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.

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.