  These lecture notes are intended as reader's digest of recent work on a
diagram-free approach to the renormalized centered model in Hairer's regularity
structures. More precisely, it is about the stochastic estimates of the
centered model, based on Malliavin calculus and a spectral gap assumption. We
focus on a specific parabolic partial differential equation in quasi-linear
form driven by (white) noise.
  We follow a natural renormalization strategy based on preserving symmetries,
and carefully introduce Hairer's notion of a centered model, which provides the
coefficients in a formal series expansion of a general solution. We explain how
the Malliavin derivative in conjunction with Hairer's re-expansion map allows
to reformulate this definition in a way that is stable under removing the
small-scale regularization.
  A few exemplary proofs are provided, both of analytic and of algebraic
character. The working horse of the analytic arguments is an ``annealed''
Schauder estimate and related Liouville principle, which is provided. The
algebra of formal power series, in variables that play the role of coordinates
of the solution manifold, and its algebra morphisms are the key algebraic
objects.
