Problems in Low-Dimensional Topology

9802945 Cooper Cooper将与Kerckhoff（斯坦福大学）和Hodgson（墨尔本大学）合作研究Thurston的Orbifold定理的证明以及组合群论和Kleinian群之间的联系。Long的项目包括继续与Cooper合作，研究3流形中不可压缩浸入表面的结果，以及建筑物上的编织群作用。此外，他计划研究在试图理解有限体积流形的哪些数场可以是迹场时所产生的问题。Scharlemann现在的主要兴趣集中在3流形的Heegaard分裂的结构上，特别是它们在几何或其他大尺度结构下的表征，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。一个类比是在化学中研究晶体的形状。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初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得几何中可能存在的晶体。由于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得几何，但在相对较小的人类存在范围内，这种几何非常接近欧几里得。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这个特殊宇宙的大规模结构。＊＊＊

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. ***