# Colloquium by Laith Rastanawi (Freie Universität Berlin): On the Dimension of the Realization Spaces of Polytopes

The study of realization spaces of convex polytopes is one of the oldest subjects in Polytope Theory. Most likely, it goes back to Legendre (1794). A lot of progress took place since that time. However, many questions remained open. In general, computing the dimension of the realization space $\mathcal{R}(P)$ of a d-polytope $P$ is hard, even for $d = 4$, as shown by Mnëv (1988) and Richter-Gebert (1996).

In this presentation, we will discuss two criteria to determine the dimension of the realization space, and use them to show that $\dim \mathcal{R}(P) = f_1(P) + 6$ for a 3-polytope $P$, and $\dim \mathcal{R}(P) = df_{d-1}(P)$ (resp. $\dim \mathcal{R}(P) = df_0(P)$ ) for a simple (resp. simplicial) $d$-polytope $P$. We will also discuss the realization spaces of some interesting 2-simple 2-simplicial 4-polytopes. Namely, we will consider the realization space of the 24-cell and of a 2s2s polytope with 12 vertices which was found by Miyata (2011), and give a better bound for/determine its dimension.

### Time & Location

Jul 02, 2018 | 04:00 PM

Freie Universität Berlin

Institut für Informatik

Takustr. 9

14195 Berlin

Room 005 (ground floor)