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Jerusalem Analysis and PDEs seminar
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(Second of two talks this week. See a separate announcement of the first talk.)
 
Speaker: Sergei Yakovenko (Weizmann)
 
Date:    November 24th
 
Time:    ((14:00))   14:15  
 
Place:   Manchester   209.  
 
         (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.
 
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Technion Math Net-2 (TECHMATH2)
Editor: Michael Cwikel   <techm@math.technion.ac.il> 
Announcement from: Dan Mangoubi   <mangoubi@math.huji.ac.il>