Mathematical Sciences: Low Dimensional Topology

9501105 Freedman Freedman关注低维几何拓扑中的各种问题，在四维中研究导致手术的结构和s协共定理，同时试图在结构似乎失败的情况下制定唯一的四维手术障碍。在双曲3-流形理论中，利用最小面积片和最小长度网研究了非紧化3-流形的端点。一种猜想是，这些终点是平淡无奇的。这就暗示了阿尔福斯猜想。在4流形上的工作导致了3流形群的无扭转幂零商的“分类”。随着时间维度的增加，我们生活的世界是一个四维流形，这使得四维流形理论中出现各种异常的地方变得更加有趣。弗里德曼是这一理论的主要推动者之一。例如，在所有其他维度中，在n空间中只有一种可微结构，即一种做微积分的方法。但是在四维空间中，有无限多种不同的微积分方法。弗里德曼的工作继续照亮了流形理论的这些黑暗角落，并探索了它们对数学物理的意义。＊＊＊

9501105 Freedman Freedman is concerned with a variety of problems in low dimensional geometric topology, investigating in dimension four, constructions leading to surgery and s-cobordism theorems, while attempting to formulate a uniquely 4-dimensional surgery obstruction in the case where the constructions appear to fail. In the theory of hyperbolic 3-manifolds, he studies the ends of non-compact 3-manifolds, using least area laminations and least length webs. A conjecture is that these ends are tame. This would imply the Ahlfors conjecture. The work on 4-manifolds leads to a "classification" of the torsion-free nilpotent quotients of 3-manifold groups. With the addition of a time dimension, the world we live in is a four-dimensional manifold, which makes it that much more intriguing that dimension four is where various anomalies in manifold theory occur. Freedman was one of the prime movers in this theory. For example, in all other dimensions there is only one differentiable structure on n-space, i.e. one way of doing calculus. But in dimension four, there are infinitely many different ways of doing calculus. Freedman's work continues to shed light on these dark corners of manifold theory as well as to explore their significance for mathematical physics. ***