Mathematical Sciences: Topology, Geometry and Arithmetic

沃尔特·诺伊曼将继续他的研究（之前由DMS-8805551支持）复杂曲线和曲面的拓扑结构及其奇点，以及低维流形的几何和算术。Alan Reid将研究双曲3流形的算术和几何方面，特别强调不可压缩曲面的存在性问题，等谱性，结补的特殊算术和几何性质，以及Lehmer猜想。低维流形是非常自然的几何对象，值得彻底理解。物理世界是一个三维或四维流形，取决于是否包括时间维度。双曲度规在某些方面比欧几里得度规更自然，即使在局部我们的世界似乎是欧几里得的。

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.