Low-Volume Questions for Hyperbolic 3-Manifolds

[801736]迈耶霍夫我们生活的三维宇宙具有非常好的几何性质。两千年来，人们一直认为其中涉及的几何是欧几里得几何。但是，在19世纪初，c·f·高斯和n·罗巴切夫斯基独立地推测宇宙可能具有非欧几里得几何结构（这种几何结构是通过否定欧几里得第五公设得到的，通常被称为“双曲几何”）。此外，他们还进行了测量，以确定他们的推测是否正确。高斯的测量来自地球上的测量，而罗巴切夫斯基的测量是天文上的，使用了天狼星的视差。他们的测量结果没有定论，但今年，各种天文研究提供了大量证据，证明宇宙具有双曲几何结构。正如高斯和罗巴切夫斯基推测的那样，宇宙是一个双曲三维物体，从技术上讲，是一个双曲三维流形。有很多很多不同类型的双曲3流形。我们的宇宙是哪一种？回答这个问题是一项艰巨的任务，但似乎有一个重要的简化：一些理论和美学信息表明，如果可以进行几何测量，结果是——在与宇宙膨胀相关的重新缩放之后——宇宙在尺寸上是“小”的，也就是说，它是一个小体积的双曲3流形。因此，我们想要了解所有的小体积双曲3流形。这是一个难题，多年来进展缓慢，但最近的研究为解决这个问题提供了强大的新工具。其中一些工具包括使用严格的计算机程序来分析参数空间和研究双曲空间中自然几何物体的构型。这个项目将使用这些工具并开发新的工具来引导我们最终确定小体积双曲3流形的目标。将这些理论信息与计划中的MAP（微波各向异性探测器）空间项目的数据相结合，可以精确地确定宇宙是哪一个双曲3流形。＊＊＊

9801736 Meyerhoff The 3-dimensional Universe in which we live has very nice geometric properties. For 2000 years, it was believed that the geometry involved was Euclidean geometry. But, at the beginning of the 19th century, C. F. Gauss and N. Lobachevskii independently speculated that the Universe might have a non-Euclidean geometric structure (this geometry is obtained by negating Euclid's fifth postulate, and it is generally called "hyperbolic geometry"). Further, they took measurements to determine if their speculations were correct. Gauss's measurements came from surveys on the earth, while Lobachevskii's measurements were astronomical and used the parallax for the star Sirius. Their measurements were inconclusive, but this year, a variety of astronomical studies have provided substantial evidence that the Universe has a hyperbolic geometric structure. Just as Gauss and Lobachevskii speculated, the Universe is a hyperbolic 3-dimensional object, technically, a hyperbolic 3-manifold. There are many, many different types of hyperbolic 3-manifolds. Which one of these types is our Universe? Answering this question is a daunting task, but there appears to be one significant simplification: some theoretical and esthetic information indicates that if geometric measurements could be made, it would turn out---after a re-scaling related to the expansion of the Universe---that the Universe is "small" in size, that is, it is a hyperbolic 3-manifold of low volume. Thus, we want to understand all hyperbolic 3-manifolds of low volume. This is a hard problem, and for many years progress was slow, but recent research has provided powerful new tools for attacking the problem. Some of the tools involve the use of rigorous computer programs to analyze parameter spaces and to study configurations of natural geometric objects in hyperbolic space. This project will use these tools and develop new ones to lead us towards the goal of definitively identifying the low-volume hyperbolic 3-manifolds. Coupling this theoretical information with data from the planned MAP (microwave anisotropy probe) space project could lead to the determination of precisely which hyperbolic 3-manifold the Universe is. ***