Mathematical Sciences: K-Theory and Cyclic Cohomology Related to Operator Algebras

夏目漱石教授的项目涉及算子代数的函数式不变量，称为K-理论和循环上同调。要解决的问题涉及几何学思想的非对易分支。一个这样的问题是几何地证明了与闭Riemann曲面的基本群相关的某些算子代数的K-零的显式非平凡生成元，以及在适当的稠密子代数上找到与该生成元非平凡配对的循环上循环。该项目的另一个方面涉及到多叶流形和由此产生的算子代数。夏目漱石教授所从事的数学研究领域的基本见解如下：关于空间结构的完整信息(比方说一个表面或一些更高维的类似物)存储在该空间上适当的函数代数中。(函数将数值分配给空间中的点。)例如，人们想要说的关于紧致流形的几何的一切，都可以用流形上无限可微复函数的代数来表示。与数学中许多有趣的对象一样，这种代数可以有效地表示为希尔伯特空间上的算子代数。通常，对光滑或连续函数的代数有意义的几何或拓扑概念也或多或少对任意算子代数有意义。代数工作有两个优点，一是澄清思想，二是允许使用机器，如K理论，它在代数环境中比在空间环境中工作得更好。夏目漱石教授的项目将考虑几个具体的问题，以追求通过算符-代数透镜观察几何现象的议程。

Professor Natsume's project has to do with functorial invariants, called K-theory and cyclic cohomology, for algebras of operators. The problems to be tackled concern noncommutative ramifications of ideas from geometry. One such problem is to exhibit geometrically an explicit nontrivial generator for K- nought of certain operator algebras associated to the fundamental group of a closed Riemann surface, and to find as well a cyclic cocycle on an appropriate dense subalgebra that pairs nontrivially with this generator. Another facet of the project involves foliated manifolds and the operator algebras to which they give rise. The fundamental insight of the area of mathematical research in which Professor Natsume works is the following: complete information about the structure of a space (a surface, say, or some higher dimensional analogue) is stored in an appropriate algebra of functions on the space. (Functions assign number values to points in the space.) Everything one would want to say about the geometry of a compact manifold, for instance, can be stated in terms of the algebra of infinitely differentiable complex functions on the manifold. This algebra, like so many interesting objects in mathematics, can be represented usefully as an algebra of operators on Hilbert space. Quite often, geometrical or topological notions that make sense for algebras of smooth or continuous functions also make sense for more or less arbitrary operator algebras. Working algebraically has the twofold merit of clarifying ideas, and also of permitting the use of machinery such as K-theory that works much better in the context of algebras than of spaces. Professor Natsume's project will consider several specific problems in pursuit of this agenda of looking at geometric phenomena through an operator-algebraic lens.