Pure Maths Colloquium: Arnau Padrol

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.