Mathematical Sciences: Low Dimensional Topology

弗里德曼最近一直在关注拓扑学在其他数学领域的应用，包括几何、理想MHD和信号处理，并继续他在四维流形理论中的计划。瑟斯顿的几何化猜想是后一个领域最有趣的问题，研究它的分析工具，如哈密尔顿方程，现在可能是可用的。随着时间维度的增加，我们生活的世界是一个四维流形，这使得更耐人寻味的是，四维是流形理论中各种反常发生的地方。弗里德曼是发现这些的主要推动者之一。例如，在所有其他维度中，n-空间上只有一种可微结构，即微积分的一种方法，但在维度4中，情况并非如此。

Freedman has recently been concerned with applications of topology to other areas of mathematics, including geometry, ideal MHD, and signal processing, as well as continuing his program in four-dimensional manifold theory. Thurston's geometrization conjecture is the most interesting problem in the latter area, and the analytic tools for studying it, such as Hamilton's equation, may now be available. 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 discovering these. 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 this is known not to be the case.