The ACO Seminar (2021–2022)

One of the most famous statements in graph theory is that of Kuratowski’s theorem, which states that a graph is planar if and only if it does not contain either of K_{3,3} or K_5 as a topological minor. That is, if no subdivision of either K_{3,3} or K_5 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 (topological) 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.