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

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.

1. Taylor series animations
2. Fourier series animations

“Auxiliary Lesson” שעור עזר) #9) Dec. 20, 2007

Dynamics generated by finitely generated subgroups of conformal germs

1. Generic subgroups of are non-solvable.
2. Dynamics generated by several germs. Definition of a pseudogroup. Orbits of points.
3. Periodicity of germs (finiteness of order) vs. periodicity of orbits. Cycles and limit cycles of pseudogroups.
4. Convergence of elements in pseudogroups. Closure.
5. Density of orbits. Linear subgroups. Abundance of limit cycles for generic (nonsolvable) subgroups of .
6. Topological equivalence of subgroups and pseudogroups. Conjugacy of dense linear subgroups.
7. Rigidity of nonsolvable subgroups: topological conjugacy implies holomorphic conjugacy.

Disclaimer… if somebody still needs it…
Reading: Section 6 (second part) from the book, printing disabled.

Seminar on Khovanskii-Varchenko theorem (II)

Topological properties of Abelian integrals

The second “learning in groups” meeting will be devoted to the study of the Gauss–Manin connexion in homology, which will ultimately result in a local representation of Abelian integrals as linear combinations of real powers and logarithms with analytic coefficients analytically depending on parameters.

This representation already suffices to produce local uniform bounds for the number of isolated zeros, as was explained on the previous Tuesday.

Recommended reading: Section 26 from the book (printing disabled), esp., subsections F and I-K.

Time and location: Tuesday Dec. 18, 2007, 14:00 (in place of the usual Geometry & Topology seminar time), Pekeris Room.

What it will be about:  

“Auxiliary Lesson” שעור עזר) #8) December 13, 2007

Finitely generated subgroups of , I. Formal theory. 

1. Formal normal form for a single holomorphic self-map from . Parabolic germs.
2. Bochner theorem on holomorphic linearization of finite groups.
3. Stratification of the subgroup of parabolic germs .
4. Tits alternative for finitely generated subgroups of : every such subgroup is either metabelian (its commutator is commutative, e.g., trivial), or non-solvable (all iterated commutators are nontrivial).
5. Centralizers and symmetries: formal classification of solvable subgroups.
6. Integrable germs and their holomorphic linearizability.

Recommended reading: Section 6 (first part) from the book (printing disabled)

Disclaimer applies, as usual

Seminar on Khovanskii-Varchenko theorem (I)

 We (D. Novikov and S.Y.)  launch a campaign “Learn Khovanskii–Varchenko Theorem“. A few (2-4) next weeks we will discuss in detail the proof of this remarkably simple but powerful result with a view to have a number of generalizations.

The two manuscripts (one in Russian, another in English) are available:

• Varchenko (1984) (Engl. translation)
• Khovanskii (1984), scanned (46 Mb, large!) (in Russian)
• A relevant section from Zoladek, The Monodromy Group (2006)

Time and location: Tuesdays, 16:00-18:00, Room 261 (unless otherwise announced).

The first meeting: Dec 11, 2007.

Fewnomial theory (S.Y.). This purely geometric theory starts with a multidimensional generalization of the Rolle theorem for several variables and allows to prove infinitely many both classical and new results starting from the Descartes’ rule.

If somebody has a scanned copy of the English original by Khovanskii, please post a link in comments.

“Auxiliary Lesson” שעור עזר) #7) December 6, 2007

Invariant manifolds for hyperbolic maps. Complex hyperbolicity.

1. Formal theory: cross-resonances.
2. Hadamard-Perron theorem for holomorphisms. Contracting map principle reactivated.
3. Hadamard-Perron theorem for vector fields. Complex hyperbolicity.
4. Invariant hypernolic curve for saddle-nodes.
5. Poincare resonances.
6. Center manifolds: formal but non-analytic.

Reading: Section 7 from the book (printing disabled), Section 27 (parts A-C) from the book (printing disabled)

Disclaimer is as sadly relevant as before…