Operator Theory and Complex Analysis

PI将研究复分析和算子理论之间相互作用的一系列问题。这两个数学领域已经成为共生的，每一个导致增长和发展的其他。一个问题是私家侦探。将研究的是所有零点都在单位圆上的多项式的熵。这里有一个自然的猜想，如果多项式有等间距的零，熵就最小化;这个猜想可以用算子理论的术语重新表述，在复分析中有结果，比如理解到穿孔圆盘的极值映射。私家侦探将致力于将算子理论与多元函数联系起来的有趣问题。特别是，他将寻求表征这些职能的几个变量是矩阵单调，这将扩大基本结果K。Löwner从1934年，其特点是功能的一个变量与此属性。私家侦探他还将继续与医学超声成像专家合作，通过分析熵而不是反射信号的能量来改善成像。 私家侦探最直接的影响。的工作将在超声波领域。他与M。Hughes的目标是生产可以安装在手持式超声扫描仪中的软件，该扫描仪可以在家中用于测量杜氏肌营养不良症儿童的肌肉密度，从而允许每天调整他们的药物方案。他的工作在纯数学，因为是正常的在dsicipline，将扩散更缓慢地进入更广泛的科学领域。最初的主要影响将是在纯数学，并在未来的研究人员的教育，但有一个长期的历史交叉从算子理论复杂的分析到工程领域的控制理论，并应继续下去。私家侦探与化学和电气工程师、物理学家、化学家、生物学家和医生有着成功的合作记录。他的背景，纯数学导致不同的方法来解决他们的问题，这往往是成功的。他将继续寻找科学家和工程师合作。

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.