  The dynamical classification of rational maps is a central concern of
holomorphic dynamics. Much progress has been made, especially on the
classification of polynomials and some approachable one-parameter families of
rational maps; the goal of finding a classification of general rational maps is
so far elusive. Newton maps (rational maps that arise when applying Newton's
method to a polynomial) form a most natural family to be studied from the
dynamical perspective. Using Thurston's characterization and rigidity theorem,
a complete combinatorial classification of postcritically finite Newton maps is
given in terms of a finite connected graph satisfying certain explicit
conditions.
