Moduli spaces of geometric graphs, real discriminants and Floer homology

In this talk I will investigate the structure of the "moduli space" $W(G,d)$ of a geometric graph $G$, i.e. the set of all possible geometric realizations in $R^d$ of a given graph $G$ on n vertices. Such moduli space is Spanier-Whitehead dual to a real algebraic discriminant. For instance, in the case of geometric realizations of $G$ on the real line, the moduli space $W(G, 1)$ is a component of the complement of a hyperplane arrangement in $R^n$. Numerous questions about graph enumeration can be formulated in terms of the topology of this moduli space.

I will explain how to associate to a graph $G$ a new graph invariant which encodes the asymptotic structure of the moduli space when d goes to infinity. Surprisingly, the sum of the Betti numbers of $W(G, d)$ stabilizes, as d goes to infinity, and gives the claimed graph invariant $B(G)$ -- even though the cohomology of $W(G, d)$ "shifts" its dimension (we call the invariant $B(G)$ the "Floer number" of the graph $G$, as its construction is reminiscent of Floer theory from symplectic geometry.)

This is joint work with M. Belotti and A. Newman