Multidimensional Persistence: Invariants and Parameterization

This article grew out of my Master's thesis at the Faculty of Mathematics and Information Science at Ruprecht-Karls-Universit\"at Heidelberg under the supervision of PD Dr. Andreas Ott. The content of this article is mainly based on and inspired by the work of G. Carlsson and A. Zomorodian on the theory of multidimensional persistence in 2007 and 2009. Our main goal is to carry out a complete classification and parameterization for the algebraic objects that correspond to the homology of a multifiltered simplicial complex. Moreover, we show that it is only possible to obtain a discrete and complete invariant for these algebraic objects in the case of one-dimensional persistence, and that it is impossible to obtain the same in dimension greater than one.