Georgia Institute of Technology College of Sciences

Given a surface in space with a set of curves on it, one can ask whichpossible combinatorial arrangement of curves are possible. We give anenriched formulation of this question in terms of which two-dimensionalfans occur as the tropicalization of an algebraic surface in space. Ourmain result is that the arrangement is either degenerate or verycomplicated. Along the way, we introduce tropical Laplacians, ageneralization of graph Laplacians, explain their relation to the Colin deVerdiere invariant and to tensegrity frameworks in dynamics.This is joint work with June Huh.