Mathematical Sciences: Reciprocity for Fredholm Operators

算子理论是现代分析的核心学科。它起源于20世纪初对数学物理和偏微分方程的研究。当时，人们看到平衡理论、振动理论、量子理论等许多物理问题可以通过对现象建模的积分方程进行有效的研究。因此，从希尔伯特、冯·诺伊曼和其他巨人的丰富思想中，算子理论这一学科已经发展到这些研究的中心位置，在核心数学中也是如此。该方法的核心是对算子的频谱进行深入研究。对于自伴随算子或正规算子，这个理论现在是整个分析的标准技术，谱定理为所有这些算子提供了必要的构建块。因此，当前的前沿是研究非正常算子的结构，例如在Fredholm理论中。在这一前沿领域，Carey教授和他的合作者Pincus教授在深度、概念创新、基础和跨学科研究方面享有世界领先的声誉。他们的主函数理论是在某些算符的谱上定义的，是普通指标的广泛推广。这项新的研究合作在函数代数研究中建立了与遍历流相关的函数的主函数和零分布、次正规算子的结构和闭流理论之间的联系。这项工作特别丰富，因为它直接将算子理论思想与数学的其他分支相结合，特别是交换环理论和代数几何。目前，在算子理论的其他技术应用中，设想了对交换Toeplitz算子和线性和非线性偏微分算子理论的应用。

Operator theory is a central discipline in Modern Analysis. Its origins lie in the study of mathematical physics and partial differential equations in the early twentieth century. At that time, it was seen that numerous physical problems in the theory of equilibria, vibration, quantum theory, etc. could be studied productively via the integral equations that model the phenomena. Thus, from the fertile minds of Hilbert, von Neumann, and other giants the subject of operator theory has grown to a central position in such investigations, and in core mathematics as well. At the heart of this methodology is the deep investigation of the spectrum of an operator. For self-adjoint or normal operators, this theory is now a standard technique throughout analysis, and the spectral theorem provides the necessary building blocks for all such operators. The current frontier, therefore, is the study of the structure of non-normal operators, as for example in the Fredholm theory. At this frontier, Professor Carey and his collaborator, Professor Pincus, are world leaders with established reputations for deep, conceptually innovative, fundamental, interdisciplinary work. Their theory of the principal function, defined on the spectrum of certain operators, is a broad extension of the ordinary index. This new research collaboration has established connections between the principal function and the distribution of zeroes of functions associated with ergodic flows, the structure of subnormal operators and the theory of closed currents in the study of function algebras. This work is particularly fertile as it directly combines operator theoretic ideas with other branches of mathematics, notably commutative ring theory and algebraic geometry. Currently, applications to the theory of commuting Toeplitz operators and linear and nonlinear partial differential operators are envisaged, among other technical applications in operator theory.