Realization spaces of polytopes: Dimensions, naive guesses, singularities and special examples

The space of all realizations of a convex polytope can described as an explicit intersection of real quadrics. While the "Universality Theorem" says that there exist realization spaces that are extremely complicated, most examples met in practice turn out to be smooth manifolds, and the "naive guess" for their dimensions seems to be usually correct.

I will argue that it pays off to look at specific examples - the most exciting ones will be two 4-dimensional polytopes, namely the bipyramid over a prism, and the 24-cell, whose realization space is non-trivial, but also still mysterious - and poses computational challenges.

(Joint work with Laith Rastanawi and Rainer Sinn.)