Sergey Yakovenko

Abstract:   The second part of Hilbert's 16th problem is about the maximal possible number of isolated ovals (also called limit cycles) on the phase portraits of planar polynomial vector fields. The general question still remains open. Only various local or semilocal versions of this problem seem to be accessible, every time with great efforts. In 2000 Yu. Ilyashenko gave a lecture "Centennial History of Hilbert's 16th Problem" at this colloquium (the extended version appeared in Bull. AMS), focusing on the (semi)local study of limit cycles near separatrix polygons which implies, among other things, finiteness of their number for each particular polynomial vector field. This was the main achievement of the theory by that time.

I will discuss recent progress in another direction of research going back to Petrovskii and Landis. This approach deals with limit cycles born from continuous families of (non-isolated) ovals by small perturbations. The corresponding infinitesimal Hilbert's problem was intensely studied for the last 40 years. I will introduce different versions of this problem and state the first explicit uniform global upper bound for the number of limit cycles of near-Hamiltonian polynomial vector fields. If time allows, I will also describe some very recent results for perturbations of non-Hamiltonian integrable vector fields (based on joint work with G. Binyamini and D. Novikov). The talk is aimed for a general audience.

Here are some directions to Northeastern University. Lake Hall and Nightingale Hall can be best accessed from the entrance on the corner of Greenleaf Street and Leon Street. The two halls are connected, with no well-defined boundary in between. In particular, 509 Lake Hall is on the same corridor as 544 Nightingale Hall.

There is free parking available for people coming to the Colloquium at Northeastern's visitor parking (Rennaisance Garage). The entrance is from Columbus Avenue. If coming by car, you should park there and take the parking talon. After the lecture, you may pick up the payment coupon from Andrei Zelevinsky.