Mathematical Sciences: Problems in Low Dimensional Manifolds

龙计划研究低维流形理论和群论中几个相互关联的问题。具体问题涉及一般双曲3-流形(它们的基本群是否可以子群可分)，圆上双曲曲面丛的特殊情况(同样的问题，以及与熵和体积函数的行为有关的其他具体问题)，以及曲面自同构的余边(是否有算法来确定何时层叠是圆盘的极限)。这些问题与Long对双曲流形、纽结和曲面共边理论中的几个经典问题的既定兴趣有关。这样的几何拓扑研究发展了分析几何对象的定性属性的方法，例如结节。在某些情况下，了解这些属性与了解严格的度量属性一样有价值。

Long plans to examine several interrelated questions in the theory of low dimensional manifolds and group theory. The specific problems concern general hyperbolic 3-manifolds (whether their fundamental groups can be subgroup separable), the particular case of hyperbolic surface bundles over the circle (same question, and other specific questions related to the behavior of the entropy and volume functions), and cobordism of surface automorphisms (whether there is an algorithm to determine when a lamination is a limit of discs). These problems are related to Long's established interests in several classical problems in the theory of hyperbolic manifolds and knot and surface cobordism. Geometric topological investigations such as these develop methods for analysing qualitative properties of geometric objects, e.g. knottedness. In some contexts understanding these properties is every bit as valuable as understanding rigid metric properties.