Robertson’s Conjecture in topology.

One of the most famous statements in graph theory is that of Kuratowski’s theorem, which states that a graph is non-planar if and only if it contains one of K_3,3 or K₅ as a topological minor. That is, if some subdivision of either K_3,3 or K₅ appears as a subgraph of your graph. In this case we say that the question of planarity is determined by a finite set of forbidden minors. A conjecture of Robertson, whose proof was recently announced by Chun-Hung Liu, characterizes the kinds of graph theoretic properties that can be determined by finitely many forbidden minors. In this talk I will present a categorical, or algebraic, version of Robertson’s conjecture. I will then illustrate how this conjecture, if proven true, would be able to prove many non-trivial statements in the topology of graph configuration spaces.