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SUMMARY:Antonio Lerario (SISSA)
DTSTART:20210503T160000Z
DTEND:20210503T170000Z
DTSTAMP:20260423T040741Z
UID:TG_ET/18
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/TG_ET/18/">M
 oduli spaces of geometric graphs</a>\nby Antonio Lerario (SISSA) as part o
 f Topology and geometry: extremal and typical\n\n\nAbstract\nIn this talk 
 I will investigate the structure of the "moduli space" $W(G\,d)$ of a geom
 etric graph $G$\, i.e. the set of all possible geometric realizations in $
 \\mathbb R^d$ of a given graph $G$ on $n$ vertices. Such moduli space is S
 panier–Whitehead dual to a real algebraic discriminant.\n\nFor example\,
  in the case of geometric realizations of $G$ on the real line\, the modul
 i space $W(G\, 1)$ is a component of the complement of a hyperplane arrang
 ement in $\\mathbb R^n$. (Another example: when $G$ is the empty graph on 
 n vertices\, $W(G\, d)$ is homotopy equivalent to the configuration space 
 of $n$ points in $\\mathbb R^d$.) Numerous questions about graph enumerati
 on can be formulated in terms of the topology of this moduli space.\n\nI w
 ill explain how to associate to a graph $G$ a new graph invariant which en
 codes the asymptotic structure of the moduli space when $d$ goes to infini
 ty\, for fixed $G$. Surprisingly\, the sum of the Betti numbers of $W(G\,d
 )$ stabilizes as $d$ goes to infinity\, and gives the claimed graph invari
 ant $B(G)$\, even though the cohomology of $W(G\,d)$ "shifts" its dimensio
 n.  We call the invariant $B(G)$ the "Floer number" of the graph $G$\, as 
 its construction is reminiscent of Floer theory from symplectic geometry.\
 n\nJoint work with M. Belotti and A. Newman.\n
LOCATION:https://researchseminars.org/talk/TG_ET/18/
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