We start this talk with some basic linear algebras. Consider m(A)=[A,A*] on sl(n,C). Ness studied the (normalized) square of norm |m|^2 and showed some nice properties of |m|^2 which characterize the unitary orbits inside the nilpotent orbits of sl(n,C). We generalize Ness’ results to general orbits by modifying |m|^2, motivated by the curvature formula of the symmetric spaces. As applications, we show some domination results in n-Fuchsian fibers in the Hitchin fibration of the moduli space of Higgs bundles over a Riemann surface. And we also show some existence and uniqueness results of equivariant minimal surfaces in certain product spaces, which generalize a theorem of Schoen.

This talk studies the direct and inverse scattering problem associated with a time-harmonic random Schroedinger equation with a Gaussian white noise source term. We estab-lish the well-posedness of the direct scattering problem and obtain three uniqueness results in determining the variance of the source term, the potential and the mean of the source term, sequentially, by the corresponding far-field measurements. The first one shows that a single realization of the passive scattering measurement can uniquely recover the variance of the source term, without knowing the other two unknowns. The second shows that if active scat-tering measurement is further used, then a single realization can uniquely recover the potential function without knowing the source term

In this talk, some questions on Heegaard splitting will be introduced.

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.