algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety survey; differential algebra; diophantine geometry; stability; stable groups; differentially closed fields; \(\omega\)-stable structures; differential algebraic groups; \(\delta\)-definable groups; differential algebraic geometry; algebraic group; differential Galois theory of strongly normal extensions; Galois groups; Mordell-Lang conjecture; geometric version 