  This is a survey article on Gorenstein initial complexes of extensively
studied ideals in commutative algebra and algebraic geometry. These include
defining ideals of Segre and Veronese varieties, toric deformations of flag
varieties known as Hibi ideals, determinantal ideals of generic matrices of
indeterminates, and ideals generated by Pfaffians of generic skew symmetric
matrices. We give a summary of recent work on the construction of squarefree
Gorenstein initial ideals of these ideals when the ideals are themselves
Gorenstein. We also present our own independent results for the Segre,
Veronese, and some determinantal cases.
