Mathematical Sciences: The Arithmetic of Hyperbolic 3-Manifolds Continued

研究人员将继续研究 双曲三维流形开始于DMS-9108050。 他的主要 重点是双曲型不可压缩曲面的研究 3-流形，这种基本群的表示 三维流形，以及闭测地线的几何和结构 在双曲三维流形中。 因为日常生活的空间是三维的， 理解这个维度物体的结构 可能与真实的世界特别相关。 这个世界 当然是局部欧几里得的，但在宇宙学的尺度上， 很可能是双曲线的--争论的焦点是光路的行为 非常长。 无论如何， 关于双曲流形的答案令人惊讶地 在数学和理论上的许多应用 物理学

The investigator will continue research on the arithmetic of hyperbolic 3-manifolds begun under DMS-9108050. His main emphasis is on the study of incompressible surfaces in hyperbolic 3-manifolds, representations of the fundamental groups of such 3-manifolds, and the geometry and structure of closed geodesics in hyperbolic 3-manifolds. Since the space of everyday life is three dimensional, understanding the structure of objects of this dimension is likely to be especially relevant to the real world. This world is certainly locally Euclidean, but on a cosmological scale it may well be hyperbolic -- at issue is the behavior of light paths of enormous length. In any event the techniques invented to answer questions about hyperbolic manifolds have surprisingly many applications elswehere in mathematics and theoretical physics.