-Kahler structure with the so-called mild ddbar-lemma is stable under small differentiable deformation. This solves a problem of Kodaira in his classic and generalizes Kodaira-Spencer's local stability theorem of Kahler structure. Using a differential geometric method, we solve a logarithmic dbar-equation on Kahler manifold to revisit Deligne's degeneracy theorem for the logarithmic Hodge to de Rham spectral sequence at E1-level and Katzarkov-Kontsevich-Pantev's unobstructedness of the deformations of a log Calabi-Yau pair. Finally, we will introduce a blow-up formula for Dolbeault cohomologies of compact complex manifolds by introducing relative Dolbeault cohomology. This talk is based on several joint works with Kefeng Liu, Xueyuan Wan, Song Yang, Xiangdong Yang, Quanting Zhao, etc.

I will talk about the weighted equation $$/Delta(|x|^{-/alpha}/Delta u)=|x|^{/beta}u^p {in}~ /mathbb{R}^n/backslash{/{0}/}, $$ where $n/geq5, -n1$ and $$/frac{n+/alpha}{2}+/frac{n+/beta}{p+1}=n-2.$$ First, we give the classification to the positive solutions. It is also closely related to the Caffarelli-Kohn-Nirenberg inequality, and we get some fundamental results such as the best embedding constants, the existence and nonexistence of extremal functions, and their qualitative properties. It's well-known that for $p=1$, it's relate to the Hardy-Rellich inequality, at last if time permits, I also will report new results of Hardy -Rellich Inequalities via Equalities and application of Hardy-Rellich Inequalities with remainder terms in stability.

The interplay of convex bodies and log-concave functions has attracted increasing attention in recent years and many notions in convex geometry have been extended to the set of log-concave functions. In this talk we will introduce some new connections between convex bodies and log-concave functions. This talk is based on the joint works with Prof. Jiazu Zhou.

主要针对自由流体与多孔介质耦合问题数值方法中非线性问题、多物理问题耦合与多变量耦合系统等难点问题展开讨论：对于整个多物理耦合系统，我们主要采用高效解耦方法使得耦合问题求解化为各自物理域问题求解。进一步，针对耦合问题数值方法，我们分别有：1.对非线性问题针对稳态问题采取非奇异解束理论和非稳态问题采用small data假设来解决；2.对于Robin-Robin区域分解对稳态问题使用多物理迭代解耦方法，对非稳态问题采用多物理非迭代方法解耦方法；3.利用裂解算法求解复杂自由流Stokes问题，降低计算存储和规模，使得整个多变量耦合系统转化为拟椭圆问题或泊松问题求解、大规模的科学计算化为小规模计算。最后，我们针对现场油藏开采数值模拟进行简要汇报。