Some Problems in Riemannian Geometry

The PI (principal investigator) plans to continue his research

program in metric and comparison geometry. The PI is pursuing to solve or

make progress on some basic problems in the following two areas: collapsed

Riemannian manifolds with curvature bounded from below and positively

curved manifolds with symmetry. In this research, mathematics from several

disciplines interact, such as metric comparison geometry including

Alexandrov geometry, geometric analysis and topology. This project is buit

on the early success by PI in the study of the collapsed Riemannian

manifolds with curvature bounded from above and below.

Mathematics is the foundation of the natural sciences, and

differential geometry and Riemannian geometry are among the most important

branches that interact with other natural sciences such as physics. The PI

is 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.