Mathematical Sciences: Pure and Applied Operator Theory

算子理论是现代分析中的一个中心学科。 它的起源在于研究数学物理和部分 微分方程在世纪早期。 在 与此同时，人们看到，许多物理问题，在 平衡理论、振动理论、量子理论等等， 通过模拟的积分方程进行了富有成效的研究 现象。 所以，从那些思想丰富的人 希尔伯特，冯·诺依曼，和其他巨人， 算子理论已经发展到这样的中心位置， 研究，以及核心数学。 在 这种方法论的核心是对 一个算子的谱 对于自伴算子或正规算子， 这个理论现在是整个分析的标准技术， 谱定理提供了必要的积木， 所有这些运营商。 因此，目前的前沿， 算子结构的研究属于非正规理论。 在这一前沿领域，平卡斯教授是一位世界领导者， 建立了深厚的声誉，概念创新 跨学科的基础工作。 在他早期的研究生涯中， 实验室，平卡斯教授开创了新的想法， 技术，包括理论和计算，在医疗 断层扫描 值得注意的是，他对图像的Radon变换 重建似乎是更多的 其他人的著名作品，例如诺贝尔奖的作品， 科马克 随后，他开始调查 主要功能，这一直是他的主要主题之一， 经营者理论 他的主函数理论， 定义在某些运营商的频谱上，是一个广泛的 普通指数的延伸。 这个新物体有它自己的 拓扑性质和稳定性性质。 他敏锐 调查，部分是与他的学生合作， 其他人，已经建立了校长之间的联系 函数和相关函数的零点分布 遍历流，次正规算子的结构，以及 函数代数研究中的闭流理论。 这些几何测量理论技术最近导致了 平卡斯教授介绍了一个新的指数与 生成算子代数 这一新指数具有重要意义 与Fredholm谱及其行为有关的性质 关于奇异子变种 这个理论的研究是 在目前的提案中继续。 其重点是 代数几何技术的系统介绍 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.