Dmitry Novikov of the Institute for Advanced Study presents joint work with Pavao Mardesic, Laura Ortiz-Bobadilla, and Jessie Pontigo-Herrera on Hilbert's 16th problem, which asks for an upper bound on the number of limit cycles of planar polynomial vector fields. Novikov connects this question to isolated zeros of Abelian integrals arising from polynomial perturbations of planar Hamiltonian foliations, then turns to the degenerate cases where these first-order approximations vanish. He introduces higher-order approximations built from Chen's iterated integrals and discusses their finiteness properties, tied to the monodromy orbit of a cycle in the fundamental group of the level curve. Delivered at Simonyi Hall as part of a workshop on Hodge theory and o-minimality, the talk is aimed at researchers in dynamical systems and algebraic geometry working on bifurcation theory and limit cycles.