Based on reverse isoperimetric inequalities on Grassmann manifolds, a Grassmannian Loomis-Whitney inequality and its dual are established, which provides a lower bound for the volume of an origin-symmetric convex body in terms of its lower dimensional sections.

The celebrated Legendre ellipsoid and the LYZ ellipsoid introduced by Lutwak, Yang, and Zhang in 2000 are important concepts in convex geometric analysis. These ellipsoids are generated by the cosine transform (i.e., this transform origins from the inner product of two vectors). In this talk, we will discuss two types of ellipsoids by using the sine transform (i.e., this transform is related to the cross product of two vectors), which can be considered as the sine counterparts of the Legendre ellipsoid and the LYZ ellipsoid.

In this talk, we investigate the evolution of a curvature flow in the plane, which can be regarded as the gradient flow of isoperimetric ratio, for immersed locally convex closed curves. In particular, it is shown that the flow evolves two classes of rotationally symmetric curves, i.e., highly symmetric curves and Abresch-Langer type curves, into $m$fold circles as time goes to infinity.

Nevanlinna理论是复分析、复几何中的重要研究领域，其核心是两个基本定理，特别是后者。报告人此次所做的系列报告基于最近在涉及处于次一般位置超曲面的第二基本定理研究方面最新进展及其特别的观察视角。系列报告将分三次：第一次报告将从涉及处于次一般位置超曲面的第二基本定理的背景入手并简要介绍报告人最近工作的动机、结果、例子和证明框架；后两次报告将给出基于滤过方法和Chow权、Hilbert权方法的不同证明以及如何给出含截断水平的计数函数的精细估计的细节

Hamilton-Tian conjecture says that the Kahler-Ricci flow on Fano manifolds converges to a limit space admitting Kahler-Ricci soliton outside the singularity of dimension 4. This conjecture has been proved by Chen-Wang and Bamler. Their proof depends on the metric geometry. Using Liu-Szekelyhidi's work on partial C^0 estimate, we will prove a weak version of Hamilton-Tian conjecture.

A function valued valuation is an additive map defined on convex bodies and taking values in a function space. We say a valuation is affine if it behaves “nicely” under affine transforms, e.g., volumes, Euler characteristics (constant functions), moment vectors (understand as linear functions), support functions, Minkowski functionals, and so on. In this talk, I will show some classifications of SL(n) covariant or contravariant valuations which not only characterize valuations mentioned above but also characterize some (functional) extensions of Lp projection bodies and Lp moment bodies (polar L-p intersection bodies). Some applications will be also introduced.

In this talk we will show that there is a natural Willmore deformation between minimal surfaces in H^n and S^n. The deformation of the Veronese two sphere and its generalization provide examples of complete minimal surfaces in H^4 with varying Willmore energy.

本报告系统介绍千禧问题之一: 不可压缩Navier-Stokes方程相关性质. 首先介绍该方程的来源及背景, 弱解存在性、正则性等的研究历史和现状; 其次, 介绍解的长时间渐近行为性质; 最后列出一些目前尚未解决的重要问题.