Mathematical Sciences: Torsion Invariants and the Algebraic K-Theory of von Neumann Algebras

在第一个项目中，“扭转不变量和代数K- 冯诺依曼代数理论”，一个新的拓扑不变量， 定义于具有真余紧作用的黎曼流形 可数离散群G的等距性。 概括 Reidemeister扭转的经典概念。 一个目标是找到一个 更好地理解冯·诺依曼的弱K理论 G的代数，不变量存在的地方。 校长 研究者想通过计算证明， 在一些有趣的案例中， 同调3-球面，双曲流形，晶体学 流形和对称空间 另一个问题是， 解析解释的频谱理论的条款 拉普拉斯算子。 另一个项目，“手术转移和一般签名 纤维束的公式，”是专门研究的， 外科手术转移纤维束的歧管。 一个代数 描述由主要研究者制定， 安德鲁·拉尼基。 它将用于进行显式计算， 证明了消失结果，这些结果具有几何意义， 应用程序，并建立一个通用的签名公式， 纤维束 第一个项目是一个联合项目， 芝加哥大学的梅尔文·罗森伯格是第二位 爱丁堡大学的安德鲁·拉尼基教授。 两 联合收割机拓扑，分析和真正的综合代数。 它们的主要价值可能是产生新的工具， 了解流形和它们的对称性 歧管是， 当然，非常基本的几何物体， 在数学和物理中，人们常常把它看作是 常微分方程组。

In the first project, "Torsion Invariants and Algebraic K- theory of von Neumann Algebras", a new topological invariant is defined for a Riemannian manifold with a proper cocompact action ofa countable discrete group G by isometries. It generalizes classical notions of Reidemeister torsion. One goal is to find a better understanding of the weak K-theory of the von Neumann algebra of G, where the invariant lives. The principal investigator wants to show by computation that the invariant carries a lot of information in interesting cases, including homology 3-spheres, hyperbolic manifolds, crystallographic manifolds and symmetric spaces. Another problem is to give an analytic interpretation in terms of the spectral theory of the Laplace operator. The other project, "Surgery Transfer and a General Signature Formula for Fibre Bundles," is devoted to the study of the surgery transfer of a fibre bundle of manifolds. An algebraic description was developed by the principal investigator and Andrew Ranicki. It will be used to make explicit calculations, to prove vanishing results, which have geometric meaning and applications, and to establish a general signature formula for fibre bundles. The first project is a joint project together with Professor Melvin Rothenberg of the University of Chicago, the second with Professor Andrew Ranicki of the University of Edinburgh. Both combine topology, analysis, and algebra in genuine synthesis. Their principal value may be generating new tools for understanding manifolds and their symmetries. Manifolds are, of course, very basic geometric objects and arise almost everywhere one looks in mathematics and physics, often as solutions of systems of ordinary or differential equations.