Geometry, topology and number theory of low-dimensional manifolds

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Proposal: DMS-9971705Principal Investigator: Alan ReidThis proposal is concerned with the structure of 3- and 4-dimensional manifolds. Our interests are in so-called hyperbolic manifolds of finite volume. In dimension 3 some of our main aims are to study the existence and properties of certain surfaces in hyperbolic 3-manifolds. Other parts of the proposal deal with interactions between number theory and algebra with hyperbolic 3-manifolds. Previous work has shown how certain invariants arising from algebraic data associated to a hyperbolic 3-manifold are related to the geometry and topology of the 3-manifold. In dimension 4 we will explore constructions of hyperbolic 4- manifolds and how 3-dimensional manifolds can immerse in them. This is closely connected to problems in cosmology.Manifolds are fundamental objects which appear naturally in many places in mathematics and physics. At present the study of 3 and 4 dimensional manifolds and the interplay between them is fundamental. For example, recently, the theory of 3 and 4 dimensional manifolds modeled on hyperbolic spaces has become central to understanding the spatial geometry of the Universe. Our work hopes to reveal more about these spaces. A major tool that we use in our studies comes from number theory. As in the case of the solution of Fermat's Last Theorem, the interaction between number theory and our hyperbolic spaces is significant. It is our intention to explore these connections.