UAlberta Math and Statistics Colloquium: Beatrice Vritisiou

Given a convex solid in 3D space, how many light sources do we have to place around it in order to illuminate its entire surface? What is the minimum number that works? We call this number the illumination number of the solid, and perhaps a bit surprisingly it is very strongly affected by both properties of the surface of the solid, and by other, more 'global' properties of the shape (such as symmetries of the solid).

We can even find solids arbitrarily close to each other which have very different illumination numbers (where 'closeness' is measured by whether we can fit one solid in very small enlargements of the other, and vice versa). For higher-dimensional analogues of this question, such a phenomenon becomes even more pronounced. It can be shown that for an n-dimensional parallelepiped, we need at least 2^n light sources to illuminate its entire surface, but arbitrarily close to this parallelepiped we can find convex shapes for which n+1 light sources suffice.

It was conjectured in the 1960s that the parallelepiped is the worst case, but the full conjecture is still open even in 3 dimensions. Still, from early on, a combination of arguments on related problems made it clear that, if the convex shape is symmetric about a point, then its illumination number is at most a number very close to the conjectured bound (if one is willing to ignore lower-order terms).

In my talk I will discuss several, very different approaches considered over the years towards the conjecture (many of them dealing with special families of convex shapes), and will conclude with work on two very different settings: (i) general shapes, where no symmetry is assumed, and (ii) a family of highly symmetric convex shapes (which have the same symmetries as the cube).