Mathematical Sciences: Low Dimensional Topology

小行星9501105 弗里德曼关注的是各种问题，在低维几何拓扑，调查在第四个维度，建设导致手术和s-协边定理，同时试图制定一个独特的四维手术障碍的情况下，建设似乎失败。 在双曲3流形理论中，他研究了非紧3流形的端点，使用最小面积叠层和最小长度网。 一种推测是，这些末端是驯服的。 这就意味着阿尔福斯猜想。 关于4-流形的工作导致了3-流形群的无挠幂零子的一个“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，这使得四维是流形理论中各种异常现象发生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空间上只有一种可微结构，即一种做微积分的方法。 但在四维空间中，有无穷多种不同的计算方法。 弗里德曼的工作继续揭示这些黑暗角落的流形理论以及探索其意义的数学物理。 ***

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. ***