In this talk, the qualitative dynamics of the vector fields in the complex plane defined by complex polynomials is studied. The ultimate goal is to give a description of the possible bifurcations that can occur, i.e., given an arbitrary point c in parameter space, what are the topological-equivalence classes that intersect every neighborhood of c? The goal of this talk is to describe the non-splitting bifurcations: the bifurcations that can occur when when the multiplicities of the equilibrium points are preserved under small perturbation. It will be proved that any non-splitting bifurcation can be realized as a composition of simpler bifurcations from a finite list. Speaker(s): Kealey Dias (Bronx Community College)