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

夏目教授的课题是关于算子代数的函子不变量，称为k理论和循环上同调。要解决的问题涉及几何思想的非交换分支。一个这样的问题是，从几何上展示与封闭黎曼曲面的基本群相关的某些算子代数的K- 0的显式非平凡生成器，并在适当的密集子代数上找到一个与该生成器非平凡配对的循环环。项目的另一个方面涉及叶形流形和它们所产生的算子代数。夏目姆教授所从事的数学研究领域的基本见解是：关于一个空间（比如一个曲面，或者一些高维的类似物）结构的完整信息存储在空间上的一个适当的函数代数中。（函数将数值赋给空间中的点。）例如，关于紧流形几何的一切，都可以用流形上的无穷可微复函数的代数来表述。这个代数，就像数学中许多有趣的对象一样，可以被有效地表示为希尔伯特空间上的算子代数。通常，对光滑或连续函数代数有意义的几何或拓扑概念对或多或少任意算子代数也有意义。研究代数有两方面的优点，一是澄清思想，二是允许使用k理论这样的机制，它在代数的背景下比在空间的背景下工作得更好。夏目姆教授的项目将考虑几个具体的问题，以追求这个议程，通过算子-代数的镜头看几何现象。

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.