Around the Bass' Triangulability Problem

The talk is aimed to elaborate on the Bass' Triangulability Problem concerning rational triangulability of unipotent algebraic subgroups of the affine Cremona groups.
The following topics will be discussed: a rational triangulability criterion; the existence of rationally non-triangulable connected solvable affine algebraic subgroups of the Cremona groups; stable rational triangulability of such subgroups (in particular, the affirmative answer to the Bass' Triangulability Problem in the stable range); a general theorem on invariant subfields of purely transcendental field extensions used for proving these results; a general construction of all rationally triangulable subgroups of the Cremona groups; application: the classification of all rationally triangulable connected one-dimensional unipotent affine algebraic subgroups of the Cremona groups up to conjugacy.