The monumental work of W. Thurston has shown the fundamental importance of hyperbolic 3-manifolds within the study of 3- manifolds. The proposer plans to work on a hard open problem in the theory of hyperbolic 3-manifolds, and to attempt to strengthen connections between number theory and the study of invariants of hyperbolic 3-manifolds. The hard open question is the problem of finding the closed hyperbolic 3-manifold of minimum volume. The proposer, working jointly with D. Gabai and P. Milley, 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. The connection between number theory/algebraic K-theory and invariants of hyperbolic 3-manifolds is well-known, but a new approach to understanding it is proposed. Specifically, a new method for computing the Chern-Simons invariant of a hyperbolic 3-manifold might lead to interesting properties of the dilogarithm function.
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 P. Milley, and in consultation with N. Thurston, plans to work on a computer-based approach to solving one of the hardest and most fundamental problems about hyperbolic 3-manifolds: finding the smallest one. In addition, the proposer will try to strengthen the already existing connection between hyperbolic 3-manifolds and number theory. The history of mathematics has borne out the importance of finding strong connections between (supposedly) disparate areas of mathematics.
