  Persistent homology has emerged as a popular technique for the topological
simplification of big data, including biomolecular data. Multidimensional
persistence bears considerable promise to bridge the gap between geometry and
topology. However, its practical and robust construction has been a challenge.
We introduce two families of multidimensional persistence, namely
pseudo-multidimensional persistence and multiscale multidimensional
persistence. The former is generated via the repeated applications of
persistent homology filtration to high dimensional data, such as results from
molecular dynamics or partial differential equations. The latter is constructed
via isotropic and anisotropic scales that create new simiplicial complexes and
associated topological spaces. The utility, robustness and efficiency of the
proposed topological methods are demonstrated via protein folding, protein
flexibility analysis, the topological denoising of cryo-electron microscopy
data, and the scale dependence of nano particles. Topological transition
between partial folded and unfolded proteins has been observed in
multidimensional persistence. The separation between noise topological
signatures and molecular topological fingerprints is achieved by the
Laplace-Beltrami flow. The multiscale multidimensional persistent homology
reveals relative local features in Betti-0 invariants and the relatively global
characteristics of Betti-1 and Betti-2 invariants.
