Mathematical Sciences: Topology

小行星9306240 弗兰克·雷蒙德（Frank Raymond）改进了一个以主G-丛为模型的广义塞弗特纤维化理论，使其包含任意李群G。 应用程序和例子说明如何一般理论不同的情况下，G是阿贝尔或幂零已经开发，并将投入最终形式。 塞弗特纤维化的技巧和结果将被用来研究常曲率伪黎曼流形及其框架丛的空间形式问题。 欧拉特征为零的椭圆曲面支持4维几何。 一个Teichmueller理论为这些几何和模的等距类将被构造和确定。 G.彼得·斯科特计划在三个主要领域工作：首先，他将继续他的工作对拓扑刚性的3-流形。 其次，他计划在该地区的3维庞加莱对偶群体。 长期的目标是在这里表明，任何这样的群体来自一个3流形，但他将限制他的注意力，这些群体'应该'对应于塞弗特纤维空间或哈肯流形。 第三，他计划扩大到更高的层面，他的工作产生的特点子流形的3流形。 流形是自然的几何对象，如圆、球和甜甜圈，可以用与相同维度的欧几里得空间相同类型的坐标局部描述。 它们的多样性和复杂性随着维度的增加而迅速增加，但一些最棘手的问题已经出现在三维和四维中。 我们生活在一个三维的世界里，如果考虑到时间的话，我们生活在一个四维的世界里，这使得这不仅仅是纯粹的学术兴趣。 宇宙学家并不知道物质宇宙是由哪种3维或4维流形构成的，因此研究可能发生的可能性有着充分的动机。 这两位研究者巧妙地运用几何学和代数来阐明这个问题。 他们的工作也将丰富其他人研究低维流形的工具库。 ***

9306240 Raymond Frank Raymond has refined a theory for generalized Seifert fiberings modelled on principal G-bundles to include arbitrary Lie groups G. Applications and examples illustrating how the general theory differs from the case where G is abelian or nilpotent have been developed and will be put into final form. Techniques and results from Seifert fiberings will be used to investigate space form problems for the pseudo-Riemannian manifolds of constant curvature and their frame bundles. Elliptic surfaces whose Euler characteristic are zero support 4-dimensional geometries. A Teichmueller theory for these geometries and the moduli of isometry classes will be constructed and determined. G. Peter Scott plans to work in three main areas: Firstly, he will continue his work on the topological rigidity of 3-manifolds. Secondly, he plans to work in the area of 3-dimensional Poincare duality groups. The long term aim here is to show that any such group comes from a 3-manifold, but he will restrict his attention to those groups which 'ought' to correspond to Seifert fiber spaces or to Haken manifolds. Thirdly, he plans to extend to higher dimensions his work generating the characteristic submanifold of a 3-manifold. Manifolds are natural geometric objects like circles and spheres and doughnuts that can be described locally by the same type of coordinates as a Euclidean space of the same dimension. Their variety and complexity increases rapidly as the dimension increases, but some of the most intractable problems arise already in dimensions three and four. That we live in a world of three dimensions, or four if time is considered, makes this of more than purely academic interest. Cosmologists do not know which 3- or 4-manifold constitutes the physical universe, so studying the possibilities that can occur has ample motivation. These two investigators are making ingenious use of geometry and of algebra to shed light on the question. Their work w ill also enrich the arsenal of tools available to others for studying low-dimensional manifolds. ***