Implicit planar webs and singularities

Planar web geometry deals with families of foliations by curves on surfaces. In the complex setting, locally in ${\mathbb C}^2$ or on ${\mathbb P}^2({\mathbb C})$, a $d$-planar web ${\cal W}(d)$ is given by the generic family of integral curves of an analytic or algebraic differential equation $F(x,y,y')=0$ with $y'$-degree $d$. Invariants of these configurations as abelian relations (related to Abel's addition theorem) or infinitesimal symmetries will be discussed, in the nonsingular case and through the singularities and their residues as well. This viewpoint enlarges the qualitative study of such equations. Basic examples will be given from different domains including classic algebraic geometry and WDVV-equations. Standard results and some open problems will be mentioned. By using connections methods “à la Cartan-Spencer” or “à la Chern” new results on regularity and monodromy questions will be presented.