Mathematical Sciences: Pure and Applied Operator Theory

算子论是现代分析的中心学科。它起源于二十世纪初对数学物理和偏微分方程式的研究。同时，可以看到，平衡理论、振动理论、量子理论等中的许多物理问题都可以通过模拟这些现象的积分方程组进行富有成效的研究。因此，从希尔伯特、冯·诺伊曼和其他巨人的丰富头脑中，算子理论的主题已经成长为此类研究的中心位置，在核心数学中也是如此。这种方法的核心是深入研究操作员的频谱。对于自伴或正规算子，这一理论现在是整个分析的标准技术，而谱定理为所有这类算子提供了必要的基础。因此，目前研究算子结构的前沿是非正态理论。在这一前沿领域，平卡斯教授是一位世界领导者，在深入、概念性的创新基础和跨学科工作方面享有盛誉。在他早期的研究生涯中，在布鲁克海文实验室，平卡斯教授开创了医学断层扫描的新想法和新技术，无论是理论上的还是计算上的。值得注意的是，他将Radon变换用于图像重建似乎是其他人更知名的工作的先驱，例如科马克的诺贝尔奖工作。随后，他开始了对主函数的研究，这一直是他在算子理论生涯中的主要主题之一。他的主函数理论定义在某些算子的谱上，是普通指数的广泛推广。这个新对象有它自己的拓扑和稳定性属性。他的深入研究，部分与他的学生和其他人合作，建立了主函数和与遍历流有关的函数的零点分布、次正规算子的结构以及函数代数研究中的闭流理论之间的联系。最近，这些几何测度论的技巧使Pincus教授引入了一个与生成的算子代数相关的新指标。这一新指标具有与Fredholm谱及其在奇异子簇上的行为有关的重要性质。对这一理论的研究在本提案中继续进行。它的重点是系统地将代数几何的技术引入算子理论。

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 the same 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. So it has been, that from the fertile minds of Hilbert, von Neumann, and other giants that the subject of operator theory has grown to a central position in such investigations, and in the 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, in the study of the structure of operators is in the non-normal theory. At this frontier, Professor Pincus is a world leader with an established reputation for deep, conceptualy innovative fundamental, interdisciplinary work. In his early research career, while at Brookhaven Laboratories, Professor Pincus pioneered new ideas and techniques, both theoretical and computational, in medical tomography. Notably, his use of the Radon transforms for image reconstruction appears to have been the precursor of the more well known work of others, for example the Nobel prize work of Cormack. Subsequently, he began his investigation of the principal function, which has been one of the main themes of his career in operator theory. His theory of the principal function, defined on the spectrum of certain operators, is a broad extension of the ordinary index. This new object has its own topological and stability properties. His penetrating investigations, in part in collaboration with his students and others, have established connections between the principal functions 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. These geometric measure theory techniques have recently led Professor Pincus to introduce a new index associated with the generated operator algebras. This new index has important properties connected with the Fredholm spectrum and its behavior on singular subvarieties. The investigation of this theory is continued in the current proposal. Its focus is on the systematic introduction of techniques from algebraic geometry into operator theory.