This project is in the area of complex and harmonic analysis. It is concerned with the set of problems in harmonic analysis that fall under the heading of the Uncertainty Principle, a relative of the principle of the same name that arises in quantum mechanics. Stemming from the work of Norbert Wiener in mathematics and Werner Heisenberg in physics, the area of the Uncertainty Principle still presents many mathematical challenges, and the field has numerous applications to adjacent fields. Several classical problems associated with the Uncertainty Principle, problems posed decades ago by such prominent mathematicians as Norman Levinson, Andrei Kolmogorov, and Norbert Wiener, remain open. Certain of those problems will be studied in this project. Modern methods of complex and harmonic analysis that appeared in the last thirty years suggest new approaches to the classical challenges of the Uncertainty Principle. Two such problems, the so-called gap and type problems, have recently been solved by the principal investigator. These problems have a number of important applications in approximation theory, prediction theory, spectral theory of differential operators, and mathematical physics.
A new approach in the area of the Uncertainty Principle was developed in recent papers of Nikolai Makarov and the principal investigator. One of the main ingredients of the new approach is the use of Toeplitz operators, which explains the title of the project. Indeed, the next stage of the application of the Toeplitz approach in harmonic analysis and spectral theory is the focal point of this project. It contains several classical open problems. Among the topics of research to be pursued are the following: general completeness problems, which will expand the results on the "type" problem mentioned earlier; spectral problems for Schroedinger and Dirac operators; and a Toeplitz operator version of the so-called Krein-de Branges theory, which was designed to connect complex and spectral analysis. Successful completion of this research will create a new and systematic view of the large variety of problems in the area of the Uncertainty Principle. A part of this project will be done in collaboration with the principal investigator's graduate students at Texas A&M University. Several mini-courses on the Toeplitz approach to the Uncertainty Principle were recently given by the principal investigator at U.S. and international research centers. All of the courses are oriented towards young researchers and graduate students. Together with Nikolai Makarov and Mikhail Sodin, the principal investigator is working on a book that will contain a modern account of the Krein-de Branges theory and other topics related to the Uncertainty Principle.
