Limit: the first encounter with infinity

• Introduction and logistics
• Goals of the course
• Infinity: the name of the game in Analysis
• Some examples and counterexamples
  1. Sets and their subsets that have “the same number” of elements
  2. Summation of infinitely many terms: success and failure
  3. Limits in geometry: how to measure the lengths?
  4. Pathological curves on the plane
• Numerical sequences and their limits: the first real encounter with infinity.
  1. “Almost all” vs. “infinitely many“
  2. Intervals and operations on them
  3. Geometric definition of the sequence limit
  4. “Instrumental” definition of the limit
  5. “Standard” definition:
  6. First theorems about limits (limits and arithmetic operations, limit and rearrangements, limit and boundedness, limits and monotonicity)
• Step back: number systems. Integer, rational and real numbers. Sealing the gaps. Completeness

This post marks beginning of the new “blogged” (blog-accompanied) course “Analysis for High School Teachers” within the framework of the Caesarea-Rothschild Program. I will place here brief content of the forthcoming lectures and some relevant material. It may come in a variety of electronic formats, of which most popular are pdf and dejavu. e-Readers for these formats are freely available from the internet.

Blogging rules:

For those not familiar with the blogging subculture (are there still such people?). You are welcome to leave your comments/questions/remarks next to the relevant posts. Please introduce yourself when commenting. My wet dream is having mathematical discussions between the students attending the course on these pages. Don’t be afraid to express yourself and teach others. I promise not to abuse my rights as a moderator here. This venue for interaction is especially important since it is convenient for students which stay away from the teachers for most of the time.

Any language is accepted, though I strongly urge to write in the l.c.d. (= Simple English).

This platform (WordPress) allows for easy insertion of LaTeX code, which makes mathematical discussions here especially pleasant and easy to maintain.

The course will be accompanied by guided seminars led by Gal Binyamini: these seminars will be devoted to discussion of problems and their solutions as well as complimentary material to the main lectures. Gal will also post to this blog.

One of the sources for the course will be the excellent book which equally fascinates both professional mathematicians and high school children. It is in English and you can download an (illegally scanned) copy strictly for your personal use here (21 Mb in pdf format: beware!). I will also try to upload separate relevant sections of the book next to posts on specific lectures.

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).

A couple of weeks ago some two-thirds of the conspirators coworkers attended the workshop Equations aux dérivées partielles et théorie de Galois différentielle dit Malgrangefest in Luminy and delivered a talk on their work.

Slides from this talk are now available (static pdf, 2 Mb) for everybody to see.

Best wishes for the New 5769 (תשס”ט) Year to all readers of this blog!

שנה טובה

The number of limit cycles that can be born from periodic solutions of a polynomial Hamiltonian planar system by a small polynomial perturbation

not increasing the degree , is explicitly bounded by a double exponent , where is an explicit polynomial in 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 the bound can be improved to with an explicit constant in the term . 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 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.

Actually, I forgot to tell the last time that the academic year is over. Congratulations to all the survivors who made it till the end. Hope you don’t regret.

Logarithmic singularities

1. De Rham division lemma (and its generalization)
2. Definition of a logarithmic pole: (scalar case). Residues.
3. Logarithmic complex: principal lemma on Λ-closedness.
4. Principal example: logarithmic complex for the normal crossings. Saito theorem.
5. Closed logarithmic 1-forms: complete description. Darbouxian foliations.
6. Matrix casse. Conjugacy of the residues along the polar locus. Residues on the normal crossings.
7. Schlesinger system: flat connexions with logarithmic poles along the diagonal.
8. Flat connexions with first order poles are almost always logarithmic, yet resonances may spoil the pattern.

Recommended reading: the same notes, sect. 3-4.

Meromorphic flat connexions on holomorphic manifolds: Integrability, monodromy, classification

1. Pfaffian systems and their integrability
2. From local to global solutions: monodromy
3. Geometric language: covariant derivative and its curvature
4. Meromorphic functions, meromorphic forms
5. Example: multidimensional Euler system
6. Regular singularities
7. Flat connexions vs. isomonodromic deformations

Recommended reading: D. Novikov & S.Y., Lectures on meromorphic flat connexions, sect. 1-2.

No classes today, as 50% of the students are speaking on a conference elsewhere.