In this talk, we will present some recent progress on the geometry of strictly nef vector bundles, with focus on the characterization of projective spaces.

In this talk, we will establish asymptotic behaviors at infinity of solutions of general fully nonlinear elliptic equations and then will use it to study some concrete equations, especially, to Monge-Ampere equations. The domain will be the whole spaces or the half spaces.

In this talk, I give an introduction on Sobolev and Moser-Trudinger type inequalities for the complex Monge-Ampere equation. In particular, I will present a PDE proof to these inequalities. These inequalities can be applied to a PDE approach to a priori estimates for solutions to the equation with the right-hand side in $L^p$ for any given $p>1$. Our proof uses various PDE techniques but not the pluri-potential theory.

In this talk, we will first recall some background and research history of Chern's conjecture, which asserts that a closed, minimally immersed hypersurface of the unit sphere Sn+1(1) with constant scalar curvature is isoparametric.

三维流形的几何与拓扑一直是数学领域主流研究方向之一，较为典型的研究方法有二：Thurston通过几何剖分考察流形的双曲化，并提出三维流形的几何化纲领；Hamilton-Perelman等数学家用Ricci流研究Thurston的几何化纲领。组合Ricci流综合了他们的想法，在带剖分的三维流形上直接演化一组耦合剖分的组合结构、度量结构、角结构的常微分方程组，目标是寻找三维流形的几何剖分以及典型度量，因此组合Ricci流提供了研究三维几何拓扑的新思路。此报告将介绍组合Ricci流的基本理论以及在三维几何拓扑领域的应用，将重点关注三维流形的双曲化定理以及体积猜想。

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. The last one shows that if full measurements are used, then both the potential and the random source can be uniquely recovered.

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