Operator Theory and Complex Analysis

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The PI will study a range of problems on the interplay between complex analysis and operator theory. These two areas of mathematics have become symbiotic, with each leading to growth and development in the other. One problem the P.I. will study is the entropy of polynomials that have all their zeroes on the unit circle. There is a natural conjecture here that the entropy is minimized if the polynomial has equally spaced zeroes; this conjecture, which can be reformulated in operator theoretic terms, has consequences in complex analysis, such as understanding extremal maps into punctured disks. The P.I. will work on the intriguing problem of linking operator theory with functions of several variables. In particular, he will seek to characterize those functions of several variables that are matrix monotone; this will extend the fundamental results of K. Löwner from 1934 that characterized the functions of one variable with this property. The P.I. will also continue his collaboration with specialists in medical ultrasound imaging on ways to improve the imaging by analyzing the entropy rather than the energy of the reflected signal. The most immediate impact of the P.I.'s work will be in the field of ultrasound. His work with M. Hughes is aimed at producing software that can fit into a handheld ultrasound scanner that can be used in the home to measure the muscle density of children with Duschenne muscular dystrophy, and thus allow daily adjustments of their drug regimen. His work in pure mathematics, as is normal in the dsicipline, will diffuse more slowly into broader fields of science. Initially the main impact will be in pure mathematics, and in the education of future researchers, but there is a long history of crossover from operator-theoretic complex analysis into the engineering field of control theory, and that should continue. The P.I. Has a successful record of collaborating with chemical and electrical engineers, physicists, chemists, biologists and doctors. His background in pure mathematics leads to a different approach to their problems, which has often been successful. He will continue to seek out scientists and engineers to collaborate with.