REU Site: Inverse Problems for Electrical Networks

University Of Washington, Seattle WA

Investigators

Abstract

The REU program on Inverse Problems for Electrical Networks will bring eight undergraduates to the University of Washington in each of the summers of 2015-2017 to work on inverse problems for electrical networks and related problems on graphs. The students will also investigate problems in combinatorics and special topics in number theory. The simplest inverse problem for an electrical network is to determine the conductance of each of the conductors in an electrical network when the boundary current response of the network to boundary voltages is known, but no interior knowledge is available. Imagine a box in which there is an unknown electric circuit with leads protruding from the box at which current and voltage can be measured. The solution of the problem can be interpreted as providing an electrical version of an x-ray. A simplified version of the inverse problem is the problem of mine detection with an electrical field. The intent of the program is to guide students into independent research projects which can be completed in the eight week period. The students will be given reading material before the program starts and introduce them to research problems in the first week. Students will learn how to formulate and attack research problems in a supportive, collaborative atmosphere. An emphasis will be placed on communicating results. Prior students will work as TAs and mentor current students. Alumni of the program will be frequent visitors and provide additional help. There are difficult technical questions that students have asked and answered. In recent years students have discovered exotic electrical networks that cannot be recovered by boundary data but are almost uniquely determined. In other words there are only a finite number of possibilities for networks compatible with the given boundary measurements. These are called networks n-to-1 networks. Students in the program invented these networks and discovered their properties. New examples are constructed by gluing certain "four-star" graphs in a pattern that looks something like a cactus or shoelace graph. There are now many different families of such graphs, which will be investigated. Another topic is properties of permutations, especially peak points of permutations. Students have recently found interesting properties of peak polynomials, for instance that all peaks in a peak set are zeros of its corresponding peak polynomial. These and other topics will be the building blocks of this program.