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.

I will talk about the weighted equation $$/Delta(|x|^{-/alpha}/Delta u)=|x|^{/beta}u^p {in}~ /mathbb{R}^n/backslash{/{0}/}, $$ where $n/geq5, -n</alpha<n-4$ and $(p, /alpha,/beta, n)$ belongs to the critical hyperbola with $p>1$ 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.

In this talk, we first give a brief introduction to the optimal transport problem, and then its extension to nonlinear case with applications in geometric optics. Last, we introduce some recent results on the optimal partial transport problem, which is based on joint work with Shibing Chen (USTC) and Xu-Jia Wang (ANU).

We will discuss a class of Monge-Ampere type equations defined on the unit hypersphere, which are related to the Orlicz-Brunn-Minkowski theory in modern convex geometry. These equations are fully nonlinear partial differential equations, and could be degenerate or singular in different cases. We will talk about some recent results about the existence and non-uniqueness of solutions to these equations.

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。

High order nonlinear weighted essentially non-oscillatory finite difference (WENO) schemes, which have the capability of capturing shocks essentially non-oscillatory while re-solving small scale structures efficiently, are popular for solving hyperbolic conservation laws. However, the WENO scheme is fairly complex to implement, computationally expen-sive and too dissipative for certain classes of problems. A natural way to alleviate some of these difficulties is to construct a hybrid scheme conjugating a nonlinear WENO scheme in the non-smooth stencils with a linear method in the smooth stencils. In this talk, I will briefly introduce the difficulties and recent development in the hybrid schemes.