Bloc I (1 week): Basic concepts on singular foliations and reduction of singularities. a. Singular foliations and vector fields in dimension two. b. Blowing-up vector fields. simple singulaities. c. Separatrices and integral curves. Briot-Bouquet Theorem. d. Seidenberg's result on desingularization of vector fields. e. Camacho-Sad theorem of existence of separatrices.

Bloc II (1 week): Some applications of the reduction of singularities of codimension one foliations. a. Simple singularities in codimension one. Behavior under blow-up. b. The statement of reduction of singularities in dimension three. Consequences on the existence of invariant hypersurfaces. c. About the dicriticalness. d. Singular Frobenius I. e. Singualr Frobenius II.

Bloc III (1 week): Technics for the reduction of singularities. a. The reduction of the singularities of surfaces as a model for low dimensional problems. b. The main invariants used in the control: Multiplicities, Newton polygons and resonancies. c. Reduction of the singularities of codimension one foliations in dimension three. d. The valuative aproach for vector fields. The birational problem of reduction of singularities. Globalization. e. Local Uniformization of Vector Fields.