Uploaded March 2026 | Updated September 2026, 3 weeks ago
Workshop on Recent Developments in Hodge Theory and O-minimality
4:00pm|Simonyi Hall 101
Topic: Melnikov Functions Appearing in Polynomial Hamiltonian Perturbations
Speaker: Dmitry Novikov
Affiliation: Institute for Advanced Study
Date: March 12, 2026
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+ϵω
of planar polynomial foliations, this is closely related to isolated zeros of the Abelian integrals ∫δω.
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 ∫δωω′. We are trying to understand their finiteness properties, which are closely related to the monodromy orbit of δ in π1({H=t}).
Workshop on Recent Developments in Hodge Theory and O-minimality
4:00pm|Simonyi Hall 101
Topic: Melnikov Functions Appearing in Polynomial Hamiltonian Perturbations
Speaker: Dmitry Novikov
Affiliation: Institute for Advanced Study
Date: March 12, 2026
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+ϵω
of planar polynomial foliations, this is closely related to isolated zeros of the Abelian integrals ∫δω.
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 ∫δωω′. We are trying to understand their finiteness properties, which are closely related to the monodromy orbit of δ in π1({H=t}).
