Mathematics

这项拟议研究的目的是验证我最近观察到的一个相当惊人的现象。现在已经有相当多的证据表明，一大类相当复杂的数学对象-几乎所有希尔伯特空间中自然产生的范数闭代数(包括数学物理中感兴趣的那些，以及在其中考虑复杂系统的其他数学分支，如动力系统理论和叶理论)-可以用简单的数据来完整地描述-所谓的K理论不变量，现在有时被称为埃利奥特不变量。一方面，这是最出人意料的。(但证据确凿。)另一方面，这一发现对所涉及的对象类别(技术术语中，可服从C*-代数的类别)以及这些对象所在的相关数学和物理领域都具有深远的影响。(例如，这种简单的行为可以在动力系统的轨道结构中检测到。)

The objective of the proposed research is to verify a rather striking phenomenon that I have recently observed. Considerable evidence has now been assembled that a large class of quite complicated mathematical objects---virtually all naturally arising norm-closed algebras in Hilbert space (including those of interest in mathematical physics, and in other branches of mathematics in which complex systems are considered, such as the theory of dynamical systems and the theory of foliations)---can be completely described in terms of simple data---the so-called K-theory invariant, sometimes now referred to as the Elliott invariant. On the one hand, this is most unexpected. (But the evidence is overwhelming.) On the other hand, this discovery has far-reaching implications, both for the class of objects involved (in technical terms, the class of amenable C*-algebras), and, it appears likely, for the related areas of mathematics and physics in which these objects appear. (For instance, this simple behaviour can be detected in the orbit structure of dynamical systems.)