  This work concerns the theoretical foundations of persistence-based
topological data analysis. We develop theory of topological inference in the
multidimensional persistence setting, and directly at the (topological) level
of filtrations rather than only at the (algebraic) level of persistent homology
modules.
  Our main mathematical objects of study are interleavings. These are tools for
quantifying the similarity between two multidimensional filtrations or
persistence modules. They were introduced for 1-D filtrations and persistence
modules by Chazal, Cohen-Steiner, Glisse, Guibas, and Oudot. We introduce
generalizations of the definitions of interleavings given by Chazal et al. and
use these to define pseudometrics, called interleaving distances, on
multidimensional filtrations and multidimensional persistence modules.
  We present an in-depth study of interleavings and interleaving distances. We
then use them to formulate and prove several multidimensional analogues of a
topological inference theorem of Chazal, Guibas, Oudot, and Skraba. These
results hold directly at the level of filtrations; they yield as corollaries
corresponding results at the module level.
