Some Problems in Riemannian Geometry

首席研究员计划继续他在度量和比较几何方面的研究项目。PI致力于解决或取得以下两个领域的一些基本问题：曲率从下有界的坍缩黎曼流形和对称的正弯曲流形。在这项研究中，来自几个学科的数学相互作用，如度量比较几何，包括亚历山德罗夫几何，几何分析和拓扑。本课题是PI在研究具有上下有界曲率的坍缩黎曼流形方面取得的初步成功。数学是自然科学的基础，而微分几何和黎曼几何是与物理学等其他自然科学相互作用的最重要的分支。pii正在努力解决上述领域的一些基本问题，这将产生广泛的智力影响。PI将继续积极寻求与美国和国外其他数学家的合作，并在未来三年内在一些国内和国际会议上发言。

The PI (principal investigator) plans to continue his researchprogram in metric and comparison geometry. The PI is pursuing to solve ormake progress on some basic problems in the following two areas: collapsedRiemannian manifolds with curvature bounded from below and positivelycurved manifolds with symmetry. In this research, mathematics from severaldisciplines interact, such as metric comparison geometry includingAlexandrov geometry, geometric analysis and topology. This project is buiton the early success by PI in the study of the collapsed Riemannianmanifolds with curvature bounded from above and below.Mathematics is the foundation of the natural sciences, anddifferential geometry and Riemannian geometry are among the most importantbranches that interact with other natural sciences such as physics. The PIis pursuing to solve some basic problems in the above fields, which will have a broad intellectual impact. The PI will continue to actively pursue collaborations with other mathematicians in the United States and abroad, and speak at several national and international conferences in the next three years.