Sergei Yakovenko's blog: on Math and Teaching

Tuesday, January 22, 2008

Exam for the Winter Semester 2007/8

Filed under: Analytic ODE course,problems & exercises — Sergei Yakovenko @ 4:59
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Contrary to the pessimistic expectations voiced in the previous post, the strike is over and everybody can come out of the closet.

The course “Geometric and Analytic Theory of Differential Equations” is declared a guided reading course based on this weblog. The winter semester for this course is over: the classes will resume some time on the last week of February, 2008.

 Those interested in grades or in controlling how well they digested  the material, are welcome to pass the exam. The rules of the game are simple: the exam is take-home, the deadline for submission is February 28.

Problems for Semester I are available online.  Any questions (if they appear)  can be left in the comments to this post.

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Thursday, January 17, 2008

End of non-semester

Filed under: Analytic ODE course — Sergei Yakovenko @ 5:42

Announcement

The non-lecture today was the last one in the first non-semester of the academic year that perhaps will also be declared non-existing (see the Disclaimer).

There will be no meetings on January {24, 31} and February {7, 14}, as I am going to participate in the Carnival in Rio-de-Janeiro. The preliminary date for the next meeting is set February 21, 2008, unless there will be a complete closure of all universities, mass dismissal of the academic staff etc. Follow the announcements!

Andrei Gabrielov @ W.I.S. – 2nd seminar

Filed under: research seminar — Sergei Yakovenko @ 5:23
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Andrei will continue his story on complexity of various classes of problems in tame (e.g., semialgebraic) geometry.

Venue: Room 261,

Tuesday, January 22, 2008, 

14:00-16:00 (the ordinary time for the seminar).

Wednesday, January 9, 2008

Andrei Gabrielov @ W.I.S.

Filed under: lecture,research seminar — Sergei Yakovenko @ 10:52
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Mini-programme on real analytic/algebraic/o-minimal geometry

Andrei Gabrielov (Purdue U.) is visiting us for a month (until February 8). Among other things, he will explain his recent work with N. Vorobjov on topology of o-minimal sets via approximation.

The exposition, split into several lectures, will serve also as an initiation to the field of semianalytic/subanalitic geometry, accessible to newcomers.

Recommended reading:

  1. E. Bierstone & P. Milman, Semianalytic and subanalytic sets.

The first meeting:

Tuesday, January 15, 2008, 16:00-18:00 (note the unusual time), Room 261 (unless suddenly changed).

Wednesday, January 2, 2008

“Auxiliary Lessons” שעור עזר) 11-13) January {3, 10, 17}, 2008

Blow-up and Desingularization Theorem(s)

We will start a long discussion of one of the most useful results in the local analysis of complex analytic objects. It will take at least two (perhaps, three) non-lectures (see the Disclaimer) which are guaranteed to extend beyond the Judgement Day January 13, 2008. Follow the new postings!

  1. Trigonometric blow-up and its effect on the simplest singularities. Advantages and drawbacks.
  2. Algebraic blow-up in the real and complex domain. Exceptional divisor and its exceptionality. Complex Möbius band.
  3. Blow-up of analytic curves and singular foliations. Dicritical and nondicritical cases. Computation in the local coordinates.
  4. Intersections and their multiplicity.
    • Divisors, their local representations and transformations by holomorphic maps.
    • Divisors and cocycles. Effective divisors
    • Isolated intersection between effective divisors. Intersection multiplicity (algebraic, geometric and deformational  construction).  
    • Principal theorem of Singularity theory. An idea of the proof.
    • Intersection index and its property as a bilinear form.
    • Intersection index and blow-up. Self-intersection index (of the exceptional divisor). 
    • Multiplicity of foliation and its blow-up (principal formula)
  5. Proof of the Desingularization theorem: the main part.
  6. Cleaning the field: desingulation of cuspidal points, elimination of resonant nodes and tangencies with the exceptional divisor.

 Recommended reading: Section 8 from the book (printing disabled).Auxiliary reading:

  1. D. Mumford, Algebraic Geometry I. Complex projective manifolds. Springer, 1976, §2(A-B).
  2. V. Arnold, S. Gusein-Zade, A. Varchenko, Singularities of differentiable mappings, vol. 1, §5 (Multiplicity of holomorphic maps).

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!

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