  The theory of persistence modules is an emerging field of algebraic topology
which originated in topological data analysis. In these notes we provide a
concise introduction into this field and give an account on some of its
interactions with geometry and analysis. In particular, we present applications
of persistence to symplectic topology, including the geometry of
symplectomorphism groups and embedding problems. Furthermore, we discuss
topological function theory which provides a new insight on oscillation of
functions. The material should be accessible to readers with a basic background
in algebraic and differential topology.
