~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Jerusalem Analysis and PDEs seminar ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ (Second of two talks this week. See a separate announcement of the first talk.) Speaker: Sergei Yakovenko (Weizmann) Date: November 24th Time: ((14:00))14:15Place: Manchester209.(Note change of room.) Title: Infinitesimal Hilbert 16th Problem Abstract: The Hilbert 16th problem, one of the two remaining open problems from the Hilbert's list of 1901, asks a question on the number of "limit cycles", closed (periodic) isolated orbits of polynomial vector fields on the plane. The history of this problem in the XXth century was full of deep developments and errors, yet still we know precious few about it. Thus different relaxations were suggested as an intermediate step. One such relaxation is the problem on the number of isolated orbits born by perturbation of integrable systems, whose orbits are all closed. This is a typical "small parameter problem". In the linear approximation the answer depends on vanishing of an integral of the rational 1-form over certain cycles. If these cycles are algebraic (i.e., the unperturbed system admits a rational first integral), the finite answer was obtained in a series of works joint with G. Binyamini and D. Novikov. The general case of Darbouxian integrability is considerably more delicate, and only the existential finiteness results are available (M. Bobienski - Novikov - P. Mardesic). Besides, the first steps of the higher order theory are developed (S. Benditkis - L. Gavrilov - Novikov). I will try to present the global picture and outline some possible directions. The talk is geared (for the most part) for advanced undergraduate and graduate students. --------------------------------------------------------- Technion Math Net-2 (TECHMATH2) Editor: Michael Cwikel <techm@math.technion.ac.il> Announcement from: Dan Mangoubi <mangoubi@math.huji.ac.il>