Invariants of Hyperbolic 3-Manifolds and Applications

dms - 0204311 g。瑟斯顿（W. Thurston）的不朽著作表明了双曲型3流形在3流形研究中的根本重要性。拟研究双曲3-流形理论中的一个难解问题，并尝试加强数论与双曲3-流形不变量研究之间的联系。难解的问题是求体积最小的闭双曲3流形的问题。提议者与D. Gabai和P. Milley合作，并与N. Thurston协商，提出了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理论与双曲3流形不变量之间的联系是众所周知的，但提出了一种新的理解方法。具体地说，一种计算双曲3流形的chen - simons不变量的新方法可能会得到二对数函数的有趣性质。大约200年前，J.博利亚、C.高斯和N.罗巴切夫斯基革新了数学，他们声称可以通过采用欧几里得的五个经典公设并否定第五个公设（平行公设）来构造一个合法的几何。此外，他们还从理论上推断，这种新的神秘的非欧几里得几何（现在被称为“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研究中是极其重要的。例如，在“三维流形”的研究中，双曲几何比欧几里德几何重要得多（我们的三维宇宙就是三维流形的一个例子）。另一个例子是，我们的宇宙很可能遵循非欧几里得几何定律，而不是欧几里得几何定律。这位提议者与D. Gabai和P. Milley合作，并与N. Thurston协商，计划研究一种基于计算机的方法来解决关于双曲3流形的最难和最基本的问题之一：找到最小的问题。此外，作者将尝试加强双曲3流形与数论之间已经存在的联系。数学的历史已经证明了在不同的数学领域之间寻找紧密联系的重要性。

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.