  The geometric evolution equations provide new ways to address a variety of
non-linear problems in Riemannian geometry, and, at the same time, they enjoy
numerous physical applications, most notably within the renormalization group
analysis of non-linear sigma models and in general relativity. They are divided
into classes of intrinsic and extrinsic curvature flows. Here, we review the
main aspects of intrinsic geometric flows driven by the Ricci curvature, in
various forms, and explain the intimate relation between Ricci and Calabi flows
on Kahler manifolds using the notion of super-evolution. The integration of
these flows on two-dimensional surfaces relies on the introduction of a novel
class of infinite dimensional algebras with infinite growth. It is also
explained in this context how Kac's K_2 simple Lie algebra can be used to
construct metrics on S^2 with prescribed scalar curvature equal to the sum of
any holomorphic function and its complex conjugate; applications of this
special problem to general relativity and to a model of interfaces in
statistical mechanics are also briefly discussed.
