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.
With partial support from the grant, the PI wrote 9 papers in pure mathematics, and 7 in applied mathematics. In addition, Ph.D. students Kelly Bickel, Ben Passer and Cheng Chu, who were also partially supported by the grant, wrote 6 papers in pure mathematics. The main thrust of the papers in pure mathematics was to investigate how certain phenomena changed when going from one to two dimensions. The two dimensional situation is much more complicated, and naive generalizations from one dimension are rarely true. However, if carefully interpreted, one can recover traces of the one variable behavior. For example, a self-adjoint matrix A is called positive if all its eigenvalues are positive. Given two self-adjoint matrices A and B, we say that A <= B if B - A is positive. In 1934, K. Lowner characterized functions f with the property that if A <= B then f(A) <= f(B). These functions are called matrix monotone. (If A and B are both positive, then f(z) = sqrt{z}$is matrix monotone, but g(z) = z^2 is not). The P.I., with collaborators J. Agler and N. Young, gave a characterization for functions of two variables. In applied mathematics, the P.I. collaborated with members of the Washington University School of Medicine on mathematical algorithms that improve the resolution of ultrasound images. 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 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. We have shown that these ``information", or ``entropy'', images often reveal features that are completely missed by conventional, energy images. One approach we used was to calculate the Renyi entropy of pieces of the signal, and develop a gray-scale picture using these valeus. The problem with the Renyi entropy was that it could not be computed in real-time. We discovered that the approximation of taking the log of the sum of the absolute value of the second derivative at the critical points was extremely effective, and could be calculated in real time. It has been used for nanoparticle detection in vivo, and also has promise for detection of hidden security risks. Alumina is produced by heating bauxite and caustic soda to 250 degrees Celsius, where they undergo a chemical reaction. This requires enormous amounts of energy. Typical plants use several GigaJoules per tonne. The key to efficiency is to recover as much as possible of the heat from the finished product to heat up the incoming slurry. This is accomplished by an array of flash tanks. In the paper ``Model based methodology development for energy recovery in flash heat exchange systems'', with lead author A. Korobeinikov, we model this, so that the cost-effectiveness of adding an extra tank to the cascade can be calculated.
