Sergei Yakovenko's blog: on Math and Teaching

Monday, November 21, 2016

Lecture 3, Nov 21, 2016

Filed under: Calculus on manifolds course,links — Sergei Yakovenko @ 4:51
Tags: ,

Concept of Manifold

The entire lecture was devoted to motivation and examples of C^\infty-smooth manifolds (submanifolds of \mathbb R^n, spheres, tori, projective spaces, matrix groups etc).

Slightly more detailed plan of the lecture is here.

If you want to read more (which is most highly welcome), here are a few recommendations:

  1. F. Warner, Foundations of Differentiable Manifolds and Lie Groups
  2. W. M. Boothby, Introduction to Differentiable Manifolds and Riemannian Geometry
  3. N. J. Hicks, Notes on Differential Geometry, Van Nostrand
  4. B. A. Dubrovin, Differential Geometry. Notes from SISSA course (Trieste, Italy)

A few thoughts on how to use these books. The subject (calculus on manifolds) is difficult because it involves both complicated concepts and the new language describing these concepts, and there is no way to learn these things but in parallel. One possibility to practice in the new language is to read as many texts about familiar subjects, as possible. This is what I suggest: if you believe you understand certain things, try to read about them in different books and make sure that different notation adopted by different authors does not detract you from the core.

A little bit more specific note. A closely related beautiful subject, Algebraic Geometry, was born from studies of how subsets of real or complex Euclidean space may look like. For some time it developed using mostly geometric/analytic tools, but eventually it was realized that to avoid problems with singularities, “double points”, “points at infinity” etc., one should start with the algebra of polynomials in one and several variables, its ideals, the quotient algebras and schemes in general. This approach brought tremendous achievements.

In my attempt to present the basic constructions of Calculus on Manifolds and, more generally, Differential Geometry, I decided to make the first several steps in a similar spirit and build objects from the algebra of C^\infty-smooth functions on a manifold. Of course, these algebras are very different from the algebras of polynomials (in particular, they are not Noetherian), which makes life some times easier, some times more difficult.

See you in a week.

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Monday, March 7, 2016

Wonderful interview with Sir Michael Atiyah

Filed under: links — Sergei Yakovenko @ 4:48
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M. Atiyah

One of the most exquisite minds of our times shares his insights of how a Mathematician perceives the worlds of Mathematics and Physics.

A must read for all ages and all specializations, from students to retirees.

Tuesday, January 26, 2016

Lecture 13, Jan 26, 2016

Questions and answers

Questions concerned integrability of discontinuous functions, notions of improper integrals (how and when they can be defined), topological properties (equivalent definitions of compactness, connectedness etc.)

Here are some textbooks that I recommend for preparing when working on the exam. Keep them on your virtual bookshelf: they cover much more that I explained in the course, but who knows what questions related to analysis you might have.

  1. V. Zorich, vol. 1: Chapters 1-6, pp.1-371.
  2. V. Zorich, vol. 2: Parts of Chapter 9 (continuous maps) and the first part of Chapter 18 on Fourier series.
  3. W. Rudin: Chapters 1-6, pp.1-165.

The problems for exam will be posted on February 1st (at least the English version).

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.

Click for full size photo

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.

Sunday, March 23, 2008

2-Sphere eversion in 3D-space

Filed under: links — Sergei Yakovenko @ 12:23
Tags: , , ,

If a smooth curve  embedded in the plane \mathbb R^2  is deformed allowing self-intersections but remaining smooth, then there is a natural integral invariant, the rotation number, which prevents eversion of a circle (deformation of the oriented circle into another circle with an opposite orientation). For two-dimensional surfaces smoothly embedded in \mathbb R^3 a similar invariant of deformations exists, yet this invariant does not preclude eversion of the sphere inside out.

 The possibility of such deformation was discovered bt S. Smale in 1958. Relatively recently W. Thurston invented a general algorithm of smoothening, which yields an explicit sphere eversion. All these spectacular things are discussed on the level accessible to high school students in the most fascinating animation (21 min.) discovered on the web by Dmitry Novikov (thanks!). A much shorter animation (mere 22 sec.) does not easily reveal the mistery, so the longer one is really worth its time!

Tuesday, January 1, 2008

Happy Rat Year 2008!

Filed under: links — Sergei Yakovenko @ 12:58

I wish to all readers of this experimental blog the wisdom, resilience and staunchness of that beautiful creature!

Wednesday, December 26, 2007

“Auxiliary Lesson” שעור עזר)#10) December 27, 2007

Filed under: Analytic ODE course,lecture,links — Sergei Yakovenko @ 5:56

Because of the dismal failure to meet the schedule in Lesson 9, the next meeting will deal with the items from the previous list that are not rendered in blue.

In the meantime you may enjoy the funny animations illustrating  differences in the convergence patterns of Taylor and Fourier series for various functions (thanks to D. K. for pointing me to the site). Note the appearance of the Gibbs phenomenon for Fourier series of discontinuous functions.

Saturday, November 10, 2007

Visualization of holomoprhic maps

Filed under: links — Sergei Yakovenko @ 11:05
Tags: ,

Though it is difficult to visualize holomorphic maps and objects, some enthusiasts succeeded in this very impressively.

  1. Conformal maps  (interactive Java applet) by Jonathan Foote.
  2. Many more Java applets written by Terrence Tao, Fields prize winner, great mathematician and terrific math blogger.
  3. Animated gifs of conformal maps by Douglas Arnold.

Take a tour: it is funny and very instructive.

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