Principal Investigator: Xiaochun Rong
Positive curvature and non-positive curvature have been frequent subjects in Riemannian geometry, where mathematics from several disciplines interact; such as differential geometry, analysis and partial differential equations, transformation group theory and topology. The PI is pursuing a research program concerning some basic problems in these areas: 1. Interplay between positive curvature (with Abelian symmetry) and topology. This project is amplified by the amazing fact that among the manifolds of the same dimension whose sectional curvature is between two positive constants, all but finitely many admit (large) Abelian symmetry. 2. The semi-rigidity of the moduli space of non-positively curved metrics on a closed manifold.
Mathematics is the foundation of natural sciences, and differential geometry and Riemannian geometry is one of the most important branches of mathematics. The PI is pursuing solving some basic problems in this field which would have a broad intellectual impact. PI will continue to actively pursue collaborations with other mathematicians in US and abroad and to speak at several national and international meetings a year.
The PI organized Riemannian geometry seminars in summers of 2001-2004, which were designed to provide a forum for the dissemination of new ideas, in particular for graduate students. The PI has had three papers with his graduate students and postdoctoral fellow which were the products of the summer seminars. The PI will continue the Riemannian geometry summer seminars in the next three years. The PI has been writing two advance graduate textbooks for three years (titled: ``The convergence and collapsing theory in Riemannian geometry'' and ``Positive curvature, symmetry and topology''). He wishes to finish the two books in 2-3 years. This will not only benefit graduate students and researchers from these areas (due to the lack of graduate textbooks in these topics) but also help to advertise and disseminate Riemannian Geometry to students and mathematicians in other related fields, by giving them a quick picture of some of the research directions of Riemannian geometry, and by showing them how the subject vitally interacts with differential geometry, analysis and PDE, compact transformation group theory and topology.
