“…Tropical geometry is an emerging tool in algebraic geometry that can transform certain questions into combinatorial problems by replacing a variety with a polyhedral object called a tropical variety. It has had striking applications to a range of subjects, such as enumerative geometry [Mik05, FM10, GM08, AB13], classical geometry [CDPR12,Bak08], intersection theory [Kat09, GM12,OP13], moduli spaces and compactifications [Tev07, HKT09, ACP15, RSS14], mirror symmetry [Gro10, GPS10,Gro11], abelian varieties [Gub07,CV10], representation theory [FZ02,GL13], algebraic statistics and mathematical biology [PS04,Man11] (and many more papers by many more authors). Since its inception, it has been tempting to look for algebraic foundations of tropical geometry, e.g., to view tropical varieties as varieties in a more literal sense and to understand tropicalization as a degeneration taking place in one common algebro-geometric world.…”