Abstract for the talk on 01.12.2020 (17:45 h)Nonlinear Algebra Seminar Online (NASO)
Antonio Lerario (Scuola Internazionale Superiore di Studi Avanzati (SISSA), Triest)
Moduli spaces of geometric graphs, real discriminants and Floer homology
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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