9801736 Meyerhoff The 3-dimensional Universe in which we live has very nice geometric properties. For 2000 years, it was believed that the geometry involved was Euclidean geometry. But, at the beginning of the 19th century, C. F. Gauss and N. Lobachevskii independently speculated that the Universe might have a non-Euclidean geometric structure (this geometry is obtained by negating Euclid's fifth postulate, and it is generally called "hyperbolic geometry"). Further, they took measurements to determine if their speculations were correct. Gauss's measurements came from surveys on the earth, while Lobachevskii's measurements were astronomical and used the parallax for the star Sirius. Their measurements were inconclusive, but this year, a variety of astronomical studies have provided substantial evidence that the Universe has a hyperbolic geometric structure. Just as Gauss and Lobachevskii speculated, the Universe is a hyperbolic 3-dimensional object, technically, a hyperbolic 3-manifold. There are many, many different types of hyperbolic 3-manifolds. Which one of these types is our Universe? Answering this question is a daunting task, but there appears to be one significant simplification: some theoretical and esthetic information indicates that if geometric measurements could be made, it would turn out---after a re-scaling related to the expansion of the Universe---that the Universe is "small" in size, that is, it is a hyperbolic 3-manifold of low volume. Thus, we want to understand all hyperbolic 3-manifolds of low volume. This is a hard problem, and for many years progress was slow, but recent research has provided powerful new tools for attacking the problem. Some of the tools involve the use of rigorous computer programs to analyze parameter spaces and to study configurations of natural geometric objects in hyperbolic space. This project will use these tools and develop new ones to lead us towards the goal of definitively identifying the low-volume hyperbolic 3-manifolds. Coupling this theoretical information with data from the planned MAP (microwave anisotropy probe) space project could lead to the determination of precisely which hyperbolic 3-manifold the Universe is. ***
