Problems in Metric Riemannian Geometry

主要研究人员计划继续他在黎曼度量几何方面的研究，特别是两个领域的一些基本问题：(1)截面曲率有界的折叠黎曼流形(绝对值或以下)和(2)Alexandrov几何中的刚性和稳定性问题。在他的研究中，来自几个学科的数学相互作用，如度量黎曼几何、分析和偏微分方程、紧变换群理论和拓扑学。在截面曲率由下有界而直径由常量有界的同一维流形中，除了有限多个流形外，几乎所有的流形都是折叠的，并且接近亚历山大空间。数学是自然科学的基础，而微分几何/黎曼几何是数学中最重要的分支之一。国际和平研究所正在寻求解决这一领域的一些基本问题，这些问题将产生广泛的智力影响。国际数学家协会将继续积极寻求与美国和国外其他数学家的合作，并每年在几个国内和国际会议上发言。

The principal investigator plans to continue his research in Riemannian metric geometry; in particular some basic problems in two area: (1) collapsed Riemannian manifolds with bounded sectional curvature (in absolute value or from below) and (2) rigidity and stability problems in Alexandrov geometry. In his research, mathematics from several disciplines interact, such as metric Riemannian geometry, analysis and partial differential equations, compact transformation group theory and topology. This project is amplified by the fact that among the manifolds of same dimension whose sectional curvature is bounded from between below and whose diameter is bounded above by a constant, all but finitely many are collapsed and are close to some Alexandrov spaces.Mathematics is the foundation of the natural sciences, and differential geometry/ Riemannian geometry are among the most important branches of mathematics. The PI is pursuing solving some basic problems in this field that would have a broad intellectual impact. The PI will continue to actively pursue collaborations with other mathematicians in the United States and abroad and to speak at several national and international meetings a year.