Poncelet’s theorem

Imagine nested two ellipses C, D ⊂ R²  in the ordinary real plane. Poncelet asks: are there any n-sided polygons in the plane inscribed in C and circumscribed about D?

Poncelet’s theorem gives a striking answer to this question: it asserts that, depending on C and D, there are either no such polygons or a continuously varying family of them. In this talk I’ll describe a modern proof of the theorem, in which we work in complex projective space and use the topology of associated loci. There’s also a less well-known Poncelet-type theorem for polyhedra in 3-space, which I’ll describe.