Title: Stanley Decompositions and Involutive Bases Author: Apel, Joachim Affiliation: MSRI Berkeley, California Abstract: In 1982 Richard P. Stanley conjectured that a graded module M over a graded k-algebra R can be decomposed in a particular way in a direct sum of finitely many free modules over suitable subalgebras of R. Besides homogeneity conditions the most important restriction which such a decomposition has to satisfy is that the subalgebras must have at least dimension depth(M). For monomial ideals I of a polynomial ring R such Stanley decompositions of M1=I and M2=R/I are strongly related to decompositions of the set of derivatives occurring in the Riquier/Janet method for solving systems of PDE's. We will show how general involutive bases may be applied in order to prove some particular cases of Stanley's conjecture and to provide algorithms for the computation of Stanley decompositions.