Invariants of Hyperbolic 3-Manifolds and Applications

DMS-0204311G。罗伯特·迈耶霍夫W.瑟斯顿已经表明了基本的重要性，双曲3-流形内的研究3-流形。 提议者计划致力于双曲三维流形理论中的一个困难的开放问题，并试图加强数论与双曲三维流形不变量研究之间的联系。 硬开问题是求最小体积的闭双曲三维流形的问题。 建议者与D. Gabai和P. Milley，并与N. Thurston有一个基于计算机的方案来解决这个问题。 该方法的计算方面本身就很有趣。 数论/代数K-理论和双曲3-流形的不变量之间的联系是众所周知的，但提出了一种理解它的新方法。 特别地，一种计算双曲三维流形的Chern-Simons不变量的新方法可能会导致双对数函数的有趣性质。Gauss和N.罗巴切夫斯基革命性的数学声称，一个合法的几何可以构造的五个经典假设的欧几里得和否定第五公设（平行公设）。 此外，他们还推论这种新的神秘的非欧几里德几何（现在称为“双曲几何”）将有重要的应用。 他们的理论已经被证实：双曲几何在现代几何学研究中至关重要。 例如，在研究“三维流形”（我们的三维宇宙就是三维流形的一个例子）时，双曲几何比欧几里得几何重要得多。 作为另一个例子，我们的宇宙很可能遵循非欧几里德几何定律，而不是欧几里德几何定律。Gabai和P. Milley，并与N. Thurston计划研究一种基于计算机的方法来解决双曲三维流形中最困难和最基本的问题之一：找到最小的一个。 此外，提议者将试图加强双曲三维流形和数论之间已经存在的联系。 数学的历史证明了在（据说）不同的数学领域之间找到强有力的联系的重要性。

DMS-0204311G. Robert MeyerhoffThe 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 vitallyimportant in the modern study of geometry. For example, hyperbolicgeometry turns out to be much more important than Euclidean geometryin the study of "3-dimensional manifolds" (our 3-dimensionalUniverse is an example of a 3-dimensional manifold). As anotherexample, it is quite possible that our Universe adheres to the lawsof 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, theproposer will try to strengthen the already existing connection between hyperbolic 3-manifolds and number theory. The history of mathematicshas borne out the importance of finding strong connections between(supposedly) disparate areas of mathematics.