For a convex $d$-polytope $P$ its realisation space is a space of all (modulo affine transformations) $d$-polytopes having the same face lattice as $P$. From Steinitz's characterisation of edge graphs of $3$-polytopes (these are precisely planar $3$-connected graphs) it follows that for a $3$-polytope with $e$ edges its realisation space is a smooth open ball of dimension $e - 6$. However, in higher dimensions realisation spaces of polytopes cease to be topologically trivial and, in fact, for any $d>3$$d$-polytopes are universal in the sense of Mnëv (J. Richter-Gebert & G.M. Ziegler, 1995) meaning that questions related to realisation spaces are as difficult as the corresponding problems for general systems of polynomial inequalities.
In the talk we will first recall all the required definitions (such as basic primary semialgebraic sets, stable equivalences, oriented matroids and their realisation spaces, etc.) Then we will prove Mnëv's university theorem for oriented matroids, namely, that for every basic primary semialgebraic set $V$ defined over integers there is an oriented matroid whose realisation space is stably equivalent to $V$. After that we will deduce from this theorem some universality theorems for polytopes.