Science is littered with examples where progress in one area has led to progress in other sometimes completely different areas thanks to similar underlying mathematical structures. Here we show that many of the linear wave-equations of physics can be reformulated in a way that is similar to the static or quasistatic equations of composites. The reformulation of Schrödinger’s equation, for example, leads to new minimization variational principles. Desymmetrizing it leads to FFT methods for solving the multielectron Schrödinger equation which only requires FFT transforms in 2 spatial variables. Unlike density functional theory it is applicable even to excited states.
The theory of composites carries over to the response of inhomogeneous bodies, and integral representations and bounds for the dynamic response can be used in an inverse way to say something about what is inside the body from boundary measurements. Even the concept of functions can be generalized, based on the theory of composites. The new functions, superfunctions, have applications to accelerating FFT methods. A superfunction can be thought as a collection of subspaces, and there are associated rules for multiplication, division, addition, and subtraction of such superfunctions. This talk is based on a book I am editing, by the same title with four chapters coauthored with Maxence Cassier, Ornella Mattei, Mordehai Milgrom, and Aaron Welters.
Further information can be found on Professor Milton’s webpages here and here.