Workshop on Recent Developments in Hodge Theory and O-minimality

Abstract: Joint project with Pavao Mardesic, Laura Ortiz-Bobadilla, and Jessie Pontigo-Herrera.

Hibert's 16th problem asks for an upper bound on the number of limit cycles of planar polynomial vector fields. For polynomial perturbations $\dH+\epsilon\omega$ of planar polynomial foliations, this is closely related to isolated zeros of the Abelian integrals $\int_\delta\omega$.

However, in degenerate cases, the first-order approximation given by Abelian integrals vanishes, and one should consider higher-order approximations given by Chen's iterated integrals like $\int_\delta\omega\omega'$. We are trying to understand their finiteness properties, which are closely related to the monodromy orbit of $\delta$ in $\pi_1(\{H=t\})$.