Christiane Rousseau - Finiteness part of Hilbert's 16th problem for quadratic vector fields

In 1991 Dumortier, Roussarie and Rousseau presented a program to prove the finiteness part of Hilbert's 16th problem for quadratic vector fields, namely the existence of a uniform bound for the number of limit cycles of a quadratic vector field. The program reduced the proof to 121 local problems consisting in proving the finite cyclicity of 121 graphics arising in quadratic systems. We present the recent progress in this program, what is done and what difficulties still remain.