Dimension formulas for period spaces via motives and species

Abstract: We apply the structure theory of finite dimensional algebras in order to deduce dimension formulas for spaces of period numbers, i.e., complex numbers defined by integrals of algebraic nature. We get a complete and conceptually clear answer in the case of $1$-periods, generalising classical results like Baker's theorem on the logarithms of algebraic numbers and partial results in Huber--W{\"u}stholz \cite{huber-wuestholz}. The application to the case of Mixed Tate Motives (i.e., Multiple Zeta Values) recovers the dimension estimates of Deligne--Goncharov \cite{deligne-goncharov}.