Mathematical Sciences: Pure and Applied Operator Theory

点击翻译按钮获取中文摘要

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.