Composite materials sometimes have properties unlike those of their constituents and in this context they are known as metamaterials.  A famous example is the Roman Lysurgis glass cup in the British museum which may appear either green or red depending on whether the light source is inside or outside the glass.  Regular arrays of spheres of silica give opals their color.  Wave propagation in static but inhomogeneous media has been well studied.  Now there is great interest in exploring wave propagation in dynamic media, i.e. "space-time materials," where the material properties vary with both time and space.  This class will explore space-time materials for which waves propagate in ordered patterns in space-time.  These are known as field pattern materials.  Previous research has only scratched the surface and there is much to investigate here.  More examples of field pattern materials are needed.  This is just ray tracing, but one needs to position the interfaces so the rays form patterns.  What relations are possible between the periodicity of the material and the periodicity of the field pattern?  (The field pattern can break symmetry just as symmetry is broken in "time crystals.")  If there is no field pattern, but rather just a complicate cascade of wave scattering, can one find something analogous to an ergodic hypothesis to make predictions?  After reviewing the literature, the class will start with easy problems then move on to more challenging problems.  It is anticipated that all students making significant contributions will be coauthors of a publication resulting from the work.  Writing up the paper will be part of the class.  For further background see the SIAM news article, "New Horizons in the Study of Waves in Space-Time Microstructures" and references therein.  Please email the instructor (graeme.milton@utah.edu) if you are interested in taking this class.

Linear Algebra (MATH 2270 or MATH 2250) and Partial Differential Equations (MATH 3150 or MATH 5440).