“Auxiliary Lesson” שעור עזר) #6) November 29, 2007

Holomorphic normalization

1. Poincaré and Siegel domains. Different types of resonances.
2. Fixed point equation and its linearization.
3. Invertibility of the homological operator.
4. Majorant norm and its properties.
5. Poincaré theorem on holomorphic linearization of vector fields in the Poincaré domain.
6. Further results: Poncare-Dulac polynomial normal form in the Poincare domain. Siegel and Brjuno theorems. Yoccoz counterexample. Divergence dychotomy.
7. Normal forms of the self-maps. Schröder-Kœnigs theorem.

Disclaimer, alas, is still relevant…

Reading: Section 5 from the book (printing disabled).

“Auxiliary Lesson” שעור עזר) #5) November 22, 2007

Formal linearization and obstructions. Poincare theorem

1. Formal equivalence of formal vector fields (total recall)
2. (Additive) Resonances
3. Poincare formal linearization theorem
4. Proof of the Poincare theorem:
   • Homological equation
   • Commutator with diagonal linear vector field
   • Stabilization of the series
5. Resonant monomials. Resonant normal form. Poincare–Dulac paradigm.
6. Formal classification of formal self-maps. Multiplicative resonances.
7. Survey of further results. Formal types of line and planar singularities.

Reading material: Section 4 from the book (printing disabled).Disclaimer (alas, still required) .

“Auxiliary Lesson” שעור עזר) #4) November 15, 2007

Formal series, formal vector fields: Flows and embedding

1. Formal series algebra . Formal vector fields (derivations) . Formal equivalence of vector fields. Truncation. Convergence in the formal algebra  .
2. Formal inverse function theorem. Geometric series.
3. Integration and formal flow of vector fields. Exponent.
4. Embedding in the flow. Linear case. Matrix logarithms.
5. Embedding in the flow and formal logarithms.

Reading Section 3 from the textbook.

Disclaimer

Given the continuing strike of the senior academic staff and temporary suspension of lectures, classes and all other forms of the frontal eductation, this weblog changes itself into a Web-based Research Project for Masters Students at the Faculty of Mathematics and Computer Science, Weizmann Institute of Science.

Participation in the project is advised to first and second year students, yet Ph. D. students are also encouraged to attend.

Participants in the project are assumed to read weekly portions of research texts (monographs and articlkes)  regularly published on these pages.

There is a weekly meeting between the participants and the head of the project (S.Y.) every Thursday between 9:00 and 11:00. During this meeting the participants receive explanations and extended comments on especially difficult instances of the text and answers to their questions. Meetings are open to everybody.

For disambiguation purposes it should be understood that:

1. This research project is in no way related to the course “Analytic and Geometric Theory of Ordinary Differential Equations”, announced earlier. The regular course will start with the end of the strike.
2. The students will not receive credit points for participation in the project: its purpose is to familiarize the prospective students with an ongoing research.

Don’t hesitate to ask any questions if you feel confused by this disambiguation

Visualization of holomoprhic maps

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.

Announcement

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.

For disambiguation see the Disclaimer at the top of this blog.

“Auxiliary Lesson” שעור עזר) #3) November 8, 2007

Holomorphic singular foliations

1. Definition of foliation. Different types of equivalence of foliations.
2. Equivalence of foliations vs. equivalence of vector fields. Foliations as “phase portraits” of holomorphic vector fields.
3. Examples of foliations: Cartesian products, bundles, constant flow on the torus.
4. Holonomy of foliations: construction, invariance.
5. Singular foliations. Singular locus, its structure. Singular loci of planar foliations. Germs of singular foliations. 
6. Digression: local structure of complex analytic subsets of .
7. Complex separatrices and their holonomy.
8. Example: holonomy of foliations generated by linear vector fields.
9. Suspension: realization of a given holonomy by a foliation.

Preliminary reading before the meeting: Textbook section 2 (printing disabled) andFirst Aid (sections A6-A8).