Mathematics

这项研究的目的是验证我最近观察到的一个相当惊人的现象。大量的证据表明，一大类相当复杂的数学对象-实际上所有自然产生的Hilbert空间中的范数闭代数-都是由它们所构成的（包括那些对数学物理感兴趣的人，以及考虑复杂系统的其他数学分支，如动力系统理论和叶理理论）-可以完全用简单的数据来描述-所谓的K理论不变量，有时被称为艾略特不变量一方面，这是最出乎意料的。(But证据是压倒性的）。另一方面，这一发现有着深远的影响，既对所涉及的对象类（用专业术语来说，顺从的C*-代数类），也很可能对这些对象出现的数学和物理学的相关领域产生影响。(For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.)