Problems in Low-Dimensional Topology

低维拓扑中的问题

• 批准号: 9802945
• 负责人: Daryl Cooper
• 金额: --
• 依托单位: University of California-Santa Barbara
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-08-01 至 2002-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9802945&HistoricalAwards=false
• 关键词: Problems; Low; Dimensional; Topology

9802945 Cooper Cooper will be working in collaboration with Kerckhoff (Stanford) and Hodgson (Melbourne) on a proof of the Orbifold Theorem of Thurston as well as connections between combinatorial group theory and Kleinian groups. Long's projects include the continuation of joint work with Cooper on results concerning incompressible immersed surfaces in 3-manifolds and on braid group actions on buildings. In addition, he plans to study the questions that arise from trying to understand which number fields can be trace fields for finite volume manifolds. Scharlemann's main interest now centers on the structure of Heegaard splittings of 3-manifolds, particularly their characterization in the presence of geometric or other large-scale structures, and with problems of uniqueness and stabilization. The orbifold theorem describes the ``shape'' of certain ``spaces.'' An analogy is the shape of crystals studied in chemistry. Besides the specific chemical properties, there are certain geometric properties that control much of the crystal's properties. The field of crystallography uses the mathematical theory developed by Bieberbach at the turn of this century that describes the possible shapes for crystals. The orbifold theorem describes what crystals are possible in non-Euclidean geometry. As a result of Einstein's theory of general relativity, we know that our universe has a non-Euclidean geometry, but at the comparatively small scale of human existence, this geometry is very nearly Euclidean. The orbifold theorem is therefore not of interest to chemists, but it may one day perhaps help shed light on the large-scale structure of our particular universe. ***

小行星9802945库珀 库珀将与Kerckhoff（斯坦福大学）和霍奇森（墨尔本）合作，证明了瑟斯顿的Orbifold定理以及组合群论和Kleinian群之间的联系。 龙的项目包括继续联合工作与库珀的结果有关不可压缩的浸没表面在3流形和编织群行动的建筑物。 此外，他计划研究所产生的问题，试图了解哪些号码领域可以跟踪领域的有限体积流形。 Scharlemann的主要兴趣现在集中在结构Heegaard分裂的3流形，特别是他们的表征存在的几何或其他大型结构，并与问题的唯一性和稳定性。 轨道定理描述了某些空间的“形状”。一个类比是化学中研究的晶体形状。 除了特定的化学性质外，还有某些几何性质控制着晶体的大部分性质。 晶体学领域使用比伯巴赫在本世纪之交发展的数学理论来描述晶体的可能形状。 轨道定理描述了在非欧几里德几何中可能存在的晶体。 根据爱因斯坦的广义相对论，我们知道我们的宇宙有一个非欧几里德几何，但在人类存在的相对较小的尺度上，这种几何非常接近欧几里德。 因此，化学家对轨道定理并不感兴趣，但也许有一天，它会有助于揭示我们特定宇宙的大尺度结构。 ***