Hyperbolic 3-Manifold Theory

亚当斯9803362 双曲三维流形理论对低维拓扑学领域产生了巨大的影响。 随着双曲线技术的应用，杰出的作品一个接一个地倒下了。 低维拓扑的整个方法已经发生了转变，因为研究人员现在使用严格的几何参数来证明拓扑定理。 两位主要研究人员打算继续他们的研究，无论是单独还是联合，双曲3流形的理论和计算。 通过一系列项目，许多项目在很大程度上依赖于他们在计算机上计算显式示例的能力，研究人员计划进一步理解双曲三维流形之间的联系，双曲三维流形理论中的不变量，以及三维流形理论中的传统拓扑不变量。 他们还将考虑应用3-流形理论的空间宇宙的宇宙学模型，与宇宙学家一起工作。 本科生将参与与本研究相关的原创性研究项目。 一个二维的表面可以有正曲率像一个球体，零曲率像一个平面欧几里德平面，或负曲率像一个马鞍。 几乎所有的曲面都可以被赋予负曲率，这意味着它们在局部上表现得像上世纪由罗巴切夫斯基、高斯和波尔约发现的双曲平面。 对有限三维空间的类似理解几十年来一直困扰着研究人员，但在20世纪70年代中期，W。P. Thurston指出，三维流形，二维表面的一个维度的模拟，可以切割成碎片，其中每一个都有八个几何之一。 他证明了猜想的大量3流形。 到目前为止，三维流形最常见的几何是双曲几何。 由此产生的“双曲结构”是丰富的，并已被用来解决大量的拓扑和几何问题。 目前的项目在理论和计算上继续这项工作。 理论的进步使计算成为可能，显式计算的例子指导理论的进一步发展。 在过去的一年里，该项目已经找到了一个新的和非常令人兴奋的应用。 我们生活的空间宇宙是三维流形的一个例子。 美国宇航局的微波各向异性探测器（MAP）的数据可能在2001年揭示空间宇宙的范围是有限的。 如果是这样的话，本项目的算法将用于确定宇宙的确切形状。 研究人员目前正在扩展他们的软件，以容忍MAP数据中存在的高水平噪声，并处理球形和欧几里得的情况以及双曲线。 初步数据表明，真实的宇宙是双曲线的，但其他可能性还不能排除。 ***

9803362 Adams Hyperbolic 3-manifold theory has had a tremendous impact on the field of low-dimensional topology. Outstanding conjectures have fallen, one after the other, with the application of hyperbolic techniques. There has been a shift in the entire approach to low-dimensional topology, as researchers now use rigid geometric arguments to prove topological theorems. The two principal investigators intend to continue their research, both individually and jointly, on the theory and computation of hyperbolic 3-manifolds. Through a set of projects, many relying heavily on their ability to compute explicit examples on a computer, the researchers plan to further the understanding of the connections between hyperbolic 3-manifolds, the invariants that come out of hyperbolic 3-manifold theory, and the traditional topological invariants in 3-manifold theory. They will also consider applications of 3-manifold theory to cosmological models of the spatial universe, working in conjunction with cosmologists. Undergraduates will be involved in original research projects related to this research. A 2-dimensional surface may have positive curvature like a sphere, zero curvature like a flat Euclidean plane, or negative curvature like a saddle. Almost all surfaces can be given negative curvature, meaning that locally, they behave like the hyperbolic plane discovered by Lobachevsky, Gauss, and Bolyai in the last century. An analogous understanding of finite 3-dimensional spaces eluded researchers for decades, but in the mid-1970's W. P. Thurston conjectured that 3-manifolds, the analog of 2-dimensional surfaces one dimension up, can be cut into pieces, each of which has one of eight geometries. He proved the conjecture for vast quantities of 3-manifolds. By far the most commonly occurring geometry for 3-manifolds is hyperbolic geometry. The resulting "hyperbolic structures" are rich and have been used to solve a great many topological and geometrical questions. T he present project continues this work both theoretically and computationally. The theoretical advances make the computations possible, and the explicitly computed examples guide the further development of the theory. In the past year the project has found a new and very exciting application. The spatial universe within which we live is an example of a 3-manifold. Data from NASA's Microwave Anisotropy Probe (MAP) may, in the year 2001, reveal the spatial universe to be finite in extent. If so, algorithms from the present project will be used to determine the exact shape of the universe. The investigators are currently extending their software to tolerate the high levels of noise that will be present in the MAP data, and to handle the spherical and Euclidean cases as well as the hyperbolic one. Preliminary data suggests the real universe is hyperbolic, but the other possibilities cannot yet be ruled out. ***