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

Kaminker教授将研究一系列问题：(I)算子代数和循环上同调在椭圆算子的指标理论中的应用；(Ii)分叶流形上的叶状Toeplitz扩张；(Iii)用于获得关于代数K-理论的信息的更高不变量理论；(Iv)循环上同调和椭圆边值问题；以及(V)用于交换n元组算子的次级不变量。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。椭圆算子在这项工作中扮演着重要的角色。这样的算符出现在许多应用数学和物理的偏微分方程式中。

Professor Kaminker will investigate a number of problems: (i) the use of operator algebras and cyclic cohomology in the index theory of elliptic operators, (ii) the leafwise Toeplitz extension on a foliated manifold, (iii) the theory of higher eta invariants which will be applied to obtain information about algebraic K- theory, (iv) cyclic cohomology and elliptic boundary value problems, and (v) secondary invariants for commuting n-tuples of operators. These problems all involve Hilbert space operators and their relation to the geometric properties of manifolds, the higher dimensional analogues of surfaces. Elliptic operators play a large role in this work. Such operators occur in many of the partial differential equations of applied mathematics and physics.