The emptiness inside: Finding gaps, valleys, and lacunae with geometric
  data analysis
  Discoveries of gaps in data have been important in astrophysics. For example,
there are kinematic gaps opened by resonances in dynamical systems, or
exoplanets of a certain radius that are empirically rare. A gap in a data set
is a kind of anomaly, but in an unusual sense: Instead of being a single
outlier data point, situated far from other data points, it is a region of the
space, or a set of points, that is anomalous compared to its surroundings. Gaps
are both interesting and hard to find and characterize, especially when they
have non-trivial shapes. We present methods to address this problem. First, we
present a methodological approach to identify critical points, a criterion to
select the most relevant ones and use those to trace the `valleys' in the
density field. We then build on the observed properties of critical points to
propose a novel gappiness criterion that can be computed at any point in the
data space. This allows us to identify a broader variety of gaps, either by
highlighting regions of the data-space that are `gappy' or by selecting data
points that lie in local under densities. We also explore methodological ways
to make the detected gaps robust to changes in the density estimation and noise
in the data. We illustrate our methods on the velocity distribution of nearby
stars in the Milky Way disk plane, which exhibits gaps that could originate
from different processes. Identifying and characterizing those gaps could help
determine their origins.
