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. ***
