Invariants of Hyperbolic 3-Manifolds and Applications

DMS-0204311G。罗伯特·迈耶霍夫W.瑟斯顿的不朽工作表明了双曲三维流形在三维流形研究中的基本重要性。作者计划研究双曲三维流形理论中的一个难解问题，并试图加强数论与双曲三维流形不变量研究之间的联系。难解问题是寻找具有最小体积的闭双曲3-流形的问题。提出者与D·加拜和P·米利共同工作，并与N·瑟斯顿协商，提出了一个基于计算机的方案来解决这个问题。这种方法的计算方面本身就很有趣。数论/代数K-理论与双曲三维流形的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。近200年前，J.Bolyai，C.Gauss和N.Lobachevsky声称，通过取欧几里得的五个经典公设并否定第五个公设(平行公设)可以构造合法的几何，这是一场数学革命。此外，他们还推测，这种新的、神秘的非欧几里德几何(现在称为“双曲几何”)将会有重要的应用。他们的理论已经得到证实：双曲几何在现代几何研究中至关重要。例如，在研究“3维流形”时，双曲几何比欧几里得几何重要得多(我们的3维宇宙就是3维流形的一个例子)。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，而不是欧几里得几何定律。提出者与D.Gabai和P.Milley合作，并与N.瑟斯顿协商，计划用计算机来解决双曲三维流形最困难和最基本的问题之一：找到最小的一个。此外，提出者将试图加强双曲3-流形和数论之间已经存在的联系。数学史证明了在(据说)不同的数学领域之间找到紧密联系的重要性。

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.