主讲嘉宾：关 波 杰出校友、俄亥俄州立大学数学系终身教授
报告时间：2020年9月22日 10：00
参会方式：“HEU”会议 账号：6151776598密码：541991
报告人及内容简介：
关波，美国俄亥俄州立大学数学系终身教授。研究方向为完全非线性偏微分方程和几何分析,曾经从事一些重要问题的研究,包括一般区域上实和复蒙日-安培方程的Dirichlet问题的解的存在性；具有常高斯曲率曲面的Plateau问题；闵可夫斯基-陈省身-亚历山大类型问题；双曲空间中具有常曲率和给定渐近边界的完备曲面；以及完全非线性偏微分方程的一般理论。部分研究成果发表在Advances in Mathematics, Annals of Mathematics, Communications on Pure and Applied Mathematics, Duke Journal of Mathematics, Journal of Differential Geometry, Journal of European Mathematical Society等国际专业数学期刊上。
Partial differential equations play very important roles in geometry as well as other fields of science and technology. In fact, using PDE methods to solve difficult problems in geometry has been a great achievement in mathematics since the mid-twentieth century, marked by the proofs of Bernstein Theorem, Calabi conjecture, Positive Mass Theorem in general relativity, Yamabe problem, and Poincare conjecture, to name a few.
In our talk we will start with some simple applications of PDE methods to geometric problem, including proofs of the isoperimetric inequality, Alexanderov's theorem, and if time permits a proof of Sobolev inequality using Monge-Ampere equation. We then present some classical problems in geometry and related fully nonlinear PDEs, such as Weyl problem of isometric embeddings, Minkowski problem and extensions by Alexanderov and Chern, Calabi conjecture, and Donaldson conjecture, etc. Hopefully we can also discuss the famous Plateau problem which was first raised by Lagrange in 1760, and solved in 1930's independently by Jesse Douglas who received the first Fields medal in 1936 because of this work, and Tibre Rado who was a professor at Ohio State University until retirement. Along the line we hope to be able to briefly mention some of our own related work.
