Education

Abstract

Television. Cell phones. Space flight. An invisibility cloak? All of these inventions were envisioned by science fiction writers well before they became reality, and each has evolved from crude prototypes to become part of our quotidian existence. Well, all except for the invisibility cloak, which is the subject of this issue's Education section offering, "Impedance Imaging, Inverse Problems, and Harry Potter's Cloak", by Kurt Bryan and Tanya Leise. The authors report on a body of recent work examining the theory needed to build a nonfictional cloaking device. The authors examine the question of impedance tomography, basically asking what can be deduced about the interior of a region by applying an electrical current to its boundary and observing the resulting potential. They show through a clever change of variables that if one can change the conductivity of the region's interior, then one can hide a void there and consequently cloak anything in the void's interior (which whimsically might be Harry Potter or a Romulan warship). Mathematically, the paper draws on vector calculus, Fourier series, and a bit of electromagnetic theory, primarily the ideas of conductivity and electromagnetic potential. Buried just beneath the surface are elliptic partial differential equations and the definition of an inverse problem—gems waiting to be unearthed by the industrious student. The paper nicely describes a link between an active area of research and the theory learned in a first course on Fourier analysis and partial differential equations. It provides an assortment of exercises, several of which easily could become undergraduate research projects and which potentially could lead a student to pursue a graduate degree or make a career out of trying to render objects invisible. I'll leave you now to daydream uses of that invisibility cloak, once we work out those pesky technological and mathematical details.