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.
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 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).
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:
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The first meeting:
Tuesday, January 15, 2008, 16:00-18:00 (note the unusual time), Room 261 (unless suddenly changed).
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!
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Trigonometric blow-up and its effect on the simplest singularities. Advantages and drawbacks.
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Algebraic blow-up in the real and complex domain. Exceptional divisor and its exceptionality. Complex Möbius band.
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Blow-up of analytic curves and singular foliations. Dicritical and nondicritical cases. Computation in the local coordinates.
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Intersections and their multiplicity.
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Divisors, their local representations and transformations by holomorphic maps.
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Divisors and cocycles. Effective divisors
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Isolated intersection between effective divisors. Intersection multiplicity (algebraic, geometric and deformational construction).
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Principal theorem of Singularity theory. An idea of the proof.
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Intersection index and its property as a bilinear form.
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Intersection index and blow-up. Self-intersection index (of the exceptional divisor).
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Multiplicity of foliation and its blow-up (principal formula)
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Proof of the Desingularization theorem: the main part.
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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:
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D. Mumford, Algebraic Geometry I. Complex projective manifolds. Springer, 1976, §2(A-B).
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V. Arnold, S. Gusein-Zade, A. Varchenko, Singularities of differentiable mappings, vol. 1, §5 (Multiplicity of holomorphic maps).
I wish to all readers of this experimental blog the wisdom, resilience and staunchness of that beautiful creature!
