Mathematical Sciences: Geometry, Topology and Arithmetic of Hyperbolic 3-Manifolds

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9625958 Reid One of the fundamental problems in low-dimensional topology is to understand the geometry and topology of 3-dimensional manifolds admitting a complete hyperbolic structure of finite volume. Of particular interest are properties of incompressible surfaces in hyperbolic 3-manifolds, and interactions between number theory and algebra on the one hand and hyperbolic 3-manifolds on the other. The former involves the study of the existence and construction of incompressible surfaces in hyperbolic 3-manifolds, and the latter concentrates on how certain invariants arising from algebraic data associated to a hyperbolic 3-manifold are related to the geometry and topology of the the 3-manifold. The study of surfaces in 3-manifolds plays an important role in 3-dimensional topology. Indeed, many of the deepest results in 3-manifold topology have arisen from the study of how surfaces map into 3-manifolds. Apart from their application in 3-manifold topology, the study of surfaces in 3-manifolds has had many interesting applications in physics. The number theoretic connections alluded to above have proved important in recent years in proving theorems about hyperbolic 3-manifolds. One of the simplest number theoretic connections is through the volume of a hyperbolic 3-manifold. This is closely connected to the Dilogarithm function, which appears in many guises in other branches of mathematics and physics. ***