This award is being co-funded OISE and DMS and will partially support collaborative research between Professor A. Koranyi of CUNY, H H Lehman College, and Professor G. Misra if Indian Institute of Science. The research will focus on homogeneous operators. These form an important class of operators on Hilbert space. Professor G. Misra is the leading expert on them. There are important open questions in the field, such as the classification and explicit description of all homogeneous operators in the Cowen-Douglas class. A. Koranyi and G. Misra have been doing joint research on this problem for the last three years. Two publications have resulted from this, giving a full classification in an important special case and proving various properties. The continuation of this research is expected to lead to a complete classification in the general case and to results about related problems, in particular about dilation theory. After this, it will be attempted to extend at least some of the results to the case of n-tuples of commuting operators associated to bounded symmetric domains.
The intellectual merit of the proposal consists in solving some long open problems of operator theory by the combination of ideas from different branches of mathematics. This will constitute substantial progress in operator theory. The broader impacts include advancing operator theory and widely distributing the results in a wy to be useful to all working in the field. This will furnish a method that may have applications in physics and will have a beneficial effect on teaching and research activities both at the CUNY Graduate Center and at the Indian Institute of Science in Bangalore. The project is large enough for students and junior researchers to participate in it.
The project concerned research in pure mathematics, the study of so-called homogeneous operators. This is a subject of great theoretical interest for a group of pure mathematicians working in related fields, but at the moment it does not have practical applications. It should be noted, however, that the larger mathematical context of this subject, namely operator theory and Lie group representation theory have important physical applications, in fact they lie at the foundations of modern physics. The project was done in joint research of the Principal Investigator and of Prof. G. Misra of Bangalore, India. Prof. Misra is the founder of the subject of homogeneous operators and has important previous results about it. For the specific problems of the present report the knowledge of the Principal Investigator about representation theory complemented the knowledge of Prof. Misra. The research was performed in part by correspondence, but also on several intensive two or three week sessions alternatingly in New York and in Bangalore, India. The work accomplished a large part of what it was intended to accomplish. It resulted in two publications in peer-reviewed professional journals. Some further results will be published shortly, after making them as complete as possible. Beside having advanced the subject, which is of interest to a certain number of researchers in the world, the project also contributed to mathematical education: three Ph. D. students of Prof. Misra have thesis subjects related to parts of the project, furthermore, the results were spread among mathematicians by the Principal Investigator and Prof. Misra in a number of mathematical seminar lectures.
