  Since the pioneering work of Maxwell and Boltzmann in the 1860s and 1870s, a
major challenge in mathematical physics has been the derivation of macroscopic
evolution equations from the fundamental microscopic laws of classical or
quantum mechanics. Macroscopic transport equations lie at the heart of many
important physical theories, including fluid dynamics, condensed matter theory
and nuclear physics. The rigorous derivation of macroscopic transport equations
is thus not only a conceptual exercise that establishes their consistency with
the fundamental laws of physics: the possibility of finding deviations and
corrections to classical evolution equations makes this subject both
intellectually exciting and relevant in practical applications. The plan of
these lectures is to develop a renormalisation technique that will allow us to
derive transport equations for the kinetic limits of two classes of simple
dynamical systems, the Lorentz gas and kicked Hamiltonians (or linked twist
maps). The technique uses the ergodic theory of flows on homogeneous spaces
(homogeneous flows for short), and is based on joint work with Andreas
Str\"ombergsson.
