I will give an introduction to the scattering theory for
the Helmholtz equation, and then discuss the inverse source and inverse scattering problems.

The basic problem is to describe a source, or scatterer, of a wave based on observations of the far field (a solution to the Helmholtz equation far away from the source or scatterer).

Neither the inverse source problem nor certain versions of the inverse scattering problem have unique solutions. That is, there can be many different sources that produce the same measured data. In order to compute some meaningful information, one must either assume the source has a special form (e.g. a sum of point sources, or the indicator function of convex set), or alternatively, identify something that all sources that produce that data must have in common.\\

I will take the second approach and describe some
notions of scattering support. These are sets which
support a source that can produce the measured data, and are minimal among a restricted class of sets (e.g. convex sets).

For over four hundred years, natural scientists have been intrigued and mystified by patterns appearing on plants and by the appearance of Fibonacci sequences when one counts the numbers of arms in the families of spirals on which the primordia of the plant surfaces lie. To date, there has been no widely accepted mechanistic explanation for these observations. I hope that this lecture goes some way towards providing answers.