Event Type
Lectures
A new theory of graphs embedded on surfaces: from complete walks to “topological ergodicity’’
Friday, November 11, 2022 2:30 PM
François Lalonde (Université de Montréal)

One motivation for this work is to launch a rocket from Earth carrying a satellite to be put in orbit on some other planet. There is then a main global Hamiltonian through Lagrange points, and a minimal set of smaller local Hamiltonians. In dimension 2, that main Hamiltonian (a highway) is a complete leftward walk through a graph G embedded on the surface of genus g. Here the phase space T(G) of G is defined as all pairs (v,e) where v is a vertex and e is an edge adjacent to it (this corresponds to putting an orientation on an edge). A leftward walk is determined by an initial condition (v,e) by turning left at each vertex or making a rebound if there is no other edge. The leftward walk is called complete if it visits all edges, not necessarily in both directions. Here the left motion of the walk corresponds to Hamiltonian dynamics on level surfaces. The main theorem is that if the average valence of the graph is at least  1 + \sqrt{6g +1}, the graph has no complete leftward walk. On the other hand, it is a very difficult problem to produce sharp examples of graphs with a complete leftward walk and average valence just below that bound. One way of doing this is to blow-up monographs (a graph with one vertex and as many loops as we wish) by a series of well-chosen local blow-ups, while keeping the global topology the same. It turns out that part of this new theory goes back to articles by Heffter (1860), and a century later by and Ringle, Edmonds and Youngs (1963 and 1965) for the 4-colour problem in all genera.  We will introduce, finally, a new notion, the one of topological ergodicity that is somehow opposed to ordinary ergodicity, and much more subtle. This is joint work with Dustin Connery-Grigg and Jordan Payette.

You can learn more about Professor Lalonde here.