Mathematical Sciences: Torsion Invariants for Commuting Systems of Operators

9502154凯里，这位研究者正在研究算子代数。更具体地说，他正在研究被称为斯坦伯格符号的东西，这是此类代数中乘法交换子的推广。斯坦伯格符号是通过Koszul复形结构来分解的，这就产生了一个新的不变量--关节扭转。这个不变量与交换算子代数中的局部或极大理想指标有关，它为非交换情形下的主函数提供了新的理解。还将研究涉及更高代数K-群的类似构造。Wiener-Hopf积分方程的研究始于六十多年前，与电磁波或声波的绕射等物理问题有关。自然运算符W出现了。这位研究人员的贡献始于引入了一个自然配对的运算符U，对于这种运算符，称为行列式的两个运算符的组合呈现出一种特别简单和有用的积分形式。利用这个行列式，他发现可以深化对Wiener-Hopf算子W的研究，可以找到原始积分方程组的新解，并且可以将对算子对的函数h(W，U)的研究置于新的几何背景下。这是现代数学中极为成功的新领域发展的早期一步，该领域有时被称为“非对易几何”。现在，非对易几何集合了几个不同的数学领域，而“代数K-理论”就是其中的突出人物。在代数K-理论的背景下，对上述行列式的结构的进一步分析导致了一个新的对象，称为对{W，U}的联合扭转。反过来，这个对象又以另一种方式与几何学联系在一起，出现在微分方程式和动力系统的研究中。***

9502154 Carey The investigator is studying operator algebras. More specifically, he is studying what are known as Steinberg symbols, a generalization of multiplicative commutators in such algebras. Steinberg symbols are factored by means of a Koszul complex construction, and this gives rise to a new invariant, the joint torsion. This invariant is related to a local or maximal ideal index in the setting of commutative operator algebras, and it provides new understanding of the principal function in the non- commutative case. Similar constructions involving higher algebraic K-groups will also be investigated. The study of Wiener-Hopf integral equations began more than sixty years ago in connection with physical problems such as the diffraction of electromagnetic or sound waves. A natural operator W arises. The investigator's contributions began with the introduction of a naturally paired operator U for which a combination of the two operators known as the determinant took on a particularly simple and useful integral form. Using this determinant, he found that the study of the Wiener-Hopf operator W could be deepened, that new solutions could be found to the original integral equations, and that the study of functions h(W,U) of the pair of operators could be put into a new geometric context. This was an early step in development of the hugely successful new area of modern mathematics sometimes known as "non-commutative geometry." Now non-commutative geometry has drawn together several diverse areas of mathematics, and "algebraic K-theory" figures prominently among them. In the context of algebraic K-theory, the further analysis of the structure of the determinants mentioned above has led to a new object called the joint torsion of the pair {W,U}. This object in turn is linked to geometry in still another way and occurs in the study of differential equations and in the study of dynamical systems. ***