I will present two theorems.
"Invisibility via reflecting coating": There exists a subset of the unit disc consisting of a finite number of line segments such that if it is painted black than the shadow of this set in most directions is almost the same as the shadow of the disc. If the set is painted with reflective coating then the light rays emerge on the other side of the set with arbitrarily small distortion.
"Invisible mirror": For every pair of directions, there exists a set consisting of a finite number of line segments which sends a bundle of light rays coming from the first direction, of unit width, to the other direction. The set has almost no shadow in almost all other directions.
I will present (a sketch of) the proof of the second theorem. Time permitting, I will present (a sketch of) the proof of the first theorem, motivation, etc.
This is joint work with Tadeusz Kulczycki.