Topological data analysis and diagnostics of compressible MHD turbulence
  The predictions of mean-field electrodynamics can now be probed using direct
numerical simulations of random flows and magnetic fields. When modelling
astrophysical MHD, it is important to verify that such simulations are in
agreement with observations. One of the main challenges in this area is to
identify robust \it{quantitative} measures to compare structures found in
simulations with those inferred from astrophysical observations. A similar
challenge is to compare quantitatively results from different simulations.
Topological data analysis offers a range of techniques, including the Betti
numbers and persistence diagrams, that can be used to facilitate such a
comparison. After describing these tools, we first apply them to synthetic
random fields and demonstrate that, when the data are standardized in a
straightforward manner, some topological measures are insensitive to either
large-scale trends or the resolution of the data. Focusing upon one particular
astrophysical example, we apply topological data analysis to HI observations of
the turbulent interstellar medium (ISM) in the Milky Way and to recent MHD
simulations of the random, strongly compressible ISM. We stress that these
topological techniques are generic and could be applied to any complex,
multi-dimensional random field.
