How to prove universality theorems

Given a procedure that associates to a combinatorial object x a geometric object X, one can ask how complicated the geometric objects obtainable by this construction can be. A *universality theorem* is a precise formulation of "as complicated as it could possibly be". That is, the combinatorial objects x cause the spaces X to exhibit all (bad) geometric features, in a certain sense.

Theorems of this form abound in non-linear algebra. They hold for realization spaces of matroids and polytopes in discrete geometry, for totally mixed Nash equilibria in economics, for conditional independence models in statistics. Or, as Ravi Vakil once put it: Murphy's law holds in algebraic geometry [https://arxiv.org/abs/math/0411469].

In this talk I want to present the surprisingly simple geometric idea behind all of these proofs and, time permitting, discuss a conjectured universality theorem that I have so far failed to prove.