Pure Maths Colloquium -- Arnau Padrol (Jussieu)

On Moser's shadow problem

In a famous list of problems in combinatorial geometry from 1966, Leo Moser asked for the largest s(n) such that every 3-dimensional convex polyhedron with n vertices has a 2-dimensional shadow with at least s(n) vertices. I will describe the main steps towards the answer, which is that s(n) is of order log(n)/log log(n), found recently in collaboration with Jeffrey Lagarias and Yusheng Luo, and which follows from 1989 work of Chazelle, Edelsbrunner and Guibas. I will also report on current work with Alfredo Hubard concerning higher-dimensional generalizations of this problem.