  We present, in this dissertation, a pedagogical review of the formalism for
Fermi liquids developed in [Delacretaz et al., arXiv:220305004] that exploits
an underlying algebro-geometric structure described by the group of canonical
transformations of a single particle phase space. This infinite-dimensional
group governs the space of states of zero temperature Fermi liquids and thereby
allows us to write down a nonlinear, bosonized action that reproduces Landau's
kinetic theory in the classical limit. Upon quantizing, we obtain a systematic
effective field theory as an expansion in nonlinear and higher derivative
corrections suppressed by the Fermi momentum $p_F$, without the need to
introduce artificial momentum scales through, e.g., decomposition of the Fermi
surface into patches. We find that Fermi liquid theory can essentially be
thought of as a non-trivial representation of the Lie group of canonical
transformations, bringing it within the fold of effective theories in many-body
physics whose structure is determined by symmetries. We survey the benefits and
limitations of this geometric formalism in the context of scaling, diagrammatic
calculations, scattering and interactions, coupling to background gauge fields,
etc. After setting up a path to extending this formalism to include
superconducting and magnetic phases, as well as applications to the problem of
non-Fermi liquids, we conclude with a discussion on possible future directions
for Fermi surface physics, and more broadly, the usefulness of diffeomorphism
groups in condensed matter physics. Unlike [Delacretaz et al.,
arXiv:220305004], we present a microscopic perspective on this formalism,
motivated by the closure of the algebra of bilocal fermion bilinears and the
consequences of this fact for finite density states of interacting fermions.
