Mathematical Sciences: Pure and Applied Operator Theory

数学科学：纯粹与应用算子理论

• 批准号: 8704684
• 负责人: Joel Pincus
• 金额: $ 11.97万
• 依托单位: SUNY at Albany
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-07-01 至 1990-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8704684&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Pure; Applied; Operator

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.

算子理论是现代分析的核心学科。 它的起源在于二十世纪初对数学物理和偏微分方程的研究。 同时，人们发现平衡理论、振动理论、量子理论等中的许多物理问题可以通过对现象进行建模的积分方程进行有效的研究。 因此，从希尔伯特、冯·诺依曼和其他巨人的丰富思想中，算子理论的主题已经发展到此类研究以及核心数学中的中心位置。 该方法的核心是对运营商频谱的深入调查。 对于自伴算子或正规算子，该理论现在已成为整个分析过程中的标准技术，并且谱定理为所有此类算子提供了必要的构建模块。 因此，当前算子结构研究的前沿是非正规理论。 在这一前沿领域，平卡斯教授是一位世界领袖，在深入、概念创新的基础性跨学科工作方面享有盛誉。 在布鲁克海文实验室的早期研究生涯中，平卡斯教授在医学断层扫描领域开创了理论和计算方面的新思想和技术。 值得注意的是，他使用氡变换进行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，这一直是他算子理论生涯的主题之一。 他的主函数理论是在某些算子的范围上定义的，是普通指数的广泛扩展。 这个新物体有自己的拓扑和稳定性属性。 他的深入研究，部分是与他的学生和其他人合作，在函数代数研究中建立了主函数和与遍历流相关的函数零点分布、次正规算子的结构以及闭流理论之间的联系。 这些几何测度理论技术最近促使 Pincus 教授引入了与生成的算子代数相关的新索引。 这个新指数具有与 Fredholm 谱及其在奇异子品种上的行为相关的重要属性。 当前提案中继续对该理论的研究。 它的重点是将代数几何技术系统地引入算子理论。