Mathematics

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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.)