报告人：桂长峰教授（University of Texas at San Antonio and Hunan University ）

报告题目：The Sphere Covering Inequality and its application to a Moser-Trudinger type inequality and mean field equations

报告人简介：桂长峰，任湖南大学数学研究所所长，同时也是美国University of Texas at San Antonio数学系教授，博士生导师。研究领域：非线性偏微分方程及图像分析和处理。

桂长峰教授： 1984年本科毕业北京大学，1987年得到北京大学硕士学位，1991年在美国明尼苏达大学获博士学位。曾任纽约大学库郎研究所讲师，加拿大哥伦比亚大学助教、副教授，美国康尼迪格大学副教授、教授。他在偏微分方程基础理论及应用等方面，特别是对Allen-Cahn方程的研究上取得了一系列在国际上有影响的工作。同时他在图像处理方面也有很好的工作， 他与合作者撰写的论文Distance Regularized Level Set Evolution and Its Application to Image Segmentation 获得了2015年IEEE SIGNAL PROCESSING SOCIETY颁发的最佳论文奖。

报告摘要：In this talk, I will introduce a new geometric inequality: the Sphere Covering Inequality. The inequality states that the total area of two {\it distinct} surfaces with Gaussian curvature less than 1, which are also conformal to the Euclidean unit disk with the same conformal factor on the boundary, must be at least $4 \pi$.  In other words, the areas of these surfaces must cover the whole unit sphere after a proper rearrangement. We apply the Sphere Covering Inequality to show the best constant of a Moser-Trudinger type inequality conjectured by A. Chang and P. Yang. Other applications of this inequality include the classification of certain Onsager vortices on the sphere, the radially symmetry of solutions to Gaussian curvature equation on the plane, classification of solutions for mean field equations on flat tori and the standard sphere, etc. The resolution of several open problems in these areas will be presented. The talk is based on joint work with Amir Moradifam from UC Riverside.

报告时间：2017年6月12日（星期一）上午8：30—9：30；

报告地点：科技楼南楼702学术报告厅