Sergei Yakovenko
The Gershon Kekst Professorial Chair
The subject that intrigues me can be described as analytic and
geometric properties of objects (functions, curves etc.) defined by
polynomial ordinary differential equations. Despite being in general
transcendental, they sometimes behave as algebraic, in particular,
exhibit certain finiteness properties. The most known paradigm in
this area is the Hilbert 16th problem, the last unsolved question
from the known Hilbert's list of 1900. This problem consists in
determining the number of isolated periodic solutions (limit cycles,
closed integral curves) of a polynomial differential equation with
two variables.
Very recently, together with Gal Binyamini and Dmitry Novikov, we solved an infinitesimal version of the Hilbert problem, which also stood open for about half a century. We proved that the number of limit cycles which appear by non-integrable perturbation of a Hamiltonian integrable vector field is bounded by the double exponential expression in the degree of the system.
Recent publications
- [with D. Novikov] Lectures on meromorphic flat connections. In: Normal forms, bifurcations and finiteness problems in differential equations, 387--430, NATO Sci. Ser. II Math. Phys. Chem. 137, Kluwer Acad. Publ., Dordrecht, 2004.
- Oscillation of linear ordinary differential equations: on a theorem of A. Grigoriev. J. Dyn. Control Syst. 12 (3) (2006) 433-449.
- [with Yu. Ilyashenko] Lectures on analytic differential equations. Graduate Studies in Mathematics 86, American Mathematical Society, Providence, RI, 2008. xiv+625 pp. ISBN: 978-0-8218-3667-5.
- [with G. Binyamini] Polynomial bounds for oscillation of solutions of Fuchsian systems and Abelian integrals. In preparation (2008).
- [with G. Binyamini and D. Novikov] Constructive solution of Infinitesimal Hilbert 16th problem. In preparation (2008).