Topics in Complex and Harmonic Analysis

该研究项目继续开发数学技术，这些技术的动机是应用于大型网络上的流量分析和医学断层扫描。 所研究的问题需要使用的想法，从复杂的分析在一个和几个变量，积分几何，调和分析，和交换代数。 一个统一的因素是，有两个技术元素，几乎无处不在，剩余和氡变换。 近年来，依赖于任意特征域上剩余性质的思想被用来研究求解多变量多项式方程组的复杂性。 这些结果已应用于机器人，系统工程和通信。 正在研究的问题之一是监测大型网络，以发现干扰或故障的早期迹象。 该项目采用了类似于层析成像的思想，更准确地说，是双曲平面上的逆电导率问题和Radon变换。该项目进一步发展了在层析成像和通信网络分析中发挥核心作用的复杂数学技术。 这些结果有可能应用于改善三维医学成像，并增加对互联网流量模式和安全威胁的了解。

This research project continues development of mathematical techniques that are motivated by applications to analysis of traffic on large networks and to medical tomography. The problems under study require the use of ideas from complex analysis in one and several variables, integral geometry, harmonic analysis, and commutative algebra. A unifying factor is that there are two technical elements that appear almost everywhere, residues and Radon transforms. In the recent past, ideas dependent on the properties of residues over fields of arbitrary characteristic were employed to study the complexity of solving large systems of polynomial equations in many variables. These results have applications to robotics, systems engineering, and communications. One of the problems under study is the monitoring of large networks to detect early signs of jamming or failure. The ideas used are akin to those in tomography, more precisely, the inverse conductivity problem and the Radon transform in the hyperbolic plane.This project further develops sophisticated mathematical techniques that play a central role in tomographic imaging and analysis of communication networks. The results have potential application to improvement of three-dimensional medical imaging and to increased understanding of internet traffic patterns and security threats.