Problems in Metric Riemannian Geometry

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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.