  After an introductory chapter on the quantum supersymmetric string, in which
particular attention will be devoted to the techniques via which
phenomenologically viable models can be obtained from the ultraviolet
microscopic degrees of freedom, and a brief review of the swampland program,
the technical tools required to deal with geometric flows will be outlined. The
evolution of a broad family of scalar and metric bubble solutions under
Perelman's combined flow will be then discussed, together with their asymptotic
behaviour. Thereafter, the geometric flow equations associated to a generalised
version of Perelman's entropy function will be derived and employed in defining
the action-induced flow associated to a given theory for a scalar field and a
dynamical metric. The problem of preserving Einstein field equations along the
corresponding moduli space trajectories will be cured by allowing a
supplementary energy-momentum tensor term to appear along the flow. In a
particular example, such contribution will be shown to precisely reproduce the
infinite tower of states with exponentially dropping masses postulated by the
distance conjecture.
