Mathematical Sciences: K-Theory, C*-algebras and Index Theory of Elliptic Operators

9401457卡明克这个项目将追求几个方向，涉及使用算子代数来获得椭圆算子、代数K-理论和动力系统的不变量。我们将提出更高谱流的概念。将定义指数理论不变量，它将检测复数的代数K理论的元素。将研究与双曲动力系统相关的算子代数，并将使用K-理论对偶关系来获得动力信息。此外，还将研究与Novikov猜想和C*-代数上的广义同调理论有关的一些问题。这些问题都涉及Hilbert空间算子及其与流形的几何性质的关系，流形是曲面的高维类似物。偏微分算子在这项工作中扮演着重要的角色。这些运算符自然而然地出现在应用数学和数学物理的研究中。***

9401457 Kaminker This project will pursue several directions involving the use of operator algebras to obtain invariants for elliptic operators, algebraic K-theory and dynamical systems. a notion of higher spectral flow will be developed. Index theoretic invariants will be defined which will detect elements of algebraic K-theory of the complex numbers. Operator algebras associated to hyperbolic dynamical systems will be studied and a K-theoretic duality relation will be used to obtain dynamical information. Work will also be done on some questions related to the Novikov conjecture and generalized homology theories on C*- algebras. These problems all involve Hilbert space operators and their relation to the geometric properties of manifolds, the higher dimensional analogues of surfaces. Partial differential operators play a large role in this work. These operators occur naturally in the study of applied mathematics and mathematical physics. ***