Hyperbolic 3-Manifold Invariants and Applications

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The monumental work of G. Perleman and W. Thurston has shown the fundamental importance of hyperbolic 3-manifolds within the study of 3-manifolds. Hyperbolic 3-manifolds constitute a complicated and mysterious class of 3-manifolds, and it is important to understand them to the fullest extent possible. The simplest tool for analyzing hyperbolic 3-manifolds is the volume invariant. The proposer plans to work on a grandiose open problem in the theory of hyperbolic 3-manifolds, and to use the knowledge gained from this work to better understand the connections between the topology and the geometry of hyperbolic 3-manifolds. The grandiose open problem is to find the first infinite string of volumes of 1-cusped hyperbolic 3-manifolds; that is, to find all 1-cusped hyperbolic 3-manifolds with volume less than that of the minimum volume 2-cusped hyperbolic 3-manifold(s). The proposer, working jointly with D. Gabai, and in consultation with N. Thurston, has a computer-based scheme to attack this problem. The computational aspects of the approach are interesting in their own right. Almost 200 years ago, J. Bolyai, C. Gauss, and N. Lobachevsky revolutionized mathematics by claiming that a legitimate geometry could be constructed by taking the five classical postulates of Euclid and negating the fifth postulate (the parallel postulate). Further, they theorized that this new and mysterious non-Euclidean geometry (now called "hyperbolic geometry") would have important applications. Their theorizing has been borne out: hyperbolic geometry is vitally important in the modern study of geometry. For example, hyperbolic geometry turns out to be much more important than Euclidean geometry in the study of "3-dimensional manifolds" (our 3-dimensional Universe is an example of a 3-dimensional manifold). As another example, it is quite possible that our Universe adheres to the laws of non-Euclidean geometry rather than the laws of Euclidean geometry.The proposer, working jointly with D. Gabai, and in consultation with N. Thurston, plans to work on a computer-based approach to understanding hyperbolic 3-dimensional manifolds by studying their volume, or size. This understanding of volumes of hyperbolic 3-manifolds should lead to a variety of insights about hyperbolic 3-manifolds, including connections between their topology (shape) and geometry (measurement). Such connections between seemingly unrelated areas often have profound implications for mathematics.