GADEPs focused conference: Abelian and iterated integrals and Hilbert 16th problem
IMPA, Rio de Janeiro, 26-30 September 2022
The seminar “Geometry, Arithmetic and Differential Equations of Periods” (GADEPs), started in the pandemic year 2020 and its aim is to gather people in different areas of mathematics around the notion of abelian integrals and periods which are certain multiple integrals. This is the first GADEPs conference focused on Hilbert 16th problem. It asks on the uniform bound for the number of limit cycles of planar differential equations, and it is still challenging even for quadratic differential equations. Attempts to approach this conjecture have given origin to many results in holomorphic foliations on complex surfaces and classification of their singularities. The birth of limit cycles after a perturbation of foliations with a first integral is controlled by the zeros of abelian integrals which are the first Melnikov functions of the deformed holonomy map (Poincaré first return map). The infinitesimal Hilbert 16th problem asks for finding a realistic bound for the number of zeros of abelian integrals, and attempts to understand this, have produced many results in Picard-Lefschetz theory of fibrations and Picard-Fuchs equations. When an abelian integral is identically zero, higher Melnikov functions are expressed as Chen’s iterated integrals and their study is closely related to the topology of leaves of holomorphic foliations near to those with a first integral. The conference main core will consist of 5 lecture series which aim to introduce main methods and conjectures of the topic to a broad public and in particular graduate students.
The conference will be in hybrid format and it will be transmitted through IMPA’s YouTube channel.
Hossein Movasati (IMPA, Brazil) – Deformation of foliations with a first integral and Abelian and iterated integrals – Abstract
Note: The speakers marked with an * will give online lectures.
César Camacho (FGV, Brazil)
Marcin Bobieński (University of Warsaw, Poland) – Abstract
Colin Christopher (University of Plymouth, UK) – Darboux Relative Exactness and Pseudo-Abelian Integral – Abstract
Lubomir Gavrilov (University of Toulouse III, France) – Introduction to the center-focus problem for plane polynomial vector fields – Abstract
Daniel Lopez (IME/USP, Brazil) – The monodromy problem for hyperelliptic curves – Abstract
Yulij Ilyashenko (Cornell University, US) – Persistence problems for polynomial foliations, simultaneous uniformization and identical cycles at infinity – Abstract
Pavao Mardesic (University of Bourgogne, France) – Darboux Relative Exactness and Pseudo-Abelian Integral – Abstract
Alcides Lins Neto (IMPA, Brazil) – On the number of solutions of the equation for which – Abstract
Jorge Vitório Pereira (IMPA, Brazil) – Effective Liouvillian integration of foliations – Abstract
Jessie Pontigo (Universidad Nacional Autónoma de México) – A bound for the length of iterated integrals in the first non-zero Melnikov function Abstract
Sergei Yakovenko (Weizmann Institute of Science, Israel)* – On Fuchsian equations with a small parameter before the highest derivative – Abstract
Henryk Zoladek (Warsaw University, Poland) – Normal forms of planar vector fields – Abstract
Online Interview: Robert Roussarie (University of Bourgogne, France)
Hossein Movasati (IMPA)
Younes Nikdelan (UERJ)
Suely Lima (IMPA)
The organizing committee has a limited number of financial supports (only per diem and no travel expenses) for young researchers and students. The application deadline is 30 June 2022, with answers to be out by 15 July 2022. All applicants should fill in the registration form and follow the instructions.
Postal Address: Instituto de Matemática Pura e Aplicada
Estrada Dona Castorina 110, Jardim Botânico
Rio de Janeiro, RJ, CEP 22460-320, Brasil