**Abstract:** | Finitely generated structures are importantsubjects of study in various mathematical disciplines. Examples include finitely generated groups, finitely generated Lie algebras and C*-algebras, tuples of several linear operators on Banach spaces, etc. It is thus a fundamental question whether there exists a universal mechanism in the study of these vastly diferent entities. In 2009, the notion of projective spectrum for several elements A_1, A_2, ..., A_n in a unital Banach algebra B was defined through the multiparameter pencil A(z) = z_1A_1+z_2A_2+...+z_nA_n, where the coefficients z_j are complex numbers. This conspicuously simple definition turned out to have a surprisingly rich content. In this talk we will review some results related to group theory, complex geometry, Lie algebras, operator theory and complex dynamics. |