The talk is scheduled only in virtual setting, using Zoom platform. The talk will also be live-streamed via YouTube. During the meeting, the questions can be posed via chat or audio for participants in the meeting. Everybody interested is invited to participate in the Zoom meeting, with the limit of 100 participants, or to follow the live-broadcast.
Link to Zoom meeting:
https://us02web.zoom.us/j/89253380963
Meeting ID: 892 5338 0963
Link to YouTube live broadcast will be put here 10 minutes before the talk:
http://www.youtube.com/watch?v=O7DgbKPptOQ
The talk will be in English.
Abstract: Hilbert's 16th problem and Poincaré's center-focus problem are two classical problems on planar polynomial vector fields, the first one (in its second part) is the question about the maximal number and relative positions of limit cycles, while the second one asks for conditions to have a center. These two problems have been open for over a century, just the second one has a complete solution for the case of degree two, while the first one remains completely open. The so-called infinitesimal version of these problems deals with deformations of hamiltonian foliations; more precisely, one considers a deformation given by
$dF+e*eta=0$, where $F$ belongs to R[x,y], $eta$ is a polynomial 1-form and $e$ is a small enough parameter, and we assume that the hamiltonian foliation defined by $dF=0$ has a continuous family of periodic orbits $gamma_t$ lying in F^{-1}(t), with $t$ regular values.
Now the questions are how many limit cycles can be born from the deformation and under which conditions on $eta$ one can preserve this continuous family of periodic orbits. In this talk we will discuss some techniques to study these questions and we will see how iterated integrals are involved.
This talk is based on joint work with Pavao Mardešić, Dmitry Novikov and Laura Ortiz-Bobadilla.
