A holomorphic function of a single variable can easily be applied to matrices whose spectra lie in the domain of the function, and this has developed into a very successful and important theory. The Principal Investigator proposes to study functions of several variables applied to matrices. The theory immediately splits into the commutative and noncommutative cases. In the former, the project will investigate when functions of several variables preserve the natural order structure on matrices, and what happens when the size of the matrices becomes infinite (the operator case). Conversely, the Principal Investigator proposes to use tools from operator theory to understand functions of several variables, such as characterizing asymptotic expansions of functions in the Pick class (this is called the Hamburger problem in one variable). In the noncommutative case, he will investigate how noncommutative holomorphic functions can be obtained as limits of noncommutative polynomials. The description of noncommutative holomorphic functions will be in terms of operator theoretic realizations. He will also work on applying analytic techniques to improve ultrasound imaging.
Virtually all imaging devices today function by collecting either electromagnetic or acoustic waves and using the energy carried by these waves to determine pixel values in order to build up what is basically an "energy" picture. However, waves also carry information, and this can also be used to determine the pixel values in an image. Various studies have shown that these "information" images often reveal features that are completely missed by conventional, energy images. This project will study how to improve these images, using various different measures of information-theoretic entropy, both to render them sharper, and to enable their calculation in real time. These techniques will have applications in medical imaging and also in security scanning. The Principal Investigator will also conduct basic research in areas of pure mathematics related to control theory and signal processing.
