Classification of varieties with bounded colength: a view through recent advances

Numerical invariant sequences associated with algebras have gained increasing attention due to their role in understanding the asymptotic behavior of polynomial identities. This work aims to provide a comprehensive overview of results concerning the colength sequence in many settings of PI-algebras, particularly algebras with involution, graded algebras and algebras with graded involution or superinvolution. We begin by recalling the fundamental definitions and motivations behind the study of colength, emphasizing its connection with codimension growth and other properties of PI-algebras. Finally, we present the main results recently obtained about the colength sequence of algebras with additional structures.