Topics in Complex and Harmonic Analysis

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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.