Classification of Meromorphic Vector Fields

A complex polynomial defines a holomorphic vector field in the complex
plane. The quasi-conformal conjugacy class of the polynomial is completely
determined by a combinatorial invariant. Furthermore, within each
combinatorial class the polynomial is uniquely determined by a finite number
(settled by the combinatorial class) of complex numbers. This fundamental
classification of complex polynomial vector fields is proved using surgery.

Further developments are to classify possible bifurcations, to understand
the decomposition of parameter spaces due to the different combinatorial classes
and the bifurcations among them, and also to extend to meromorphic vector
fields arising from rational functions on the Riemann sphere.