Jeff Cheeger
New York University
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Primary Section: 11, Mathematics Membership Type:
Member
(elected 1997)
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Research Interests
Broadly speaking, my research interests concern differential geometry, especially Riemannian geometry and its connections with topology and analysis. Throughout my career I have studied the structure of Riemannian manifolds whose curvature satisfies definite constraints, such as non-negative sectional curvature, bounded sectional curvature, and Ricci curvature bounded below. My current research deals the properties of sufficiently collapsed manifolds of bounded sectional curvature, as well as with the small scale structure of spaces whose Ricci curvature is bounded below and with the analytic and topological properties of singular limits of such spaces. The latter work has applications to partial regularity theory for degenerations of Einstein manifolds. I have also been very much involved in the study of spectral properties of the Laplace operator on Riemannian manifolds, in particular, with eigenvalue estimates and with those spectral invariants such as the eta invariant and the analytic torsion which are related to index theory. Moreover, I showed that by removing the singular part and doing analysis on the resulting incomplete Riemannian manifold, these considerations can be extended to a rather general class of singular spaces. In particular, I was able to define a cohomology theory for such spaces (L2-cohomology) which satisfies Poincare duality.
