Sergei Yakovenko's blog: on Math and Teaching

Sunday, February 19, 2017

Volunteers to moonlight, any?

Filed under: research announcement — Sergei Yakovenko @ 5:36
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Israeli Ministry of Education Needs You!

As part of the matriculation exam in math (5th level), Israeli high school students are (sometimes?) required to write a “final project“, עבודת גמר.

They are looking for people qualified to read, check these compilations and grade them based on certain criteria. In particular, this presumes a 45 min meeting with the students. Each case will bring you 600 shekels. The required qualification allows to hire PhD students (and faculty members, of course ;-). Needless to say, everything is in Hebrew.

If you are interested in such a job, please write me or (better) directly to Dr. Ilya Tyomkin from the Ben Gurion University, who is overseeing this program.

Analgebraic Geometry

It so happened that at the beginning of 2016 I gave a talk on the conference “Geometric aspects of modern dynamics” in Porto, delivered a minicourse at Journées Louis Antoine in Rennes and wrote an expository paper for the European Mathematical Society Newsletter, all devoted to the same subject. The subject, provisionally dubbed as “Analgebraic geometry”, deals with algebraic-like properties (especially from the point of view of intersection theory) of real and complex analytic varieties defined by ordinary and Pfaffian differential equations with polynomial right hand sides. Thus

analgebraic = un-algebraic + analytic + algebraic (background) + weak algebraicity-like properties.

It turns out that this analgebraic geometry has very intimate connections with classical problems like Hilbert 16th problem, properties of periods of algebraic varieties, analytic number theory and arithmetic geometry.

For more details see the presentation prepared for the minicourse (or the shorter version of the talk) and the draft of the paper.

Any remarks and comments will be highly appreciated.

Tuesday, December 30, 2014

Final announcement

This is to inform the noble audience of the course that the main program of the course is completed. I will stay in Pisa for one more week (till January 8, 2015) and will be happy to discuss any subject (upon request).

Meanwhile one of the subjects discussed in this course was brought to a pre-final form: the manuscript

1. Shira Tanny, Sergei Yakovenko, On local Weyl equivalence of higher order Fucshian equations, arXiv:1412.7830,

was posted on arXiv and submitted to the Arnold Mathematical Journal, a new venue for publications molded in the spirit of  the late V. I. Arnol’d and his seminar.

Any criticism will be most appreciated. Congratulations modestly accepted.

Tanti auguri, carissimi! Buon anno, happy New Year, с наступающим Новым Годом, שנה (אזרחית) טובח, bonne année!

Monday, November 3, 2014

Analytic theory of linear differential equations: a new course

Filed under: Analytic ODE course,research announcement,schedule — Sergei Yakovenko @ 5:56
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Carissimi!

This post is to announce the midi-course (about 24 hours) that will be given in November-December in Universita di Pisa. The weekly timetable is as follows,

Lunedi 11-13 Aula 1
Venerdi 9-11 Aula 1.

The course will be based (among other) on several principal sources, all available online. Here are the links:

1. Yu. Ilyashenko, S. Yakovenko, Lectures on analytic differential equations, MR2085816 (2005f:34255). Mainly Chapter III and Section 26.
2. D. Novikov, S. Yakovenko, 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 (Preprint math.CA/0212334).
3. Yakovenko, S. Quantitative theory of ordinary differential equations and tangential Hilbert 16th problem, Preprint math.DS/0104140 (2001). On finiteness in differential equations and Diophantine geometry, CRM Monogr. Ser., vol. 24, Amer. Math. Soc., Providence, RI, 2005, pp. 41–109, MR2180125 (2006g:34062)

More specialized references will be added in the appropriate posts.

Arrivederci!

Wednesday, December 3, 2008

IH16 and friends: the final dash

Finally the two texts concerned with solution of the Infinitesimal Hilbert problem, are put into the polished form (including the publisher’s LaTeX style files). The new revisions, already uploaded to ArXiv, differ from the initial submissions only by corrected typos, a few rearrangements aimed at improving the readability of the texts, and a couple of more references added. There is absolutely no need to read the new revision if you already have read the first one.

Mostly for the reasons of “internal convenience” the complete references are repoduced here:

• G. Binyamini and S. Yakovenko, Polynomial Bounds for Oscillation of Solutions of Fuchsian Systems, posted as arXiv:0808.2950v2 [math.DS], 36 p.p., submitted to Ann. Inst. Fourier (Dec. 2008), accepted (February, 2009)
• G. Binyamini, D. Novikov and S. Yakovenko, On the Number of Zeros of Abelian Integrals: A Constructive Solution of the Infinitesimal Hilbert Sixteenth Problem, posted as arXiv:0808.2952v2 [math.DS], 57 p.p., submitted to Inventiones Mathematicae (Nov. 2008), accepted (Oct. 2009).

Friday, August 22, 2008

Infinitesimal Hilbert 16th Problem

The number of limit cycles that can be born from periodic solutions of a polynomial Hamiltonian planar system $\frac{dx}{dt}=\frac{\partial H}{\partial y}(x,y),~~\frac{dy}{dt}=-\frac{\partial H}{\partial x}(x,y)$ by a small polynomial perturbation

$\frac{dx}{dt}=\frac{\partial H}{\partial y}(x,y)+\varepsilon P(x,y),~~~~~~~~\frac{dy}{dt}=-\frac{\partial H}{\partial x}(x,y)-\varepsilon Q(x,y)$

not increasing the degree $n=\text{deg}H$, is explicitly bounded by a double exponent $2^{2^{\text{Poly}(n)}}$, where $\text{Poly}(n)$ is an explicit polynomial in $n$ of degree not exceeding 60 (fine tuning of the proof gives a better value around 5 or so, which hypothetically could be reduced to just 2). For hyperelliptic Hamiltonians of the form $H(x,y)=y^2+x^{n+1}+a_1 x^{n-1}+\cdots+a_{n-1}x+a_n$ the bound can be improved to $2^{2^{O(n)}}$ with an explicit constant in the term $O(n)$. This assertion constitutes an explicit constructive solution of the so called “Infinitesimal” Hilbert 16th Problem which first implicitly appeared in the works of Petrovskii and Landis in the 1950-s. Since mid-1960-s the problem was repeatedly formulated in many sources (starting with Arnold’s problems and as recently as in Ilyashenko’s 2008 list) as the natural step towards a still evasive solution of the complete Hilbert 16th Problem.

J’ai Nous (i.e., Gal Binyamini, Dmitry Novikov et moi-même) avons trouvé une merveilleuse démonstration de cette proposition, mais je ne peux l’écrire dans cette marge car elle est trop longue.”

La démonstration is indeed a bit too long to be reproduced here: the complete exposition is available on arXiv (50+ pages) and strongly uses another paper of 30+ pages which establishes non-uniform explicit double exponential upper bound on the number of isolated complex zeros of functions satisfying linear systems of Fuchsian differential equations, provided that all residue matrices have only real eigenvalues.
Our proof is based solely on the fact that Abelian integrals of polynomial 1-forms along cycles on complexified level curves of the Hamiltonian, satisfy an integrable system of regular Pfaffian differential equations defined over $\mathbb Q$ with quasiunipotent monodromy along all small loops.

Bookmark this page, as it will display the most up-to-date version of the text of both papers. Any comments, suggestions and spotted typos will be accepted with warmest gratitude.

Saturday, November 10, 2007

Announcement

Filed under: research announcement — Sergei Yakovenko @ 10:45
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A friendly “project” on complex algebraic geometry will be launched by Dmitry Novikov. Meetings are on Sundays, 14:00-16:00 in Room 261.  The first meeting is November 11, 2007.

It is highly recommended for all involved in the “project” on Analytic and Geometric Theory of Differential Equations.

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