Problems in Low-Dimensional Topology

9802945 Cooper Cooper 将与 Kerckhoff（斯坦福大学）和 Hodgson（墨尔本）合作证明 Thurston 的 Orbifold 定理以及组合群理论和 Kleinian 群之间的联系。 Long 的项目包括继续与 Cooper 合作研究有关 3 歧管中不可压缩浸没表面和建筑物上的编织组作用的结果。 此外，他计划研究因试图了解哪些数域可以是有限体积流形的迹域而产生的问题。 Scharlemann 现在的主要兴趣集中在 3 流形的 Heegaard 分裂结构上，特别是在几何或其他大规模结构存在的情况下的表征，以及唯一性和稳定性问题。 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的形状。 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开发的数学理论来描述晶体的可能形状。 轨道定理描述了非欧几里得几何中可能出现的晶体。 根据爱因斯坦的广义相对论，我们知道我们的宇宙具有非欧几里德几何形状，但在人类存在的相对较小的尺度上，这种几何形状非常接近欧几里德几何形状。 因此，化学家对轨道定理不感兴趣，但有一天它可能有助于揭示我们特定宇宙的大规模结构。 ***

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. ***