MATH5225 : Dynamics of complex differential equations

The study of differential equations in the complex domain is a relatively old subject which dates back to the end of the 19th century but continues to arouse great interest.

These are differentiable equations with a complex variable and algebraic coefficients.

After compactification, we come back to the study of algebraic (singular) foliations on the complex projective plane.

The aim of the course is to study the dynamics and the geometry of certain classes of such foliations.

The main ingredient is monodromy which is a linear representation of the fundamental group of the complement of the set of singularities.

Some of the topics that will be covered in this course include

•  linear case: Fuchsian equations, Fuchsian groups, Riemann-Hilbert problem (monodromy inverse problem);
•  the differential equations of Riccati and their monodromies;
•  the equations of Painlevé and their applications in various mathematical fields;
• the algebraic structure of the foliation space of the complex projective plane and (time permitting) the structural stability of the equation of Jouanolou.