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

纳塔莉教授的项目与函子有关 代数的不变量，称为K-理论和循环上同调， 的操作员。要解决的问题涉及非交换 几何思想的分支。其中一个问题是 在几何上展示了K的显式非平凡生成元， 与基本函数相关的某些算子代数的零 一个封闭的黎曼曲面群，并找到一个循环的 一个适当的稠密子代数上的上圈， 这台发电机的重要性。该项目的另一个方面 涉及到叶状流形和算子代数， 它们会产生。 数学研究领域的基本见解 其中教授纳塔利工作如下：完成 关于空间结构的信息（比如说， 一些更高维度的模拟）被存储在适当的 空间上的函数代数（函数分配编号 空间中的点）。所有人想说的 例如，关于紧致流形的几何， 用无穷可微的代数表示 流形上的复杂函数这个代数，就像很多 数学中有趣的对象，可以有效地表示 希尔伯特空间上的算子代数。很多时候， 对代数有意义的几何或拓扑概念 平滑或连续函数的连续性对更多或更少的情况也有意义。 不那么任意的算子代数用代数的方法工作， 澄清思想的双重好处，也允许使用 机械，如K-理论，工作得更好， 代数背景而不是空间背景。Natalie教授的项目 为实现这一议程，我将考虑几个具体问题 通过算子代数来观察几何现象 透镜。

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.