Mathematical Sciences: Torsion Invariants for Commuting Systems of Operators

9502154凯里研究者正在研究算子代数。更具体地说，他正在研究所谓的斯坦伯格符号，这是这种代数中乘法对易子的一种推广。Steinberg符号通过Koszul复结构进行因式分解，这就产生了一个新的不变量，即关节扭转。该不变量与交换算子代数集合中的一个局部或极大理想指标有关，为非交换情况下的主函数提供了新的认识。涉及更高代数k群的类似构造也将被研究。对维纳-霍普夫积分方程的研究始于60多年前，与诸如电磁波或声波衍射之类的物理问题有关。自然算子W出现了。研究者的贡献始于引入自然配对算子U，其中两个算子的组合被称为行列式，采用了特别简单和有用的积分形式。利用这个行列式，他发现可以深化对维纳-霍普夫算子W的研究，可以找到原积分方程的新解，并且可以将对算子的函数h（W，U）的研究放到一个新的几何背景中。这是现代数学中非常成功的新领域发展的早期一步，有时被称为“非交换几何”。现在，非交换几何汇集了几个不同的数学领域，“代数k理论”在其中占有突出地位。在代数k理论的背景下，对上述行列式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。这个对象又以另一种方式与几何联系在一起，出现在微分方程的研究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. ***