Abstract: The Hilbert 16th problem (2nd part) is about the maximal possible number of isolated ovals on the phase portraits of planar polynomial vector fields (these ovals are called limit cycles). Despite the progress of analysis, geometry and algebra in the 20th century, the general question remains open as it was hundred years before. Only various local or semilocal versions of this problem seem to be amenable, every time with great efforts.

In this talk I will describe the recent progress in another direction of research going back to Petrovskii and Landis. This approach deals with limit cycles born from continuous families of (nonisolated) ovals. The corresponding infinitesimal Hilbert problem was intensely studied for the last 40 years or so. I will introduce different flavors of this problem and formulate the first explicit uniform global upper bound for the number of limit cycles of near-Hamiltonian polynomial vector fields.

The talk (based on works by G. Binyamini, D. Novikov and the speaker) is aimed for a general audience.

Coffee will be served at 12:00 before the lecture
at Schreiber building 006