Mathematical Sciences: Topology, Geometry and Arithmetic

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Walter Neumann will continue his study (previously supported by DMS-8805551) of the topology of complex curves and surfaces and their singularities and of the geometry and arithmetic of low-dimensional manifolds. Alan Reid will study arithmetic and geometric aspects of hyperbolic 3-manifolds, with special emphasis on questions of existence of incompressible surfaces, isospectrality, special arithmetic and geometric properties of knot complements, and the Lehmer conjecture. Low-dimensional manifolds are very natural geometric objects and deserve to be thoroughly understood. The physical world is a 3-dimensional or 4-dimensional manifold, depending upon whether one includes the time dimension. Hyperbolic metrics are in some ways more natural than Euclidean metrics, even through locally our world appears to be Euclidean.