In this talk I will investigate the structure of the "moduli space" of a geometric graph , i.e. the set of all possible geometric realizations in of a given graph on n vertices. Such moduli space is Spanier-Whitehead dual to a real algebraic discriminant. For instance, in the case of geometric realizations of on the real line, the moduli space is a component of the complement of a hyperplane arrangement in . 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 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 stabilizes, as d goes to infinity, and gives the claimed graph invariant -- even though the cohomology of "shifts" its dimension (we call the invariant the "Floer number" of the graph , as its construction is reminiscent of Floer theory from symplectic geometry.)
This is joint work with M. Belotti and A. Newman
