Mathematical Sciences: Hyperbolic 3-Manifold Theory

9626780亚当斯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，一个又一个突出的猜想落空了。随着研究人员现在使用严格的几何论证来证明拓扑定理，整个方法已经转向低维拓扑。研究人员打算单独和联合继续他们对双曲三维流形的理论和计算的研究。通过一系列项目，其中许多项目在很大程度上依赖于他们在计算机上计算实例的能力，研究人员计划进一步了解双曲3-流形、双曲3-流形理论中的不变量和3-流形理论中传统的拓扑不变量之间的联系。球面、环面和两孔环面是拓扑上截然不同的二维曲面的例子。大约100年来，数学家们已经知道，唯一可能的(有限)曲面是n孔环面和相应的一族不可定向曲面(例如，对应于1孔环面的不可定向曲面是Klein瓶)。相比之下，三维空间的拓扑可能性要丰富得多。自从1978年威廉·瑟斯顿发现理解空间拓扑的最好方法是给它们提供恒曲率的形状以来，可能的三维空间之间的美丽关系已经得到了极大的澄清。本项目继续探索曲率和拓扑之间的关系。由于只有在极少数情况下才能手工计算形状，该项目还继续开发和支持计算机软件，以创建空间并研究其性质。这个由美国国家科学基金会资助的软件是免费提供的，它不仅供数学家使用，还供寻求对真实宇宙的形状进行建模的宇宙学家以及对各种动力系统的行为进行建模的物理学家使用。***

9626780 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 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 examples on the 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. A sphere, a torus and a 2-holed torus are examples of topologically distinct 2-dimensional surfaces. For about 100 years mathematicians have known that the only possible (finite) surfaces are the n-holed tori and a corresponding family of nonorientable surfaces (e.g., the nonorientable surface corresponding to a 1-holed torus is a Klein bottle). In contrast, the topological possibilities for a 3-dimensional space are vastly richer. The beautiful relationships among the possible 3-dimensional spaces have been greatly clarified since 1978, when William Thurston discovered that the best way to understand the topology of the spaces was to give them shapes of constant curvature. The present project continues to explore the relationships between the curvature and the topology. Because the shapes can be computed by hand only in a very few cases, the project also continues to develop and support computer software for creating the spaces and studying their properties. This NSF funded software is freely available, and it is used not only by mathematicians, but also by cosmologists seeking to model the shape of the real universe, and by physicists modelling the behavior of various dynamical systems. ***